General Mathematics
Feigenbaum constants
In mathematics, specifically bifurcation theory, the Feigenbaum constants are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the physicist Mitchell J. Feigenbaum.
Radicand and Index
The number under the sign is the radicand, and the number of the root you're taking is the index of the radical
Euler's Constant
e = 2.7182818284590452353602874 - a transcendental, non repeating number like π. It comes from geometry, interest on a loan, and continued fractions. The exponential function comes from Euler's constant. The definition of e is the positive real number such that: ln e = the integral from 1 to e of 1/t dt = 1
Graph the following over a one-period interval. State the amplitude, period, phase shift, and range. y = -1 -3sin(2x + π)
Amplitude: 3 Range: [-4,2] Period: π Phase Shift: left π/2 Vertical Shift: down 1 Equilibrium line: y = -1 To find the spacing, you divide the period by 4. Accordingly, the spacing is π/2: x-values: π/4, π/2, 3π/4, π x-values after P.S.: -π/2, -π/4, 0, π/4, π/2 y-values: -1, -4, -1, 2, -1
Graph the following over a one-period interval. State the amplitude, period, and range. f(x) = -4cos((π/8)x)
Amplitude: 4 Range: [-4, 4] Period: (2π)/ (π/8) = 16 To determine the spacing for the x-values, divide the period by 4 equal parts. 16/4 = 4. Cosine and sine functions start at zero. The y-values are found by making an z-y chart. Remember to ensure that the calculator is in radian mode. Accordingly, x-values: 0, 4, 8, 12, 16 y-values: -4, 0, 4, 0, -4
Derivatives of trig functions
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Double Angle Cos Identities
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Half Angle Identities
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Sum and Difference Identities
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Implicit differentiation:
See image. Note that the slope of a graph can be found implicitly.
The chain rule:
See image. Note the difference between this and the general power rule.
Derivative of ln(ln(x))
ln(u) d/dx = u' / u = (1/x) / ln(x) (Multiply numerator and denominator by x) = 1/ xlnx
Sum of angles in a polygon
sum = 180(n - 2) where n is the number of sides.
SIR equation
the SIR equation is a difference equation intended to be updated frequently to reveal disease spread in a population. SIR EQUATION: Start with S1, I1, P S[n+1]= S[n] - (r-naught * I[n] * S[n]/P) I[n+1]= I[n] + (r-naught * I[n] * S[n]/P) - (0.1 * I[n])
Combinatoric formula to account for subgroups among which order does not matter. For example, how many ways are there to arrange the letters in the word Mississippi?
(Total number of items)!/((First group)!(second group)!etc!) (11!/ (4!4!2!)) = 34650 Notice that the redundant factor in the denominator is a matter of sets.
Sine Function
- Y values start at: 0 - Period: 2π - Domain: (-inf, inf) - Range: [-1, 1] - Amplitude: 1 - Sine is an odd function, meaning that it is symmetric about the origin and sin(-x) = -sin(x)
Name the reflections and translations of the following: y = 3 - ( x + 1 )^2
- reflected over x axis (vertical reflection) - shifted left one unit - shifted up three units The graph is now a downwardly facing parabola with a maximum at (-1 , 3)
Completing the Square
1. Move constant term to the other side of the equation. 2. Find the value (b/2)^2, and add it to both sides of the equation. 3. Factor. If needed, you can solve the equation, but it isn't necessary when you're converting a circle from general to standard form.
Central Angle Vertex of a circle
A central angle's vertex is the center point of the circle. An inscribed angle's vertex is on the circle itself. When these share an arc, the inscribed angles is half the size of the central angle.
Quadrantal Angle
A quadrantal is an angle whose terminal side lies on an axis. (See image for table)
Area of a rhombus
A=1/2d1d2 where d1 and d2 are the diagonals.
If the graph of f(x)=ax^2+bx+c, a≠0, has a maximum value at its vertex, which of the following conditions must be true? A. a>0 B. a<0 C. −b/2a<0 D. −b/2a>0
B.
Cramer's Rule
Cramer's Rule ax + by = e cx + dy = f x = (ed - bf) / (ad - bc) y = (af - ec) / (ad - bc)
Write an algebraic expression of the following: tan(arccos(x/3))
Create a triangle with cos(θ) = x/3. Then, take the tangent of that. This should yield the following: tan(θ) = (sqrt(9-x^2))/x
Derivation of Formula for Total Surface Area of the Sphere by Integration
Derivation of Formula for Total Surface Area of the Sphere by Integration | Derivation of Formulas Review at MATHalino
Derivation formulae (lead to link)
Derivation of Formulas | Review at MATHalino
QR Theorem
Dividend / divisor = quotient + remainder
Forms of Linear Equations
General form: Ax +By = C Slope intercept: Y = mx + b Point slope: Y-Y1 = m(X-X1) Also, note that parallel lines have equivalent slopes, whereas perpendicular lines have negative reciprocal slope.
Present Value Equation
PMT * (1-(1+i)^-n)/i
Synthetic Division
Synthetic Division is a technique that allows one to find the quotient and remainder when dividing with polynomials. This is a list of problems. youtube.com/watch?v=FxHWoUOq2iQ&ab_channel=TheOrganicChemistryTutor
Newton's Law of Cooling
T(t) = T[sub]0 + (T[1] -T[0])e^-kt Where T[1] is the temperature of the body that is placed into T[0], the surroundings; k is a constant.
Integrals of the Six Trigonometric Functions
The integral of tan(u) can be written as ln(sec(x)) + C. The two are equivalent.
Volume and surface area of a box
V = l X w X h Surface A = 2lw + 2wh + 2lh
Right Circular Cylinder
V = π r^2 h Lateral Surface Area 2πrh Total Surface Area = 2πrh + 2πr^2
Law of Cosines
Generalization of the Pythagorean theorem relating the lengths of the sides of any triangle. If a, b, and c are the lengths of the sides and C is the angle opposite side c, then c^2 = a^2 + b^2 − 2ab(cos[C]). Use this when given 2 sides and the included angle, or use it when given all 3 sides.
Geometric Mean Example
Given the sequence 4, 12, 36, 108, 324, 972, ... Find the geometric mean of 4 and 36. Geo. mean = sqrt(4 * 36) = 12 Find the geometric mean of 12 and 972. Geo mean = sqrt(12 * 972) = 108 *Note that a number in a geometric progression could be negative. You could have: 2, +/- 4, 8, +/- 16, ...
Stretches and Compressions
Given y = af(x) If |a| < 1, the graph is compressed vertically by a factor of a. Example: y = .5sqrt(x) If |a| >=1, the graph is stretched vertically by a factor of a. Example: y = 2sqrt(x) Given y = f(bx) If |b| < 1, the graph is stretched horizontally by a factor of 1/b. If |b| < 1, the graph is compressed horizontally by a factor of 1/b.
Overlapping sets formulae
If you have two overlapping sets- N(Total) = n(A) + n(B) + n(neither) - n(both) If there are three overlapping sets- N(Total) = n(A) + n(B) + n(C) - n(sum of two-group overlaps) - 2*(three-group overlap) + n(none)
Vertical and horizontal reflections
A vertical reflection flips the graph about the x axis. A horizontal reflection flips the graph about the y axis. Given y = f(x), y = -f(x) results in a vertical reflection y = f(-x) results in a horizontal reflection
The Multiplication Principle for Counting
According to the multiplication principle, in general, if n operations O1, O2,...,On are performed in order, with possible number of outcomes N1, N2, ..., Nn, respectively, then there are N1•N2•...•Nn possible combined outcomes of the operations performed in the given order.
Linear and Angular velocity
Angular velocity [omega] = [θ]/ t measures the change of an angle [θ] (in radians) over a period of time t. Linear velocity v = s/t measures the change of an arc length s over a period of time t. Linear velocity is an angular velocity multiplied by a radius length. v= s/t = r[θ]/t = r[omega]
Antidifferentiation and Definite Integration
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Area Between Two Curves
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Definition of Continuity
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Definition of Exponential Function to Base a
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Definition of Logarithmic Function to Base a
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Definition of an Antiderivative
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Definition of concavity:
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Using the known range restrictions for inverse functions, find the following: 1. θ = arctan(-1) 2. θ = arcsec (2) 3. θ = arcsin(2) 4. θ = arccos(-(sqrt2)/ 2)
1. tan(θ) = -1 = -π/4 or -45 degrees (remember the range restriction stating that QII is not allowed). You can rewrite the QIV parts of the unit circle as negative reflections of the QI answers. 2. sec(θ) = 2 --> cos(θ) = 1/2 3. sin(θ) = 2 cannot be determined. 4. cos(θ) = -(sqrt2)/2 = 3π/4 or 135 degrees. Note the range restriction invalidating QIII answers.
Volume of a sphere Surface area of a sphere
4πr^3 Note that you could find the volume by solving the standard form of some circle for y and then integrating from -r to r with respect to x. 4πr^2
Area of a parallelogram
A = bh
Area of a trapezoid
A = h/2(b1+b2)
Area of an ellipse Circumference of an ellipse
A = πab Circumference 2π(sqrt((a^2 + b^2) / 2)
Quadratic Graphs II
A quadratic function f(x) is a polynomial of degree 2. General Form: f(x) = ax^2 +bx^2 +c Standard form: f(x) = a(x-h)^2 + k If the leading coefficient is positive, the graph opens up. If negative, the graph opens downwardly. In standard form, the vertex occurs at (h,k) and the axis of symmetry is x = h. In general form, the vertex occurs at (-b/2a, f(-b/2a)) and the axis of symmetry is x = -b/2a.
Derivative II
Geometrically, the derivative of a function can be interpreted as the slope of the graph of the function or, more precisely, as the slope of the tangent line at a point. Its calculation, in fact, derives from the slope formula for a straight line, except that a limiting process must be used for curves. The slope is often expressed as the "rise" over the "run," or, in Cartesian terms, the ratio of the change in y to the change in x. For the straight line shown in the figure, the formula for the slope is (y[sub1] − y[sub0])/(x[sub1] − x[sub0]). Another way to express this formula is [f(x[sub0] + h) − f(x[sub0])]/h, if h is used for x[sub1] − x[sub0] and f(x) for y. This change in notation is useful for advancing from the idea of the slope of a line to the more general concept of the derivative of a function.
Polynomial Graphs
Leading Coefficient: - If negative, the graph goes downwardly on the right - If positive, the graph goes upwardly on the right Degree: - If the degree is even, the ends point in the same direction - If the degree is odd, the ends point in opposite directions X-Intercepts: - If an x-intercept has an even multiplicity, the graph bounces off the x-axis at that intercept. - If an x-intercept has an odd multiplicity, the graph crosses through the x-axis at that intercept.
(2X2)(2X1) Matrix Multiplication
Let A denote the first matrix and B denote the second matrix. Notice that A is a 2×2 matrix and B is a 2×1 matrix. Let C denote the product matrix AB. In order to find the element in the first row and first column of C, c11, multiply each element in the first row of A by its corresponding entry in the first (and only) column of B. Then add these products. Next, to find the element in the second row and first column of C, c21, multiply the elements in the second row of A by the elements in the first (and only) column of B. Since AB is a 2×1 matrix, it has 2 rows and 1 column. Both of the elements of AB have been found.
Logarithmic Functions
Like many of functions, the exponential function has an inverse. This inverse is called the logarithmic function. The logarithmic form y = log[subscript b] of x can be written in exponential form b^y = x. Common log is base 10. Ex: log(2x + 5) is base 10. Natural log is base e. Ex: ln(2x + 5) The Change of base formula is log[sub]b of a = (log a)/ (log b) = lna/lnb To find the domain of a logarithmic function, set the inside greater than 0 and solve. Solve the resulting inequality, plot the numbers on a number line, and keep the ones that render the inequality true.
DMS 1. Convert 73d 5' 42'' to decimal degrees 2. Convert 32.159 to DMS
One Minute (1') is 1/60 of a degree. One second (1'') is 1/60 of a minute and 1/3600 of a degree. 1- 73 + 5(1/60) + 4(1/3600) = 73.095 degrees 2- 32 degrees 0.159(60) = 9.54 Minutes .54(60) = 32.4 Seconds This becomes 32d 9' 32''
Definition of Polynomial
See image. The graph of every polynomial function is both smooth and continuous. The fact that it is smooth means that the graph contains no sharp corners or cusps. The fact that it is continuous means the graph has no gaps or holes and can be drawn without lifting pencil from paper.
Quadratic Graphs, III
The domain is always (-infinity, infinity), and the range of a quadratic depends on the opening and vertex. The maximum and minimum occurs at the vertex.
Definition of the Natural Logarithmic Function
The natural logarithmic function has the following properties. 1. The domain is (0, infinity) and the range is (-infinity, infinity) 2. The function is continuous, increasing, and one-to-one. 3. The graph is concave downward.
Phase Shifts and translations
The phase shift measures a horizontal shift of a period. If d is a number, d > 0 is a shift left, and d < 0 is a shift to the right (as is usually the case with shifting left or right). Similarly, the equilibrium position of a function is moved upwardly or downwardly with the shift up or down. This occurs outside of the function (again, as usual).
(1X2)(2X1) Matrix Multiplication
To start, multiply the (1,1)th entry of the first matrix by the (1,1)th entry of the second. Then, multiply the (1,2)th entry by the (2,1)th entry of the second. Finally, add the two results together.
Construct a right triangle to find an exact value for the following: sec(arcsin(-2/5))
from arcsin to sin(θ) = -2/5. Therefore: x = sqrt(21) y = -2 r = 5 x is found by following the Pythagorean theorem with the realization that sec(θ) would have a negative y-value in QIV. sec = x/r = (5sqrt(21))/21
Range restrictions for inverse functions
y = sin^-1(x) interval [-π/2, π/2] Quadrants I, IV y = cos^-1(x) interval [0,π] Quadrants I, II y = tan^-1(x) interval (-π/2, π/2) Quadrants I, IV y = cot^-1(x) interval (0,π) Quadrants I, II y = sec^-1(x) interval [0,π], y!= 0 Quadrants I, II y = csc^-1(x) interval [-π/2, π/2] y!= 0 Quadrants I, IV
How to find the derivative of a variable raised to a variable...
y = x^x lny = lnx^x (take ln of both sides) lny = xlnx (log rules) 1/y dy/dx = x/x + ln(x) (implicit differentiation) dy/dx = (1 + ln(x))y (multiply by y) dy/dx = (1 + ln(x))x^x (substitute for original y value. This process is summarized by the following formula: To find the derivative of y = f^g y' = f^g [g'(ln(f)) + g*(f'/f)]
Work, Time, Rates Example II A given doctor performs brain surgery in 4 hours when given a personal assistant. If it takes the doctor 6 hours to do this on their own, how long does the assistant take (assuming equal skill)?
(1 / doc) + (1 / assistant) = (1/ both) (1/ 6) + (1/ assistant) = (1/ 4) 1 / assistant = (3/12) - (2/12) assistant = 12
Cotangent Function
- Period: π - Domain: {x| x != nπ} - Range: (-inf, inf) - Amplitude: None - Cotangent is odd, meaning that it is symmetric about the origin and cot(-x) = -cot(x). The periods for the sin, cos, sec, and csc function was 2π. Therefore, you don't need to use 2π/b for the period of Tangent and Cotangent. Rather, use π/b, because the parent functions have a period of π. Notice that cot has the same domain restrictions as csc.
Tangent Function
- Period: π - Domain: {x| x!= nπ +π/2} - Range: (-inf, inf) - Amplitude: None - Tangent is odd, meaning that it is symmetric about the origin and tan(-x) = -tan(x) The periods for the sin, cos, sec, and csc function was 2π. Therefore, you don't need to use 2π/b for the period of Tangent and Cotangent. Rather, use π/b, because the parent functions have a period of π. Notice that tan has the same domain restrictions as sec.
Cosine Function
- Y values start at: 1 - Period: 2π - Domain: (-inf, inf) - Range: [-1, 1] - Amplitude: 1 - Cosine is an even function, meaning that it is symmetric about the y-axis and cos(-x) = cos(x)
Secant Function
- Y values start at: 1 - Period: 2π - Domain: {x| x != nπ + π/2} - Range: (-inf, -1]U[1,inf) - Secant is an even function, meaning that it is symmetric about the y-axis and sec(-x) = sec(x)
Cosecant Function
- Y values start with a vertical asymptote - Period: 2π - Domain: {x| x != nπ} (because you can't divide by zero) - Range: (-inf, -1]U[1, inf) - Cosecant is an odd function, meaning that it is symmetric about the origin and csc(-x) = -csc(x) - Amplitude: There is no amplitude- The definition of amplitude is the distance from the equilibrium line to the maximum or minimum. Therefore, there is no amplitude for Cot, Csc, Tan, or Sec. Only Cos and Sin have amplitude
Combined work formula
1/(person 1) + 1/ (person 2) = 1/T
Even and Odd functions
A function f is even if, for every number x in its domain, the number −x is also in the domain and f(−x)=f(x). A function f is odd if, for every number x in its domain, the number −x is also in the domain and f(−x)= −f(x). In other words, a function is even if and only if its graph is symmetric with respect to the y-axis. A function is odd if and only if its graph is symmetric with respect to the origin.
Ordinary Annuity
A sequence of equal periodic payments when the payments are made at the end of each time interval.
Horizontal and Vertical Stretch
A vertical stretch or compression affects the amplitude or the range. A horizontal stretch or compression instead affects the period of the function. y = sin(bx) or y = cos(bx) If you have a horizontal stretch or compression, then the period changes from 2π to (2π)/b, whatever b is.
Graph the following over a one-period interval. State the amplitude, period, phase shift, and range. g(x) = 2sec(6x) - 3
Amplitude: None Range: (-inf,-5]U[-1,inf) Period: 2π/b = 2π /6 = π/3 Phase Shift: None Vertical Shift: down 3 To find the spacing, you divide the period by 4. Accordingly, the spacing is (π/3) / 4 = π/12: x-values: 0, π/12, π/6, π/4, π/3 y-values: -1, und(asymptote), -5, und(asymptote), -1
Graph the following over a one-period interval. State the amplitude, period, phase shift, and range. g(x) = -4 + cot(x + π/6)
Amplitude: None Range: (-inf,inf) Period: π/b = π/1 = π Phase Shift: left π/6 Vertical Shift: down 4 To find the spacing, you divide the period by 4. Accordingly, the spacing is π/4 : x-values: 0, π/4, π/2, 3π/4, π x-values after P.S.: -π/6, π/12, π/3, 7π/12, 5π/6 y-values: Und, -3, -4, -5, Und
Logarithmic Functions II
Consider another definition: The logarithmic function to the base a, where a>0 and a≠1, is denoted by y=log[subscript a]x (read as "y is the logarithm to the base a of x") and is defined by the following: y=log[subscript a]x if and only if x=a^y The domain of the logarithmic function y=log[subscript a]x is {x | x > 0 } or (0, \inf) Example: The domain for f(x)=log[sub a]x, a>0, a≠1, is the set of all positive real numbers, or (0,∞) using interval notation; the range is the set of all real numbers, or (−∞,∞) using interval notation. The x-intercept of the graph is 1 (any number to the zeroth power is 1). There is no y-intercept.
Derivative I
Derivative, in mathematics, the rate of change of a function with respect to a variable. Derivatives are fundamental to the solution of problems in calculus and differential equations. In general, scientists observe changing systems (dynamical systems) to obtain the rate of change of some variable of interest, incorporate this information into some differential equation, and use integration techniques to obtain a function that can be used to predict the behaviour of the original system under diverse conditions.
Quadratic Graphs
First, rewrite the functions in the form f(x)=a(x−h)^2+k and determine the transformations needed to obtain the graph of the given function from the graph of y=x^2. Recall that the graph is reflected about the x-axis if a<0, is stretched vertically by a factor of a if |a|>1, and is compressed vertically if |a|<1. Recall also that the graph is shifted h units to the right if h>0 or h units to the left if h<0, and k units up if k>0 or k units down if k<0. Thus, the vertex is shifted to the point (h,k) and the axis of symmetry is the vertical line x = h.
Derivative III
For a curve, this ratio depends on where the points are chosen, reflecting the fact that curves do not have a constant slope. To find the slope at a desired point, the choice of the second point needed to calculate the ratio represents a difficulty because, in general, the ratio will represent only an average slope between the points, rather than the actual slope at either point (see figure). To get around this difficulty, a limiting process is used whereby the second point is not fixed but specified by a variable, as h in the ratio for the straight line above. Finding the limit in this case is a process of finding a number that the ratio approaches as h approaches 0, so that the limiting ratio will represent the actual slope at the given point. Some manipulations must be done on the quotient [f(x[sub0] + h) − f(x[sub0])]/h so that it can be rewritten in a form in which the limit as h approaches 0 can be seen more directly. Consider, for example, the parabola given by x^2. In finding the derivative of x^2 when x is 2, the quotient is [(2 + h)^2 − 2^2]/h. By expanding the numerator, the quotient becomes (4 + 4h + h2 − 4)/h = (4h + h^2)/h. Both numerator and denominator still approach 0, but if h is not actually zero but only very close to it, then h can be divided out, giving 4 + h, which is easily seen to approach 4 as h approaches 0.
If the degree of the numerator is greater than the degree of the denominator... If a rational function is proper, then _______ is a horizontal asymptote.
For a rational function R, if the degree of the numerator is less than the degree of the denominator, then R is proper. Note that if the degree of the numerator is greater than the degree of the denominator, then R is improper. y = 0: When a rational function R(x) is proper, the degree of the numerator is less than the degree of the denominator; as x→−∞ or as x→∞, the value of R(x) approaches 0.
The Mandelbrot Set III
For example, if c = 1 then the sequence is 0, 1, 2, 5, 26,..., which goes to infinity. Therefore, 1 is not an element of the Mandelbrot set, and thus is not coloured black. On the other hand, if c is equal to the square root of -1, also known as i, then the sequence is 0, i, (−1 + i), −i, (−1 + i), −i..., which does not go to infinity and so it belongs to the Mandelbrot set. When graphed to show the entire Set, the resultant image is striking, pretty, and quite recognizable. There are many variations of the Mandelbrot set, such as Multibrot, Buddhabrot, and Nebulabrot. Multibrot is a generalization that allows any exponent: z[sub n+1] = z[sub n]^d + c. These sets are called Multibrot sets. The Multibrot set for d = 2 is the Mandelbrot set.
Rational Graphs (Holes Vs. Asymptotes)
For rational functions, holes correspond to the roots (or zeros) of the denominator that cancel out entirely during simplification. Vertical asymptotes occur at places where the limit of the function is ∞ or -∞, which happen at the roots of the denominator that are left over after simplification. It's an important distinction. You find the domain before simplifying, and you find the hole and vertical asymptotes after simplifying. To find the Y value of a hole, you would simply plug the x value of that hole into the reduced form of the rational. The vertical asymptote comes from the factors in the denominator that do not cancel. The holes come from the factor in the denominator that do cancel. Precalculus, Lecture 4.4
Derivative of ln(ln(ln(x)))
From the last problem... u = ln(ln(x)) u' = 1/ xlnx Extending this principle... u' = (1/xlnx) / (ln(ln(x)) Multiply both sides by xlnx... = 1 / xlnx*ln(ln(x))
The Fundamental Theorem of Calculus
Fundamental theorem of calculus, Basic principle of calculus. It relates the derivative to the integral and provides the principal method for evaluating definite integrals (see differential calculus; integral calculus). In brief, it states that any function that is continuous (see continuity) over an interval has an antiderivative (a function whose rate of change, or derivative, equals the function) on that interval. Further, the definite integral of such a function over an interval a < x < b is the difference F(b) − F(a), where F is an antiderivative of the function. This particularly elegant theorem shows the inverse function relationship of the derivative and the integral and serves as the backbone of the physical sciences. It was articulated independently by Isaac Newton and Gottfried Wilhelm Leibniz.
The Second Fundamental Theorem of Calculus
If f is continuous on an open interval containing a, then for every x in the interval the derivative of the the integral of f(x) dx on said interval is equal to f(x)
Rolle's Theorem
In calculus, Rolle's theorem or Rolle's lemma essentially states that any real-valued differentiable function that attains equal values at two distinct points must have at least one stationary point somewhere between them—that is, a point where the first derivative (the slope of the tangent line to the graph of the function) is zero. The theorem is named after Michel Rolle.
The Extreme Value Theorem
In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval [a,b], then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in [a,b] such that: f(c) >= f(x) >= f(d) for all x elements of [a,b] The extreme value theorem is more specific than the related boundedness theorem, which states merely that a continuous function f on the closed interval [a,b] is bounded on that interval; that is, there exist real numbers m and M such that: m <= f(x) <= M for all x elements of [a,b] This does not say that M and m are necessarily the maximum and minimum values of f on the interval [a,b], which is what the extreme value theorem stipulates must also be the case. The extreme value theorem is used to prove Rolle's theorem. In a formulation due to Karl Weierstrass, this theorem states that a continuous function from a non-empty compact space to a subset of the real numbers attains a maximum and a minimum.
The Disk Method
In finding the volume of a solid of revolution, use one of the following depending on the axis of revolution - ∫ π[R(x)]^2 dx for a horizontal axis of revolution, ∫ π[R(y)]^2 dy for a vertical axis of revolution
Interval Scales
In interval scales, numbers form a continuum and provide information about the amount of difference, but the scale lacks a true zero. The differences between adjacent numbers are equal or known. If zero is used, it simply serves as a reference point on the scale but does not indicate the complete absence of the characteristic being measured. The Fahrenheit and Celsius temperature scales are examples of interval measurement. In those scales, 0 °F and 0 °C do not indicate an absence of temperature.
Eigenvalues and Eigenvectors
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by λ [lambda],[1] is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed.[2] Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated.
Difference Quotient
In single-variable calculus, the difference quotient is usually the name for the expression shown (see image). which when taken to the limit as h approaches 0 gives the derivative of the function f. The name of the expression stems from the fact that it is the quotient of the difference of values of the function by the difference of the corresponding values of its argument (the latter is (x+h)-x=h in this case). The difference quotient is a measure of the average rate of change of the function over an interval (in this case, an interval of length h). The limit of the difference quotient (i.e., the derivative) is thus the instantaneous rate of change.
Integral I
Integral, in mathematics, either a numerical value equal to the area under the graph of a function for some interval (definite integral) or a new function the derivative of which is the original function (indefinite integral). These two meanings are related by the fact that a definite integral of any function that can be integrated can be found using the indefinite integral and a corollary to the fundamental theorem of calculus. The definite integral (also called Riemann integral) of a function f(x) is denoted as: [integral sign from a to b] f(x)dx and is equal to the area of the region bounded by the curve (if the function is positive between x = a and x = b) y = f(x), the x-axis, and the lines x = a and x = b. An indefinite integral, sometimes called an antiderivative, of a function f(x), denoted by the same notation, but without the 'a to b' component, is a function the derivative of which is f(x). Because the derivative of a constant is zero, the indefinite integral is not unique. The process of finding an indefinite integral is called integration.
Name the reflections and translations of the following: h(x) = cuberoot(5-x) + 1
It helps to start by factoring the negative out of the cube root to see what is really being reflected and shifted. This should yield the following: h(x) = cuberoot(-(x-5)) + 1 -Horizontal or y-axis reflection -shift right 5 -shift up 1 This results in a cube root functions with a point of interest at (5 , 1).
Midpoint Formula
It is simply the average of the points.
Percentile Location Formula
L[sub k] = (k/100) * (n + 1) Where k is the percentile and n is the number of data items.
Definition of a Composite Function
Let f and g be functions. The function (f ◦ g)(x) = f(g(x)) is the composite of f with g. The domain of f ◦ g is the set of all x in the domain of g such that g(x) is in the domain of f.
The Mandelbrot Set II
Mandelbrot was one of the first to use computer graphics to create and display fractal geometric images, leading to his discovering the Mandelbrot set in 1979. That was because he had access to IBM's computers. He was able to show how visual complexity can be created from simple rules. He said that things typically considered to be "rough", a "mess" or "chaotic", like clouds or shorelines, actually had a "degree of order". The equation zn+1 = zn2 + c was known long before Benoit Mandelbrot used a computer to visualize it. Images are created by applying the equation to each pixel in an iterative process, using the pixel's position in the image for the number 'c'. 'c' is obtained by mapping the position of the pixel in the image relative to the position of the point on the complex plane. The shape of the Mandelbrot Set is represented in black in the image on this page.
SI prefixes
P - 10^15 (peta) Quadrillion E - 10^18 (exa) Quintillion Z - 10^21 (zetta) Sextillion Y - 10^24 (yotta) Septillion f - 10^-15 (femto) Quadrillionth a - 10^-18 (atto) Quintillionth z - 10^-21 (zepto) Sextillionth y - 10^-24 (yocto) Septillionth
Conjugate Pairs
Pairs of the form (a+b) and (a-b). When multiplied, they yield the difference of squares. (a^2 - b^2)
Ratio Scales
Ratio scales have all of the characteristics of interval scales as well as a true zero, which refers to complete absence of the characteristic being measured. Physical characteristics of persons and objects can be measured with ratio scales, and, thus, height and weight are examples of ratio measurement. A score of 0 means there is complete absence of height or weight. A person who is 1.2 metres (4 feet) tall is two-thirds as tall as a 1.8-metre- (6-foot-) tall person. Similarly, a person weighing 45.4 kg (100 pounds) is two-thirds as heavy as a person who weighs 68 kg (150 pounds).
Angle/Side relationship
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Definition of the Average Value of a Function on an Interval
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Delta Epsilon Definition of a Limit
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Derivative of Natural Logarithmic Function
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Derivatives of Inverse Trigonometric Functions
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First Derivative test:
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Graphs of Inverse Trigonometric Functions
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Greatest Integer Function
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Integration Formulas
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Limits at infinity of rational functions
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Points of inflection:
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Second Derivative test:
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Special Right Triangles
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Summation formulas
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The Mean Value Theorem for Integrals
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Double Angle Sin and Tan identity
Sin(2a) = 2sin(a)cos(a) Tan(2a) = (2tan(a)) / ( 1 - tan^2(a))
Write an algebraic expression of the following: cos(arcsin(2x))
Since θ = arcsin(2x), sin(θ) = 2x. Then, you can use the sin function to create a triangle, thereby deducing the other side. Two of the values were positive, so you know the other value will be positive (when you take the square root in the process, that is). Then, take the cosine of that result. This should reduce to sqrt(1-4x^2).
Third Side Rule
The length of any one side of a triangle must be less than the sum of the other two sides, and greater than the difference between the other two sides.
Horizontal and Vertical Stretches: y = (1/7)sin(3x) y = 3cos((2/7)x)
The negative indicates that you have an x-axis reflection. The 1/7 indicates that you have a vertical compression by a factor of 1/7. The 3 indicates that there is a horizontal compression by a factor of 1/3. Amplitude: 1/7 Range [-1/7, 1/7] Period: 2π/3 The three is a vertical stretch by a factor of three. There is a horizontal stretch by a factor of two. Accordingly, Amplitude: 3 Range: [-3,3] Period: (2π)/2/7 = 7π
A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate of 1 foot per second. When the radius is 4 feet, at what rate is the total area A of the disturbed water changing?
The variables r and A are related by A = πr^2. The rate of change of the radius is dr/dt = 1. d/dt[A] = d/dt[πr^2] Differentiate with respect to r. dA/dt = 2πr dr/dt Apply chain rule. = 2π(4)(1) When the radius is 4 feet, the area is changing at a rate of 8π.
Implications of the Mean Value Theorem
Theorem: Assume that f is a continuous, real-valued function, defined on an arbitrary interval I of the real line. If the derivative of f at every interior point [In mathematics, specifically in topology, the interior of a subset S of a topological space X is the union of all subsets of S that are open in X. A point that is in the interior of S is an interior point of S. The interior of S is the complement of the closure of the complement of S. In this sense interior and closure are dual notions.] of the interval I exists and is zero, then f is constant in the interior. Theorem 2: If f' (x) = g' (x) for all x in an interval (a, b) of the domain of these functions, then f - g is constant, i.e. f = g + c where c is a constant on (a, b). Theorem 3: If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is F(x) + c where c is an constant.
Appreciate or learn to derive Euclidean formulae.
This is the wiki for 2D Euclidean shapes. https://en.wikipedia.org/wiki/Two-dimensional_space This is the wiki for 3D Euclidean shapes. https://en.wikipedia.org/wiki/Three-dimensional_space This is the wiki for 4d shapes. https://en.wikipedia.org/wiki/Four-dimensional_space
Graph the following over a one-period interval. State the amplitude, period, phase shift, and range. y = 4 + 3sec((1/2)x+π)
To start, factor out 1/2 from the inside of the trig function. This yields y = 4 + 3sec[.5(x+2π)] Amplitude: None Range: (-inf,1]U[7,inf) Period: 2π/ b = (2π)/.5 = 4π Phase Shift: left 2 π Vertical Shift: up 4 To find the spacing, you divide the period by 4. Accordingly, the spacing is π : x-values: 0, π, 2π, 3π, 4π x-values after P.S.: -2π, -π, 0, π, 2π y-values: 7, undefined(asymptote), 1, undefined(asymptote, 7
Transcendental Functions
Transcendental function, In mathematics, a function not expressible as a finite combination of the algebraic operations of addition, subtraction, multiplication, division, raising to a power, and extracting a root. Examples include the functions log x, sin x, cos x, e^x and any functions containing them. Such functions are expressible in algebraic terms only as infinite series. In general, the term transcendental means nonalgebraic.
Ture/False : The quotient of two polynomial expressions is a rational expression.
True. A rational function is a function of the form R(x)=p(x)/q(x), where p and q are polynomial functions and q is not the zero polynomial.
Which type of asymptote will never intersect the graph of a rational function?
Vertical: Note that a line x=c is a vertical asymptote for a function f if, as x approaches c, the values f(x) either approach ∞ or −∞. That is, the function is not defined at x=c and hence the aysmptote does not intersect the function.
How to find a common multiple of two numbers (the LCM).
What is the least common multiple of 28 and 42? Take the prime factors of both numbers. This yields the following: 28 = 2 X 2 X 7 and 42 = 2 X 3 X 7. Multiply the prime factors the greatest number of times each appears in the prime factorizations. In this example, 2 appears twice, 3 appears once, and 7 appears once. Multiply 2 X 2 X 3 X 7 to receive 84. 84 is the LCM of 28 and 42.
Navigational Bearing
When a single angle is given, it implies that the bearing is measured in a clockwise direction from due north. When a north-south line is first specified followed by an acute angle and then an east-west line implies a direct interpretation. 125 degrees - Draw legend, subtract resulting QIV angle from the south position. This results in South 55 [Degrees] East N 37[Degrees] West - To convert to the single angle measure, turn clockwise from the north position. This results in the angle you subtract 37 from 360 to acquire. (323).
The mean value theorem:
Where f'(c) = slope of tangent line In mathematics, the mean value theorem states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It is one of the most important results in real analysis. This theorem is used to prove statements about a function on an interval starting from local hypotheses about derivatives at points of the interval. More precisely, the theorem states that if f is a continuous function on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in (a,b) such that the tangent at c is parallel to the secant line through the endpoints (a,f(a)) and(b,f(b)) , that is,(see image). A special case of this theorem was first described by Parameshvara (1380-1460), from the Kerala School of Astronomy and Mathematics in India, in his commentaries on Govindasvāmi and Bhāskara II. A restricted form of the theorem was proved by Michel Rolle in 1691; the result was what is now known as Rolle's theorem, and was proved only for polynomials, without the techniques of calculus. The mean value theorem in its modern form was stated and proved by Augustin Louis Cauchy in 1823. Many variations of this theorem have been proved since then
Summation notation (Sigma notation)
a concise way to express the sum of a set of numbers
How to find the derivative of a constant raised to a variable...
a^u d/dx = a^u * u' * ln(a) See image.
Cones
if A is he area of the base, V = Ah/3 Right circular cone: V = 1/3 (πh r^2) Lateral Surface Area = πr sqrt(r^2 + h^2) Realize that if you tilt the cone on the x-axis, you can find the function you would revolve on the x-axis to derive the volume: y = (r/h)x
Given f(x) = c + asin(b(x-d))...
- c moves the graph up or down (changes equilibrium) - |a| is the amplitude ( if a < 0, there is a vertical reflection) - b changes the period from 2π to 2π/b - d moves the graph left or right
Common behaviors associated with nonexistence of a limit
1. f(x) approaches a different number from the right side of c than it approaches from the left side. 2. f(x) increases or decreases without bound as x approaches c 3. f(x) oscillates between two fixed values as x approaches c
Knowing the negative-angle identities, write the following functions with no horizontal reflection using the negative-angle identities. 1. y = 2 + tan(-5x) 2. f(x) = sec(6π -7x) 3. g(x) = 5 - 9 sin(3π/2 -x)
1. y = 2 - tan(5x) 2. f(x) = sec(7x - 6π) 3. g(x) = 5 - 9sin(-x+ 3π/2) = 5 + 9sin(x-3π/2)
Amplitude
Amplitude is the distance a graph travels above or below its equilibrium position. The amplitude changes from 1 to |a|, which means the range becomes [-a, a]. Note that if a is negative, the graph has a vertical reflection (flipped over the x-axis).
Graph the following over a one-period interval. State the amplitude, period, phase shift, and range. y = 2 + sin(x - π/4)
Amplitude: 1 Range: [1,3] Period: 2π/b = 2π/1 = 2π Phase Shift: π/4 to the right Vertical Shift: up 2 Equilibrium line: changed by V.S. y = 2 To find the spacing, you divide the period by 4. Accordingly, the spacing is reduced to π/2.Also, this graph contains a phase shift, moving the x-values: x-values: 0, π/2, π, 3π/2, 2π x-values after P.S.: π/4, 3π/3, 5π/4, 7π/4, 9π/4 y-values: 2, 3, 2, 1, 2
Graph the following over a one-period interval. State the amplitude, period, phase shift, and range. y = 5 + 2cos(6x)
Amplitude: 2 Range: (affected by the vertical shift) [3,7] Period: 2π/b = 2π/6 = π/3 Phase Shift: 0 Vertical Shift: up 5 Equilibrium line: y = 5 To find the spacing, you divide the period by 4. Accordingly, the spacing is: x-values:0, π/12, π/6, π/4, π/3 y-values: 7, 5, 3, 5, 7 Note: You should always end with your period when listing your x-values.
Graph the following over a one-period interval. State the amplitude, period, and range. g(x) = 2sin(3x)
Amplitude: 2 Range: [-2,2] Period: (2π)/3 To determine the spacing for the x-values, divide the period by 4 equal parts. ((2π)/3)/4 = π/6. Cosine and sine functions start at zero. The y-values are found by making an z-y chart. Remember to ensure that the calculator is in radian mode. Accordingly, x-values: 0, π/6, 2π/6, 3π/6, 4π/6 y-values: 0, 2, 0, -2, 0
Graph the following over a one-period interval. State the amplitude, period, phase shift, and range. y = 1 + 4cos(3x - 3π/8)
Amplitude: 4 Range: [-3,5] Period: 2π/b = 2π/3 Phase Shift: right π/8 Vertical Shift: up 1 Equilibrium line: y = 1 To find the spacing, you divide the period by 4. Accordingly, the spacing is (2π/3)/4 = 2π / 12 = π/6. Given that cos begins at 0: x-values: 0, π/6, π/3, π/2, 2π/3 x-values after P.S.: π/8, 7π/24, 11π/24, 5π/8, 19π/24 y-values: 5, 1, -3, 1, 5
Graph the following over a one-period interval. State the amplitude, period, phase shift, and range. f(x) = -7tan((2.5)x)
Amplitude: None Range: (-inf, inf) Period: π/b = π/2/5 = 5π/2 Phase Shift: None Vertical Shift: None To find the spacing, you divide the period by 4. Accordingly, the spacing is 5π/8. x-values: -5π/4, -5π/8, 0 , 5π/8, 5π/4 y-values: Und, 7, 0, -7, Und Notice that tangent requires a graph centered around 0. Therefore, you must add two values to the left and two to the right.
Graph the following over a one-period interval. State the amplitude, period, phase shift, and range. y = -(5/3)csc(x-(3π/2))
Amplitude: None Range: (-inf,-5/3]U[5/3,inf) Period: 2π/b = 2π/1 = 2π Phase Shift: right 3π/2 Vertical Shift: 0 To find the spacing, you divide the period by 4. Accordingly, the spacing is 2π/4 = π/2 : x-values: 0, π/2, π, 3π/2, 2π x-values after P.S.: 3π/2, 2π, 5π/2, 3π, 7π/2 y-values: und(asymptote), -5/3, und(asymptote), 5/3, und(asymptote)
How to find common factors of two numbers.
Break both numbers down to their prime factors to see which they have in common. Then multiply the shared prime factors to find all common factors. Example: What factors greater than one do 135 and 225 have in common? Answer: 1. Find the primes of both numbers. For 135, the factors are 3 X 3 X 3 X 5. For 225, the factors are 3 X 3 X 5 X 5 . 2. The common primes are 3 X 3 X 5. 3. Multiply the common factors in every combination. This yields 3, 5, 3 X 3, 3 X 5, and 3 X 3 X 5. Thus the common factors are 3, 5, 9, 15, and 45.
The Net Change Theorem
If F'(x) is the rate of change of a quantity F(x), then the definite integral of F'(x) from a to b gives the total change, or net change, of F(x) on the interval [a,b].
Rational Zero Theorem
If P(x) is a polynomial with integer coefficients, then every rational zero has the form p/q where p is a factor of the constant term, q is a factor of the leading coefficient, and p and q have no common factors besides 1. The Rational Zero Theorem allows us to find rational zeros, but it cannot find irrational zeros, complex zeros, or detect multiplicity. youtube.com/watch?v=Iaq7z7reznM&ab_channel=TheOrganicChemistryTutor
Nominal Scales
In nominal scales, numbers, such as driver's license numbers and product serial numbers, are used to name or identify people, objects, or events. Gender is an example of a nominal measurement in which a number (e.g., 1) is used to label one gender, such as males, and a different number (e.g., 2) is used for the other gender, females. Numbers do not mean that one gender is better or worse than the other; they simply are used to classify persons. In fact, any other numbers could be used, because they do not represent an amount or a quality. It is impossible to use word names with certain statistical techniques, but numerals can be used in coding systems. For example, fire departments may wish to examine the relationship between gender (where male = 1, female = 2) and performance on physical-ability tests (with numerical scores indicating ability).
Ordinal Scales
In ordinal scales, numbers represent rank order and indicate the order of quality or quantity, but they do not provide an amount of quantity or degree of quality. Usually, the number 1 means that the person (or object or event) is better than the person labeled 2; person 2 is better than person 3, and so forth—for example, to rank order persons in terms of potential for promotion, with the person assigned the 1 rating having more potential than the person assigned a rating of 2. Such ordinal scaling does not, however, indicate how much more potential the leader has over the person assigned a rating of 2, and there may be very little difference between 1 and 2 here. When ordinal measurement is used (rather than interval measurement), certain statistical techniques are applicable (e.g., Spearman's rank correlation).
Common Pythagorean Triples
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Definition of a tangent line with slope m:
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Definition of infinite limits
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The Shell Method
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The general power rule:
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The product rule:
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The quotient rule:
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There are two special trigonometric limits (cos and sin). What are they?
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Trigonometric Functions (domain and range)
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Definition of The Natural Exponential Function
See image. Note that the function is concave upward.
Intermediate Value Theorem and the Revised Intermediate Value Theorem
The Intermediate Value Theorem states that if a and b are real numbers such that a < b and if f is a polynomial function such that f (a) /=/ f (b), f can equal any value between f (a) and f(b). The Revised Intermediate Value Theorem states that if a and b are real numbers such that a < b and if f is a polynomial function such that f(a) and f(b) are opposite signs, then at least one real zero must exist in the interval [a,b].
The Mandelbrot Set
The Mandelbrot set is an example of a fractal in mathematics. It is named after Benoît Mandelbrot, a Polish-French-American mathematician. The Mandelbrot set is important for chaos theory. The edging of the set shows a self-similarity, which is not perfect because it has deformations. The Mandelbrot set can be explained with the equation z[sub n+1] = z[sub n]^2 + c. In that equation, c and z are complex numbers and n is zero or a positive integer (natural number). Starting with z0=0, c is in the Mandelbrot set if the absolute value of zn never becomes larger than a certain number (that number depends on c), no matter how large n gets.
Tests for symmetry of a graph
The graph of an equation in x and y is symmetric with respect to the y-axis when replacing x by -x yields an equivalent equation. The graph of an equation in x and y is symmetric with respect to the x-axis when replacing y by -y yields an equivalent equation. The graph of an equation in x and y is symmetric with respect to the origin when replacing x by -x and y by -y yields an equivalent equation.
The Logistic Map
The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized in a 1976 paper by the biologist Robert May,[1] in part as a discrete-time demographic model analogous to the logistic equation written down by Pierre François Verhulst.[2] Mathematically, the logistic map is written x[sub n+1] = rx[sub n](1 -x[sub n])
Set Operations - Union, Intersection, Complement
The union of sets will give a new set whose elements are elements of A, B, or both. The intersection of sets A and B will give a new set whose elements are elements of both sets A and B. The complement of A will give us a new set whose elements are the elements in U that are not in set A.
Graph the following over a one-period interval. State the amplitude, period, phase shift, and range. h(x) = -1 + 5tan(4x-π)
To begin, factor out the b value in the function, yielding: h(x) -1 +5tan(4(x-π/4)) Amplitude: None Range: (-inf, inf) Period: π/4 Phase Shift: right π/4 Vertical Shift: down 1 To find the spacing, you divide the period by 4. Accordingly, the spacing is π/16 : x-values: -π/8, -π/16, 0, π/16, π/8 x-values after P.S.: π/8, 3π/16, π/4, 5π/16, 3π/8 y-values: und(asymptote), -6, -1, 4, und(asymptote)
Equilateral triangle Height (h): Area:
h = (sqrt(3)s)/2 A = (sqrt(3)s^2)/4
Counting Consecutive Numbers
the number of integers from A to B inclusive = B - A + 1. How many integers are there from 73 to 419, inclusive? (419 - 73) + 1. The sum is the average multiplied by the number of terms. Example: what is the sum of the integers from 10 to 50, inclusive? ((50 - 10) + 1) X ((10 + 50) / 2) = 41 X 30 = 1230.
Eigenvalues and Eigenvectors (formal definition)
uIf T is a linear transformation from a vector space V over a field F into itself and v is a nonzero vector in V, then v is an eigenvector of T if T(v) is a scalar multiple of v. This can be written as T(v) = λv where λ is a scalar in F, known as the eigenvalue, characteristic value, or characteristic root associated with v. There is a direct correspondence between n-by-n square matrices and linear transformations from an n-dimensional vector space into itself, given any basis of the vector space. Hence, in a finite-dimensional vector space, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices, or the language of linear transformations.[3][4] If V is finite-dimensional, the above equation is equivalent to[5] Au = λu where A is the matrix representation of T and u is the coordinate vector of v.
If -1 <= a <= 1, what is the solution set for cos(x) = a ? If -1 <= a <= 1, what is the solution set for sin(x) = a ? (double-check this) What is the solution set for tan(x) = a ?
{x| x = s + 2nπ} where s = arccos(a) and n is an integer. {x| x = s + 2nπ} where s = arcsin(a) and n is an integer. The solution set for tan(z)= a is {x| x = s + nπ} where s = arctan(a) and n is an integer.
Functions
A relation is a correspondence between two sets. The set of all inputs for a relation is the domain of the relation, and the set of all outputs is called the range. A function is a relation in which each element in the domain corresponds to exactly one element in the range.
Work, Time, Rates Example II Peter, Xi, and Khal can complete a task in 6, 9, and 18 hours respectively. How long will it take for them to complete the task as a team.
(1/ p) + (1 /x) + (1/k) = 1 / t (1/ 6) + (1 /9) + (1/18) = 1 / t (3 + 2 + 1) / 18 = 1/ t 1/ 3 = 1 / t t = 3
Work, Time, Rates Example I Timothy paints a room in 6 hours; Sally in 3. How long will it take for them to paint it together.
(1/3) + (1/6) = (1/t) 1/2 = 1/t t = 2
Name the reflections and translations of the following: f(x) = -2 - sqrt(-x)
- vertical reflection (x-axis reflection) - horizontal reflection (y-axis reflection) - shift down two units The graph is now a square root function pointing to negative infinity in Quadrant III. The node of the graph is at point (0 , -2)
1. How to find an arithmetic mean when a number is included or removed. 2. How to use an original and new arithmetic mean to find what was included or removed.
1. If Michael makes an average of 80 on four tests, what is his new average after making a 100 on another test? If average is 80, 4 X 80 = 320. Add the 100, and divide by 5, yielding 84. 2. If the average of five numbers is 2, and you remove one, thereby creating a new average of -3, what number was deleted? Find the original sum from the original average. 5(2) = 10. Find the new sum from the average. New sum = 4(3) = -12. The difference between these two numbers is the number deleted.
Notes on Functions
1. If looking at a series of points (a relation), non-repeating x values mean that the relation is a function. 2. If neither x nor y values repeat, it is a one-to-one function. 3. In order for a function to have an inverse, the function must be one-to-one. Otherwise, you would have repeating x-values on the inverse function (see 'inverse functions').
Steps to Rational Zero Theorem
1. Let p be all possible factors of the constant term 2. Let q be all possible factors of the leading coefficient 3. List all possibilities for +/- p/q 4. Plug each result from the previous step into the original polynomial function. If the output is 0, we have found a zero 5. If all zeros have not been found, use synthetic division and the quadratic formula to find the rest of the zeros
Useful Conversions: 1. 1 m = x ft 2. 1 kg = x lb 3. 0 C = x F 4. 0 C = x K 5. 1 C = x F 6. m/h = x km/h 7. 1 gal = x L 8. 1 hp = x W 9. 1 oz. = x L 10. 1 lb = x kg 11. 1 mi = x km
1. x = 3.2808399 2. x = 2.20462262 3. x = 32.8 4. x = 273.15 5. x = 33.8 6. x = 1.609344 7. x = 3.78541178 8. x = 745.699872 9. x = 0.02957353 10. x = 0.45359237 11. x = 1.6093E
Shortcuts for recognizing multiples of 3, 4, 6, 9, and 12
3 - the sum of the digits is a multiple of 3 4 - the last two digits are a multiple of 4 6 - if the sum of the digits is a multiple of 3, and the last digit is even 9 - the sum of the digits is a multiple of 9 12 - the sum of the digits is a multiple of 3, and the last two digits are a multiple of 4
The Unit Circle
A circle of radius 1 and center ( 0, 0 ). As such, the equation for the unit circle would be as follows: (x-0)^2 + (y-0)^2 = r^2 --> x^2 + y^2 =1 The degree measurements correspond with the radian measurements, as can be verified with the appropriate conversion factor.
Logarithmic Example I
Acids are substances that dissociate in water and release hydrogen ions (H+).Bases are substances that either take up hydrogen ions (H+) or release hydroxide ions (OH-). pH is a range of numbers expressing the relative acidity or alkalinity of a solution. In general, pH values range from 0 to 14. The pH of a neutral solution, i.e., one which is neither acidic nor alkaline, is 7. Acidic solutions have pH values below 7; alkaline, or basic, solutions have pH values above 7. A pH value provides a measure of the hydrogen ion concentration of a solution. In pure water the concentration of hydrogen ions is equal to 0.0000001, or 10−7, moles per liter. (A mole is the amount of a substance, expressed in grams, that is equal to the molecular weight, or formula weight, of the substance.) When an acid is added to pure water, the hydrogen ion concentration increases above this level. When an alkaline substance, or base, is added to pure water, the hydrogen ion concentration decreases below this level. Once the concentration is determined, the pH value is found by taking the exponent used in expressing this concentration and reversing its sign. This is expressed as pH=−log10 [H+]. For example, if the hydrogen ion concentration of a solution is 10−4, or 0.0001, moles per liter, the pH is 4. See indicators, acid-base.
Rule for matrix multiplication.
An m×n matrix has m rows and n columns. The product of an m×n matrix A and an n×p matrix B is an m×p matrix AB. To calculate the product, each row of A must have the same number of entries as each column of B. The definition of matrix multiplication states that the element in the ith row and jth column of the final matrix is found by multiplying each element in the ith row of the first matrix by the corresponding element in the jth column of the second and adding the results of these products.
The Fundamental Theorem of Algebra
An nth degree polynomial has n (not necessarily distinct) zeros. Although all of these zeros may be imaginary, a real polynomial of odd degree must have at least one real zero.
(2X2)(2X2) Matrix Multiplication
If a matrix has 1 row and n columns, and another matrix has n rows and 1 column, the product of the two matrices is an n × n matrix. The product is only defined if the first matrix has the same number of columns that the other matrix has in rows. If a matrix has m rows and p columns, and another matrix has p rows and n columns, the product of the two matrices is an m × n matrix. The product is only defined if the first matrix has the same number of columns that the other matrix has in rows. An element in the i-th row and j-th column of the product is the real number obtained from the product of the i-th row of the first matrix and the j-th column of the second matrix. To determine the (1,1)th entry of the product, multiply the first row of the first matrix by the first column of the second matrix. To determine the (1,2)th entry of the product, multiply the first row of the first matrix by the second column of the second matrix. To determine the (2,1)th entry of the product, multiply the second row of the first matrix by the first column of the second matrix. To determine the (2,2)th entry of the product, multiply the second row of the first matrix by the second column of the second matrix. Use the results of each multiplication to find the final product.
Right triangle in a circle
If one of the sides of an inscribed triangle is a diameter of the circle, then the triangle must be a right triangle.
If r is a solution to the equation f(x)=0, name three additional statements that can be made about f and r assuming f is a polynomial function.
If r is a solution to the equation f(x)=0, then the following statements are equivalent assuming f is a polynomial function. The r is a real zero of the polynomial function f, r is an x-intercept of the graph of f, and (x−r) is factor of f.
Weighted averages
If, in a classroom of boys and girls, the girls score an average of 30, and the boys (of which there are twice as many) score an average of 24, what is the class's average score? (1(30) + 2(24)) / 3 = 26 The average score is 26.
The Arithmetic Mean (wiki)
In mathematics and statistics, the arithmetic mean ( /ˌærɪθˈmɛtɪk ˈmiːn/, stress on first and third syllables of "arithmetic"), or simply the mean or the average (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection.[1] The collection is often a set of results of an experiment or an observational study, or frequently a set of results from a survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics, because it helps distinguish it from other means, such as the geometric mean and the harmonic mean. In addition to mathematics and statistics, the arithmetic mean is used frequently in many diverse fields such as economics, anthropology and history, and it is used in almost every academic field to some extent. For example, per capita income is the arithmetic average income of a nation's population. While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly influenced by outliers (values that are very much larger or smaller than most of the values). For skewed distributions, such as the distribution of income for which a few people's incomes are substantially greater than most people's, the arithmetic mean may not coincide with one's notion of "middle", and robust statistics, such as the median, may provide better description of central tendency.
Cardinality
In mathematics, the cardinality of a set is a measure of the "number of elements" of the set. For example, the set A={2,4,6} contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between the different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers.
The Geometric Mean
In mathematics, the geometric mean is a mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product, that is sqrt(2X8) = 4. As another example, the geometric mean of the three numbers 4, 1, and 1/32 is the cube root of their product (1/8), which is 1/2, that is cuberoot(4X1X1/32) = 1/2. The geometric mean applies only to positive numbers. The geometric mean is often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as a set of growth figures: values of the human population or interest rates of a financial investment over time. The geometric mean can be understood in terms of geometry. The geometric mean of two numbers, a and b, is the length of one side of a square whose area is equal to the area of a rectangle with sides of lengths a and b. Similarly, the geometric mean of three numbers, a, b, and c, is the length of one edge of a cube whose volume is the same as that of a cuboid with sides whose lengths are equal to the three given numbers. The geometric mean is one of the three classical Pythagorean means, together with the arithmetic mean and the harmonic mean. For all positive data sets containing at least one pair of unequal values, the harmonic mean is always the least of the three means, while the arithmetic mean is always the greatest of the three and the geometric mean is always in between (see Inequality of arithmetic and geometric means.)
Harmonic Mean
In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. Typically, it is appropriate for situations when the average rate is desired. The harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals of the given set of observations. Become proficient separate examples. https://en.wikipedia.org/wiki/Harmonic_mean Learn to formally write this visual proof: https://en.wikipedia.org/wiki/Pythagorean_means Here is an interesting practice problem. https://en.wikipedia.org/wiki/Crossed_ladders_problem
Inverse Functions
In order for the inverse of function f to be a function, f must be one-to-one. A function is one-to-one when f(x1) = f(x2) implies that x1 = x2. In other words, y values cannot repeat. Just as a 'vertical line test' can be used to determine if a given graph is a function, a 'horizontal line test' can be used to determine if a function is one to one. Consider it this way: If given the sets { (1,2), (2,7), (-4,7) } { (2,1), (7,2), (7,-4) } you can clearly see that this original set would no longer be a function due to x values repeating. Nonetheless, one can restrict the domain of a function that isn't one-to-one to be such that it is and thereby proceed to find the inverse (if there is one within that restriction) .
The _________ of a real zero is the number of times its corresponding factor occurs. Example: The graph of y = 5x6−3x4+2x−9 has at most how many turning points?
Multiplicity If (x−r)^m is a factor of a polynomial f and (x−r)^(m+1) is not a factor of f, then r is called a real zero of multiplicity m of f. If f is a polynomial function of degree n, then the graph of f has at most n−1 turning points. The degree of the given polynomial function is 6, so its graph has at most 6−1=5 turning points.
List the basic transformations
Original Graph: y = f(x) Horizontal shift c units to the right: y = f(x - c) Horizontal shift c units to the left: y = f(x + c) Vertical shift c units downward: y = f(x) - c Vertical shift c units upward: y = f(x) + c Reflection (about the x-axis): -f(x) Reflection (about the y-axis): f(-x) Reflection (about the origin): -f(-x)
(1X3)(3X1) Matrix Multiplication (1X2)(2X1) Matrix Multiplication
Realize that the resulting matrices will be 1X1 matrices (The process is similar to a Dot Product, but one of the matrices is simply arranged differently). Therefore, it is sum the multiplied entries.
Trig Values for special angles
See image. These can also be used for the other three quadrants of the coordinate plane, but be heedful of differences in sign.
Euclidean Distance Formula
Sqrt((x2 - x1)^2 + (y2 - y1)^2))
Arithmetic Mean Example (inserting means and sequences)
Suppose you are given the following list: 2, 5, 8, 11, 14. Note that the common difference is 3. However, what if, for statistical purposes, say, you had to make quartiles with this data, but you knew only 2 and 11? That is, you are given the numbers 2 and 11 - how would you find the two terms in between? 11 - 2 = 9. Divide 9 / 3 = 3, the common difference. Then, simply add 3 to the first number three times. This gives: 2, 5, 8, 11, ...
Angles of depression and elevation
The angle of elevation is, for example, the angle in between you and the airplane you look upwardly to. The angle of depression is the angle between the aviator and what he/she looks downwardly to. The two angles form alternate interior angles, and are, therefore, congruent. It is important to note that, if it were represented by a triangle, the angle of depression would be outside of the triangle. The corresponding congruent angle of elevation would be inside the triangle.
Arcs and Sectors
The arc length of a circle with radius r is given by s= rθ where θ is the central angle that intercepts the arc (measured in radians). If measured in degrees, the arc length s = (θ/360)2πr The area of a sector of a circle with radius r is given by A = .5 θ r^2 where θ [or x] is the central angle that intercepts the arc (measured in radians). If measured in degrees, the area A = n/360 (π)r^2
Arithmetic Sequence
The arithmetic sequence is represented by the following: A[sub]n = A[sub]1 + k(n - 1) where k is the added constant and n is the number of the desired term in the sequence.
Even and Odd trigonometric functions
The cosine and secant functions are even. cos(-x) = cos(x) sec(-x) = sec(x) The sine, cosecant, tangent, and cotangent functions are odd. sin(-x) = -sin(x) tan(-x) = tan(x) csc(-x) = -csc(x) cot(-x) = -cot(x) Also, remember that even functions are symmetric with respect to the y-axis, and odd functions are symmetric with respect to the origin.
Cardinal and Ordinal Numbers
The counting numbers. Ordinal numbers tell you order ("First, Second, Third, Fourth") Cardinal numbers signify quantity. Ordinal numbers signify order or position. Math and writing are difficult to mix well, but if you remember that ordinal numbers represent order, you will be well on your way to writing coherently about different kinds of numbers. In summary, ordinal means order, while cardinal means quantity.
What is the domain of f◦g?
The domain of f◦g is the set of all numbers x in the domain of g such that g(x) is in the domain of f.
Geometric Sequence
The geometric sequence is represented by the following: A[sub]n = r^(n-1) where r is the multiplied constant and n is the number of the desired term in the sequence.
Evaluating Triangles and Trig Functions
The relationship between trig functions and their special angles can be seen by dividing an equilateral triangle (of sides 1-1-1) in half, thereby creating a 30-60-90 triangle. The unknown side can be found with the Pythagorean theorem. Using trig functions on the resulting triangle, with θ being the 30 or 60 degree angle, results in special angles. The 45 degree angle can be found by starting with an right isosceles triangle with lengths 1 and 2^.5, which can also be found with the Pythagorean theorem.
Form of a Circle
The standard form of a circle is represented by the following equation: (x-h)^2 + (y-k)^2 = r^2 where (h,k) represents the center of the circle. The General Form of a circle is represented by the equation x^2 + y^2 +b1x + b2y + c = 0 where the x's and y's are variables, b1 and b2 are real number coefficients, and c is a real number constant. We can change an equation given in general form to center radius form using the process of completing the square.
The Pythagorean Means
The three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM)
Trigonometric Functions
sin = y/r cos = x/r Quotient Identities tan = sin/cos = y/x where x != 0 cot = cos/sin = x/y where y != 0 Reciprocal Identities sec = 1/cos = r/x where x != 0 csc = 1/sin = r/y where y != 0 cot = 1/tan = cos/sin = x/y where y != 0
Law of Sines
sinA/a = sinB/b = sinC/c Use this when given 2 angles and any side, or use it when given 2 sides and an opposite angle. Note also the area formula that this can be paired with: A = .5abSinC = .5bcSinA = .5acSinB
Pythagorean Identities
sin^2x+cos^2x=1 1+tan^2x=sec^2x 1+cot^2x=csc^2x One can derive these from a unit circle with a given radius: cos(x)^2 + sin(x)^2 = 1
