Math Terms
tan∅ =tan(∅₂-∅₁)=(tan∅₂-tan∅₁)/1-tan∅₂(tan∅₁)= |(m2 - m1) / (1 + m2(m1))|
Angle between two lines
Differential Equations
Any equation with y' involved in it
An Ellipse is a set of points on a plane that the sum of the distance can be represented by two fixed points called the foci. (x-h)²/a² + (y-k)²/b² = 1 If having a horizontal ellipse your major axis is horizontal and if you have a vertical ellipse your major axis is vertical. The line through the foci intersects the ellipse at two points (vertices). The chord joining the vertices is the major axis, and its midpoint is the center of the ellipse. The chord perpendicular to the major axis at the center is the minor axis of the ellipse. a^2=b^2+c^2 a is the distance from center two vertex b is the distance from covertex to center and c is the distance from the center to the foci 2√b^2+c^2=2a^2 That is the combined distances of both the foci the horizontal major axis is (x-h)^2/a^2+(y-k)^2+b^2=1 the major vertical axis (x-h)^2/b^2+(y-k)^2/a^2
Ellipse
U-Substitution
Let g function whose range is an interval I and f be a function that is continuous on I. if g is differentiable on its domain F is an antiderivative of f on, then. when you take u equal to something u then derive that u version which then gets u a du/dx and then u solve for dx. Then plug everything back into the interval and then you will and then keep solving until you get your answer.
ln 6= x+ 1 e^x+1=6 take the base of the log and use the other side equation to the power of the base and move the other part of the log over
Log to exponential
A width which is one complete cycle or one complete pattern, period for sin and cos= 2pi/abs(b)
Period
is determining the number of ways that elements can be arranged (in order).
Permuation
nPr= n! / (n-r)! On a TI83 or 84, go to Math, PRB, nPr
Permutation Formula
The intersection point of the two curves
Solution of system of two equations
lim (x→0) sinx/x = 1 lim (x→0) (1-cosx)/x = 0
Special Trig Limits
Square root of a Times Square root of b^2
Square root of AB^2=
f''(x)>0 for a<x<b
The graph of f is concave upward on the interval a<x<b and the graph is decreasing f(x) is also increasing.
z=a+bi a=rcosθ b=(rsinθ) r=√a^2+b^2 z=rcosθ+(rsinθ)i tanθ=b/a
Trig form of a complex number
V= |u*(v X w)| |3 -5 1 | |0 2 -2 | | 3 1 1 | you find the cross product of the v X w and then multiply u to each of its corresponding numbers
Volume of a parallelepiped
is equal to 1/(b/a) (a/b)/c a/(b/c) (a/b)/(b/c)
a/b a/b *1/c ab/c ab/bc
tangent theta equals to slope when they give an equation then we can represent the tangent theta equals the slope
arctan(theta) equals m
A segment whose endpoints lie on a circle
chord
360 divided by n each and it has to be regular polygon.
exterior angle of a polygon
rewiriting U
if U are with a U and left in the term and, then you have to rewrite in it in order to get youself able to solve the problem the best example would be the problem for homework.
does the nth term approach work, If not the series diverges.
is the series one of those special types- geometric p series telescoping or alternating, can the integral test work, ratio test or root test, then lastly can the series be compared by a special type.
Look at the answers and see their dividing. Such as 9(6)/3 but if I have 5(7)/3 then we can get a uneven number which doesn't fit the answer
prime number theory
Simple Interest Formula
principle (the initial amount invested) × rate × time I=(prt) A=P+(Prt)
anything times infinity is infinity infinity divided any number is infinity any number divided by infinity is zero
properties of infinity
a number in the form bi where b is a real number and i is the imaginary unit
pure imaginary number
an imaginary number of the form a+bi where a is 0
pure imaginary number
hole in the graph, the graph stops and starts again at the same point - the discontinuity is "removed" by factoring and canceling the factor in numerator and denominator A point on the graph that is undefined or does not fit the rest of the graph. There's a "hole" at that location when you are looking at the graph
removable discontinuity
A triangle with no even sides.
scaline triangle
PSST rule
sec->sec<-tan
an angle is in this when its initial side lies along the x-axis and its endpoint is at the origin
standard position
Is the two opposite interior angles added up, these angles are also called remote angles.
sum of exterior angles
a technique for ascertaining the self-reported attitudes or behaviors of a particular group, usually by questioning a representative, random sample of the group
survey
solving expressions with trig and inverse trig functions
tan(arcsec(pi/3)) set arcsec(pi/3)=u then build a right triangle and then once that is done you solve for the missing side then it equals squrrt(x^2-9) and then once that is done you solve for tan u which is equal too squrrt(x^2-9)/3
A line that intersects a curve once and only once
tangent line
Sin x|Cos x
tangent x
inflection point
they have steepest slopes except for horizontal, and the vertical inflection point is the the steepest out of all of them
Odd Function - Integrals you want to look if the lower limit is -x and the upper limit is x then you want to test if the function is odd or even then it makes it really easy.
this is very useful on the ap exam
Displacement
to find the displacement I need to find the integral for the function and then I need to subtract the two.
end of integral
when you have a integral that matches your initial integral you can add them both to the left side and then divide it by two which gives you an answer don't forget +C.
implicit function
An implicit function is one that is given in terms of both dependent (y) and independent (x) variables If both x and y are on the same side, then it is implicit
points that lie on the same line You can use vectors to determine whether three points P, Q, and R are collinear (lie on the same line). The points are collinear if and only if the vectors PQ and PR are parallel.
Collinear
Is one example 100 to power of zero is one -50 to power of zero is one.
Anything to the power of zero
A=P(1+r/n)^nt
Compound Interest Formula
Pe^t(rt) or also Ce^(kt) it started off as A(b)^x then it changed from this formula to the formula above which is pe^(rt) then it evolved into the Ce(kt)
Compund Continuously p= initial value/principle=C e=2.17 r=interest rate=k t=time exponential growth and exponential decay. If r is less than zero it is a decay if r is greater it is a growth
r=√(x-h)²+(y-k)²+(z-j)²
Formula for a sphere
y=kxz
Joint variation
Can't be nothing
Math operations
nPr = n!/(n-r)!
Permutation Formula
initial condtions
You actually calculate for the c value in an integral
Angles in the same place on different lines
corresponding angles
angles that share a terminal side, and differ by a multiple of 360°; for example, 32° and 32°+360°=392° are coterminal
coterminal angles
∞/∞, .0/0, -∞/∞, 0*∞, 0⁰, ∞-∞, 1^∞, ∞⁰
indeterminate forms
an angle that measures exactly 180 degrees
straight angle
nth Taylor Polynomial
this one keeps c in the equation
derivative of ln(|u|)
u'/u
A pair of opposite congruent angles formed by intersecting lines
vertical angles
Conditional convergent
when the absolute value of the series diverges and then the series itself converges calls for a conditionally convergent.
horizontal inflection point
when the point has a horizontal change in an inflection point. The slope is always zero
particular conditions
when u find the c value and then replace the c value
x^a+b
x^a+x^b
When graphing (r,θ), easiest to graph by locating the angle θ first, then counting out the distance r from the origin.
(r,θ) = (r,θ + 2nπ) = (r,θ - 2nπ) = (-r, θ + π) = (-r, θ +(2n+1)π) theta is measured from the polar axis.
Extreme Value Theorem
If f is continuous on the closed interval [a,b] then it must have both a minimum and maximum on [a,b].
preservation of inequality
If f is integrate and nonnegative on the close interval a and b then 0 ≤∫(a,b)f(x)dx if f and g are integrtable and a and b are on a closed interval a and b then and f(x)≤g(x) on all values for g(x) then ∫(a,b)f(x)d(x)≤∫(a,b)g(x)dx
y=mx+b it's a straight line
Liner function
When it is equal to a negative number.
You may have no solution when the absolute value
using implicit differentiation to find higher order derrtiaves
always replace dy/dx
derivative notation
f'(x) dy/dx y' (d f(x))/dx d/dx f(x)
Definition of Derivative
f'(x) = lim(h→0) [f(x+h) - f(x)] /h
The Squeeze Theorem for a limit says that
if f(x) ≤ g(x) ≤ h(x), and lim (as x→c) of [f(x)] = L = lim (as x→c) of [h(x)], then lim (as x→c) of [g(x)] = L -It should be the last resort because it is hard to find the two other function being squeezed inside the graphs. If graphical approach, left hand right hand limir approach and then, algebra fails then we have to go to the squeeze therom.
The only way you can add them is if they are in the same row and column, and it also has to have the same number of rows and column.
matrices addition
(x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2
midpoint formula in space
(n-2)180 one interior divide by n and has to be a regular polygon.
sum of interior angles of a polygon
Each triangle has three altitudes on a right triangle two of the altitudes overlap with the two sides of the triangle. On a obtuse triangle two of the altitudes are outside of the triangle .
triangle altitudes
trig identities in integral
trig identities in integral is is common and will happen therforen using them in integrals when all else fails is key.
Remainder for Legrange error bound
we are essentially just trying to do |f(x)-p(x)| which equals our error. our R(X) function is really just our next term which in this case would be (f^(n+1)(Z)/(n+1)!)(x-c)^n+1 |Rn(x)| <. |x-c|^n+max|f^n+1(z)/(n-1)! The Z value exists between x and c
moving objects speeding up or slowing down
we call an object speeding up when velocity and acceleration are the same sign, meaning they both positive or both negative, if their are opposite signs it means they are slowing down.
when rewriting term in a series
we can take out the negative sign in a number and make it in terms of another... for example: 2(-x)^n we can rewrite this one as a term of 2(-1)^n(x)^n This allows for easier factoring
Definite Integral
x₀=a+∆xi definite integral calculates the area under the curve between a and b only if f(x) is non-negative between a and b. and continuous If any regions are hostile between a and b it will give you a negative number, and the sum of all these regions will give you the sum of the region of this place, then it will be harmful if the region above the x-axis bigger then its positive.
DNE cases
y approaches a different number from left hand right hand approach y increases without a bound as x approaches c f(x) oscillates between two fixed values
Compound continuously
y=Pe∧rt
Euler's method formula
you want to add to the pervious y value compared to the x value you are just adding the step size. Another thing is that the step size is equivalent to deltaX and the whole equation is built on the slope formula.
calculating area under the curve using rectangles
∆x=(b-a)/n a is the start point of the interval b is the end point of the interval n is equal to the number of rectangles
p-series
∑(1/n^p) converges if p>1 Diverges if p≤1
is equal to 1
0!
Polynomial Long Division
A way to divide polynomials keeping the variable in place expressing p(x)/d(x) = q(x) + r(x)/d(x) q(x) = quotient r(x) = remainder d(x) = denominator
Parametric form for derivative
If a smooth curve C is given by the equations y=f(t) and y=g(t) then the slope at C at (x,y) is ( look at the image)
a line that intersects two or more lines
Transversal lines
standard deviation squared=σ squared
Variance
put the pencil in a vertical way and see if it hits your graph more than one time if it does then it's not a function. It's a relation.
Vertical line test
Endpoints of major axis
Vertices of ellipse
algebra rewrite how to turn a unacetable to acceptable form
a *b = a/b^-1
Limit problems part and the reason it is written twice is due to memorization
a y value that a graph approaches as x approaches a certain value. If there has a hole at that location, limit gives you the y value of the hole. lim->f(x)= f(c) table approach graph approach algebra if your graph is osciliating at 0,c then it is DNE
an alternating sequence with alternating signs (-1)^n to generate an alternating sequence we use (-1)^n (would give us alternating), (-1)^n+1 or (-1)n-1( these both would give only negative numbers) n could start with n,0 or any number
alternating sequence
integral proof
area = wh=deltaxh=(b-a/n)h a = (b-a/n)hf(x)
a coefficient matrix with an extra column containing the constant terms
augmented matrix
difference of cubes formula
a³-b³=(a-b)(a²+ab+b²)
differentiation : calculate slope using subtraction and division integration : calculate area or volume by addition
calculus topics
an angle whose vertex is the center of the circle
central angle
logaN = logbN/logbA
change of base formula
The ratio of the number of times an event occurs to the total number of trials, or times that the activity is performed. P(e) = number of times even occurs/number of times the activity is performed
experimental probability
e^3x =4 ln 4=3x
exponential to logarithms
when you have a function that does not have a variable for the second there for there us a process
first do u substitution on the variable and then the goal is to then then plug u in for your integral Once that is done u will then multiply your answer times the derrative of u
changing variable
first you do the regular u-substitution steps in order. Then once you are done with that, you then want to apply this re-write the original u substitution equation which will then allow you to to plug it back into the integral and then solve the problem.
y=arcSin(0.7)= sin y =0.7 x=arccos(0.5)=cos x = 0.5
how to rewrite inverse trig functions
Integral Properties
if f is defined at x=a, then∫(a,a)f(x)d(x)=0 If f is integrate on [a,b] then ∫(a,b)f(x)dx=-∫(b,a)f(x)dx
An infinite sequence is a function whose domain is the set of positive integers. The function values long story short it is never ending
infinite sequence
A side which is on the positive side of a graph and starts the angle of rotation
initial side
integral of 1/x
lnx
events that have one or more outcomes in common
overlapping events
All rational and irrational numbers
real numbers
speed formula
s=d/t
x^1/2
square root of x
a computed measure of how much scores vary around the mean score SD=σ=Sqrt(∑∣x−μ∣2/n)
standard deviation
(y-k) = a(x-h)² (y-k) = 4p(x-h)² a=4p
standard form of the equation of a parabola
0,1,2,3...
whole numbers
1, 2, 3, 4, 5, 6, 7, 8, 9, etc. Whole numbers that are not negative.
Natural Numbers (Counting Numbers)
x=-b/2a
X-coordinate of the vertex of a parabola
Alternating Series Test
converges if: a) decreasing b) lim∞ U(k) = 0 (pass divergence test) a(n+1) <( or equal) for all n only comparing absolute value for this test and no other test and when we compare them
2n! = 2(n!) =2(1×2×3×4...n) whereas (2n)!= 1x2x3x4... 2n (n+1)!/n! = n+1
Factorial properties
45° is equal to π/4 90°= π/2 Cos(x)= 0, x=π/2,3π/2 Tan(x)=1 or cot(x)=1, x= π/4,5π/4
Facts in trigonometry
a circle of radius 1 whose center is at the origin
Unit Circle
a circle of radius 1unit whose center is at the origin
Unit Circle
using completing the square to rewrite my integrand.
When I have a function in a square root and I want to get rid of the square root then the way I would use it is through completing the square which then allows me to get a clean integral which I then can solve for.
a triangle with 3 congruent sides
equalateral triangle
A function with a graph that is symmetric with respect to the y-axis. f(-x) = f(x).
even function
A outcome or set of outcomes
event
calculate any term in a sequence directly, an terms doesn't rely on previous terms. an = a1 + d(n - 1)
explicit formula
properties of functions defined by power series
f(x) =n=0∑∞(An(x-c))^n f'(x) =n=1∑∞ nAn(x-c)^n-1 ∫f(x)dx=C+n=1∑∞an (x-c)^n+1/(n+1)
is represented by an exclamation point, you calculate a factorial by finding the product of (multiplying) a whole number and all the whole numbers less than it down to 1, so 6 factorial is 6×5×4×3×2×1
factorial
is represented by an exclamation point, you calculate a factorial by finding the product of (multiplying) a whole number and all the whole numbers less than it down to 1, so 6 factorial is 6×5×4×3×2×1 0! = 1 special case
factorial
using distributive property to factor some polynomials having 4 or more terms 3x^2+6x+4x+8 (3x^2+6x)+(4x+8) 3x(x+2)+4(x+2) (x+2)(3x+4)
factoring by grouping
outcomes in a specified event
favorable outcomes
finding line tangent to a graph
find the two equations and set them equal to each other, then once that is done take derivative and then and plug the equation into each other and then after that you will get a value which can then be used to find graph answere
An=Am(r)^n-m
finding common ratio of a geometric sequence of two terms
an = a₁rⁿ⁻¹ an=nth term a₁=1st term r=common ratio n=nth term
geometric sequence
a logarithm with base e
natural logarithm
an exponent less than zero which causes the base and its exponent to move positions in a fraction
negative exponent
n(n-1)(n-2)...3(2)(1)=n!
number of permutations of n elements
An ordered list of numbers with either a constant difference (arithmetic) or a constant ratio (geometric).
sequence
1,3,5,7 an=2n-1
sequence of odd numbers
Sum of terms in a sequence
series
y=mx+b do not use slope intercept from unless the intercept is given.
slope-intercept form
particular solution
the particular solution is a value for the C and once the correct value for C to find the value of C the initial condition value must be known
general solution
the solution to a differential equation that includes the constant of integration c
Geometric series
the sum of a geometric sequence. Then once we are done we can write the series and then look at the standard ratio of R. The absolute value of R is the essential thing in this case. If it is between then the series converges and then after that. if it is finite then the sum will equal a(1-r^n)/(1-r) and then the reason is because
2019 question number 2
when you look into the integral when k approaches infinity and then the region approaches is inside the circle which allows us to integrate from 0 to pi/2
average in ap calc frqs
whenever it says the word average thats when you must mention the average value formula.
inflection point
where a function chan ges concavity (a function may have an inflection point where the second derivative is 0)
f(x) when solving area under curve and its a simple question
x^2+Y^2=R^2 y^2=(√p-x²)
Power Series
∑cⁿxⁿ=c₀+c₁x+c₂x²+c₃x³+... where cⁿ=fⁿ(a)/k! It goes to infinite number of terms whereas polynomials have a finite number of terms. The Power series can be centered at C.
(X-h)2/a2 +(y-k)/b2=1 h+a,k h-a,k h+c,k h-c,k h,k+b h,k-b focus should be in major axis and vertices should be on major axis. Co-certifies should be on mile axis
Major Axis equation
rule for exponents
any positive number raised to the negative or positive exponent would mean that it is always positive.
arc sin,arc cos,arc tan
arc sin and arc tan are both [-π/2,π/2] and then for arc cos its [0,π] specifically the range of the problem
inverse trig function properties
arcsin(sin(x))=x sin(arcsin(x))=x arctan(tan(x))=x tan(arctan(x))=x arcsec(sec(x))=x sec(arcsec(x))=x similar properties hold for the other arcsec(x)=arccos(1/x)
common sequences
even terms( 2^n) odd terms(2n-1) n^2, 1,4,9,16,25,36.. n^3 1,8,27,64.125,216 3^n 3,,9,27,81,243, 729, 2187 n! 1,2,6,24,120... 1/2,1/4,1/8, 1/16 (1/2^n) 1/2,1/4,1/8,1/15 (6/(n+1)(n^2-n+6) 1/2,1/4,1/8,7/62 (n^2-3n+3)/(9n^2-25n+18) 1/2,1/4,1/8,0 -n(n+1)(n-4)/6(n^2+3n-2) the first two sequences actually converge to zero then the third sequence converges to 1/9 and then the last one diverges. when you see something you can't recognize, example 3,10,29,66,127,218 x^3+2 is the sequence for this function and this is showing the alternation.
Difference of Squares Formula
a^2-b^2=(a-b)(a+b)
f(x) = {x, x≥ 0 {-x, < 0 when you work on limit functions that have absolute value simolfy before you try to find limit
absolute value for piecewise function
definite integral
an integral with specific limits, whose solution does not include an undetermined constant C, it will be a number.
indefinite integral
an integral without any specified limits, whose solution includes an undetermined constant C, or known as the anti derive, it includes plus c because its a family of anti derives.
find min/max
in general, if the slope is zero it could be min or max (If it's a horizontal inflection point it is not a min or max also when the slope is zero). if the slope is undefined then there are two cases. If the slope is dis continuous then it is undefined or if it has an undefined tangent line.
Absolute value theorem for sequence
Lim
Over lap P(A) + P(B) - P(A and B) disjoint P(A) +P(B)
P (A or B)
√a x √b
√(a×b),
Not possible And same for subtraction
√(a₊b)
√a/b
√a/b
indeterminate forms
∞/∞, .0/0, -∞/∞, 0*∞, 0⁰, ∞-∞, 1^∞, ∞⁰
properties of natural log
(0,∞) and the range is (-∞.∞) execpt for when its an x^positive then its will get a domain that is (-in,0)u(0,in) the function is continuous, increasing and one to one the graph is concave downward. doesnt matter what base they will always intersect at the x axis at (1,0) all exponential functions have a y intercept at 1F
Any area for any equilateral triangle equal to
(s^2√3)/4
x^ab
(x^a)^b
If it passes both vertical and horizontal line tests.For each x value you can only have one y value, for each y value you can only have one x value.
1-1 function
algebraic approach to find limit
1. If f(x) is a rational function then you try to use factoring to rewrite numerator and denominator then use direct subsitiusion to find the answer if your f(x) with radicals in it, multply conjugate to numerator and denominator. NEVER expand denominator
Derivative of lnx (natural log) Derivavtive of a log not natural base
1/x d/dx(logₙx)=1/x*ln(n) d/dx(logₙ(u))= (1/u*ln(n))du/dx derivative of of something like this y=2^x ln(y)=ln(2^x) 1/y(y')=ln(2) move y y'=ln(2)*y y'=ln(2)*2^x
Answer for limit
If the answer for a limit is positive or negative infinity, it must be written as Does not exist
integration stratigies
1. first, try u substitution 2. if it fails then check if it is any of the trig functions 3. then the inverse trig functions don't apply then try a partial fraction 4. then try integration by part. done for now
methods to find limits
1. table approach 2. graphing approach 3. algebraic approach
Limit Comparison Test
1.) if lim(n→∞) an/cn = 0 and cn converges, so does an 2.) if lim(n→∞) an/cn = ∞ and cn diverges, so does an 3.) if lim(n→∞) an/cn = any other real number, then both converge or they both diverge
Gravity on Earth
9.8 m/s² 32 ft/s²
Polynomial
A sum of terms/ monomials (1 or more terms) exponents have too be whole numbers and non zero that is not negative and not fractional and also no trig functions.
Arithmetic Sequence Formula
A(n)= a+(n-1)d A= ... sequence n= number in sequence d= common difference
area in polar coordinates
A=1/2∫r²,θ,α,β if it has a loop, please plug in test value between the two boundaries to find out if it is from a to b or b to a for each loop
Area of a sector
A=½r²θ how to derive an equation from scratch (A/θ)=(πr²/2π) A=1/2r²θ
Q4 Cosine gives positive , secant, Q2 sine gives positive , cosine, Q3 Tangent , cotangent all in quadrant 1
All Students Take Calculus
If two lines are cut by a transversal so that alternate exterior angles are congruent, then the lines are parallel.
Alternate Exterior Angles Converse Theoremint
FIf two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
Alternate Interior Angles Theorem
Is the half of the height of your graph.
Amplitude
Because the absolute value of any number - negative 0 or positive zero
An absolute value equation might have one equation that has no solution
explicit function
An explicit function is one that is given in terms of the independent (x) variable if x is the only variable on one side.
An equation with one or more terms with a variable in the exponent position it it has a base which cane he raised to a power that involves a variable
An exponential equation
1/2acSin b 1;2bcSina 1/2abSin c
Area of a traingle
A(n)= a1+(n-1)d A= ... sequence n= number in sequence d= common difference
Arithmetic Sequence Formula(explicit formula)
second fundamental theorem of calculus
Assume that f(x) is continuous on an open interval I containing a. Then the area function: A(x)=∫f(t)dt is an antiderivative of f(x) on I; that is, A'(x) = f(x). Equivalently, (d/dx)∫f(t)dt=f(x) *x is the upper limit for all of this* Conditions for this rule: f(t) is a derivative of an integral Derivative matches upper limit of integration Lower limit of integration is a constant. The second theory tells us derivatives and integral cancel each other out. Then it explains why t becomes x. d/dx(F(x)-F(a) d/dx(F(x))-d/dx(F(a)) f(x)-0 f(x) it is telling us that derivatives and integrals are always bound to cancel out.
1/b^a
B^-a
An expression with two terms
Binomial
cost, revenue and profit functions
C(x)= cost function R(x) = revenue function P(x) = profit functions P(x) = R(x)-C(x)
The measure of the inscribed angle (∠ABC) is always half the measure of the central angle (∠AOC) when both angles intercept the same arc AC.
Central Angle Theorem
The set of all points that are equidistant from a point called its center and the same distance from the radius.
Circle
The endpoints of the minor axis.
Co-vertices
A number in front of a variable term
Coefficient
sinx=cos(π/2-x)=cos(90-x) cosx=sin(π/2-x)=sin(90-x) tanx=cot(π/2-x)=cot(90-x) cotx=tan(π/2-x)tan(90-x) secx=csc(π/2-x)=csc(90-x) cscx=sec(π/2-x)=sec(90-x)
Cofunction Identities
is a new function formed by two fipunctiojs working in sequence.
Composite function
(f(x+h)-f(x))/h It came from the slope formula m=(y2-y1)/(x2-x1) h=x2-x1
Difference Quotient
divide by 100 5% = 5/100 = 0.05, x% = x/100
Convert percent to decimal
adjacent ÷ hypotenuse (CAH)
Cos x
A trigonometric ratio that is the reciprocal of the sine 1 / sin hyp/opp
Cosecant
X over r on a terminal side
Cosine
cotangent equals 1 divided by tangent or cosine divided by sine. 1 / tan adj/opp
Cotangent
Angles which have the same starting point and the same terminal side.
Coterminal angles
Implicit Differentiation
Differentiating both sides of the equation with respect to x and then solving the resulting equation for y' use implicit differentiation on implicit fun
1.Write a function rule function with f(x) with a y 2. Swap the x and y do so carefully especially if there is more than one apprentice of x. Each x must be replaced and there should only be one y in the original which will become x 3. Isolate y on one side of the equation leaving only x terms and constants on the other side.
Creating inverse functions
Many applications in physics, engineering, and geometry involve finding a vector in space that is orthogonal to two given vectors. In this section, you will study a product that yields such a vector. It is called the cross product, and it is conveniently defined and calculated using the standard unit vector notation. u *v = (u2v3-u3v2)i-(u1v3-u3v1)j-(u1v2-u2v1)k v*v =0 when you cross the same product
Cross Products of vectors
y=Ax^3+Bx^2+Cx+D, a third degree polynomial
Cubic equation
n!/n₁!xn₂!×.....×nκ! n=n1+n2+n3...nk n1=the number of indistinguishable objects n2=the number of indistinguishable objects of the second type use distinguishable formula if you have objects that are
Distinguishable Permutations
Only if your numerator equals zero
Division can equal zero
All the possible inputs for X.All functions have all real domain except for rational function and functions with even root.
Domain
finding domain of composite function. The domain of the inner function are also in the outer function
Domain of compsite function
X times probability of x then add all the products. X1p(x1)+X2p(x2)+Xnp(xn)=Σ(i=1,n)
Expected value
X2 /a2+Y2/b2=1 Y2/a2 +x2/b2 =1 (a,0)(-a,0) flip for vertical (C,0)(-c,0) (b,0) (-b,0)
Equation Vertices Foci Co-vertices.
They are undefined with negative number
Even root
sin (-u) = -sin u cos (-u) = cos u tan (-u) = -tan u csc (-u) = -csc u sec (-u) = sec u cot (-u) = -cot u
Even/Odd Identities
right hand and left hand limits with infity
For one sided limits you can use negative infinity and positive infinity where as with two sideded limits it is a DNE case.
Is a relations which has only one x values it will have only one y value.
Function
if f(x) is a polynomial of degree n, where n>0, then f(x) is zero then I have n solutions.
Fundamental Theorem of Algebra
(M1)(M2)(Mn)
Fundamental continuing principle
Ax+By+C=0
General equation for a line
Ax²+Bxy+Cy²+Dx+Ey+F=0
General equation for conic sections
Used as a new way to measure angles in terms of pi. Radian is just another rumor to measure angles like degrees 180 degrees = one pi radian 360 degrees = two pi radian
Radian
Multiply the number of radians 180 degrees over pi radian
Radians to degrees
H=-1/2gt^2h0 G=acceleration over gravity or 32 feet per second squared h0 initial height. h= the height of the object of the ground from different heights. g=gravity
Height function of Falling object
d=-1/2gt^2 + Vot + Ho G=acceleration over gravity or 32 feet per second squared Ho = initial height. h= the height of the object of the ground from different heights. Vo equals initial velocity.
Height function of object thrown
√s(s-a)(s-b)(s-c)
Heron's Formula
A horizontal line that the curve of a graph approaches but never reaches A horizontal line that the graph of the function approaches as x→ -∞ or x →∞ . Limit of f at infinity(check behavior model)
Horizontal Asymptote
(y-k)²=4p(x-h) p>0 (y-k)²=4p(x-h) p<0
Horizontal Parabola to the right and to the left
You put your pencil horizontally and if it crosses more than one point function has an inverse function.
Horizontal line test
All the possible inputs for Y
Range
Mean Value Theorem
If f is continuous on [a,b] and differentiable on (a,b), then ∃c∈(a,b) such that f'(c) = [f(b)-f(a)]/(b-a). The slope of a secant line connecting two endpoints which equals the mean rate of change /slopes on all the slopes on that interval. MVT is the average slope between the closed interval and that means there is an average rate of change if you find the average velocity within that interval, then we find the average rate of change of the stock price. If it is positive had a gain in that interval, if it is negative then is no gain.
done
If it comes earlier, put it as u. Logarithmic, inverse trig, algebraic, trig, exponential
it means the angle mixed with the x-axis m=tan θ to find the angle of inclination this is the formula if m ≥0 then ∅=arctan(m) because 0≤∅≤π/2 on the other hand if m<0 then you have to add pi to arctan(m) because π/2≤∅≤π
Inclination of a line
y=k/x
Indirect/ inverse Variation
All whole numbers (both positive and negative) and zero.
Integer
finding distance traveled in parametric
Integrate velocity and you get distance traveled! • Arc Length (or distance traveled) determined by: ∫(a,b) √((dx/dt)²+(dy/dt)²)dt
When a function is being solved it should always be equal to X
Inverse function test
Inverse sprunctions have a line of symmetry y=x and they are mirror images of each other. If a coordinator has a point x,y then the corresponding point is y,x. For a function to have an inverse it must have a 1-1 which it must pass the vertical line test.
Inverse functions
Log is equal to inverse of log
Inverse of log is equal to log
a²=b²+c²-2bcCosA b²=a²+c²-2acCosB c²=a²+b²-2abCosC
Law of Cosines
C=cos^-1(a^2+b^2-c^2 dvided by 2ab)
Law of Cosines derrivstive
sinA/a=sinB/b=sinC/c
Law of Sines
Is the coefficient of the variable term with the highest power( remember negative).
Leading coefficient
Limit test for sequences
Let L be a real number. L written as if the answer is DNE is diverges and if it converges it is a finete number
limits of the lower and upper sums
Let f be continuous and non negative on the interval a and b, the limits as n approaches infinity of noh lower wand upper sums exist and are equal to each other. where ∆x= b-a .n and the minium and maxium vakyes on f aare sub interval
There are 8 Paris which are touching pairs.
Liner pairs converse Theron
local min and max
Local min and max is a critical point and that is where the slope is either zero or undefined and those points A function f has a local maximum at cc if there exists an open interval II containing cc such that II is contained in the domain of f and f(c)≥f(x)f(c)≥f(x) for all x∈I.x∈I. A function fhas a local minimum at cc if there exists an open interval II containing cc such that II is contained in the domain of f and f(c)≤f(x)f(c)≤f(x) for all x∈I.. A function f has a local extremum at cc if f has a local maximum at c or f has a local minimum at c.
Performed by multiplying the elements of each row of the first matrix by the elements of each column of the second matrix. Add the products. If 3x3 and 3x3: Multiply row 1 of matrix A by column 1 of matrix B, row 1 by column 2, row 1 by column 3. Congrats! You got your first row. Repeat with rows 2,3. This is [AB]. Matrix multiplication are not commutative. # columns (n) in first matrix = # rows (m) in second matrix AB ≠BA
Matrix Multiplication
An expression with one term
Monomial
1/t1+1/t2=1/t t1 equals to time it will take first person to complete the job alone, t2 is time for second person to the job alone and then t is time if two people doing it together. This formula can be expanded to use for more than two person.
Multiple people doing one job
sin(-x)=-sinx cos(-x)=cosx tan(-x)=-tanx
Negative Angle Identities
The transformations that are not rigid are the stretches and compression. These can also be horizontal or vertical.
Nonrigid transformations
nth term test for divergence
Nth term tests only for divergence is only going to tell us when the limit is going to diverge and then as a result neglecting the other purpose
The organizational structure used to categorize numbers into complex numbers, imaginary numbers, real numbers, rational numbers, irrational numbers, integers, whole numbers, and natural numbers.
Number System
A function with a graph that is symmetrical with respect to the origin; f(-x) = - f(x)
Odd function
Can take all real numbers
Odd root
when we are given a series and then we should have the first ones memorized because when we do that it gives freedom to write our expression
Once that is done we can then solve for our expression
original price times (100% +p%) = new price After 20 percent increase new price equals 78 dollars The original price equals x(100%+20%)=78$ x(1.2)=78$ x=78 divided 1.2 x= 65
Original price after percent increase
A possible result of a experiment
Outcome
Parametric Equations and Vector Valued Functions
Parametric equations x(t) and y(t) can also be represented as vector r(t)=<x(t),y(t)> r(t) represented as a function of time veloctiy is then v(t)=r'(t)=<x'(t),y'(t)> Then to find speed Speed of the particle is the magnitude of the velocity vector || v(t) || = || r'(t) || = √((dx/dt)²+(dy/dt)²)
Has no y intercept and x intercept is 1
Parent function of log
Parent functions refers to the simplest member of a family
Parent functions
Percent increase = ( (new amount - original amount) ÷ original amount ) × 100%
Percent increase
3-4-5 5-12-13 7-24-25 8-15-17 any multiple of these triple also fit in to these categories
Phythagoream triple
a function represented by a combination of equations each corresponding to a part of the domain. The individual rules can linear of non-linear functions, or constant functions.
Piecewise Functions
Taylor polynomial approximation
Pn(x)= a₀(x-c)⁰+a₁(x-c)¹+a₂(x-c)²+a₃(x-c)³......an(-c)^x Pn(x)=f(c)(x-c)⁰/0!+f'(c)(x-c)¹/1!+f''(c)(x-c)²/2!+.... +f'(n)(x-c)^n/n! The polynomial has to go through (c,f(c)). That means the polynomial is centered at C or expanded at C. Then it had to fulfill the second requirement p prime of f'(c) has to fulfill f'(c) prime of.
Polynomial Approximation
Pn(x)= xa₀⁰+a₁(x)¹+a₂(x)²+a₃(x)³......an(x)^x
(r,θ) x=rcos(θ) y=rsin(θ) tan(θ)=x/y r²=x²+y² pole is the origin on a graph of polar coordinates r is the distance from the point to the pole theta is the angle in-between the polar axis and the point. polar coordinate is is what the points are when you graph on a polar coodinate
Polar Coordinates
likelihood that a particular event will occur, the value has to be from zero to one or 0% to 100%
Probability
Probability = (#of desired outcomes) / (#of total possible outcomes)
Probability Formula
Product Property: LOGb(M*N) = LOGb(M) + LOGb(N) Quotient Property: LOGb(M/N) = LOGb(M) - LOGb(N) Power Property: LOGb(M^N) = N * [LOGb(M)] [p. 462]
Properties of Logarithms
sin²θ + cos²θ = 1 tan²θ + 1 = sec²θ 1 + cot²θ = csc²θ
Pythagorean Identities
x = -b ± √(b² - 4ac)/2a
Quadratic Formula
a function written as a fraction with a variable in the denominator( The first five letters is ratio).
Rational function
Any number that can be expressed as a ratio of two integers
Rational number
sinθ = 1/cscθ ; cscθ = 1/sinθ cosθ = 1/secθ ; secθ = 1/cosθ tanθ = 1/cotθ ; cotθ = 1/tanθ
Reciprocal Identities
Look at term.
Reference angle cannot be negative, angle between the nearest side and closet x axis
Pairing 2 sets of numbers from another set.All values that can be put in are the domain and the outputs are the range.
Relations
If we have a repeating decimal then we should just divide the repeating numbers by the digit their asking and we should get the answer.
Repeating decimals problems
Slope of the line tangent to the curve at that point
Slope of a curve at a point
that the shape meaning that the shape of the graph will not change is just changes locations. The rigid transformations are translations or slide and reflections or flip.
Rigid transformations
1. All nonzero rows are above any rows of all zeros. 2. Each leading entry of a row is in a column to the right of the leading entry of the row above it. 3. All entries in a column below a leading entry are zeros.
Row Echelon Form
two exterior angles on the same side of the transversal
Same side exterior angles
Scalar multiplication of matrices involves multiplying each element in a matrix by the same value.
Scalar Matrix Multiplication
Secant equals 1 divided by cosine. 1 / cos hyp/adj
Secant
the term aₙ is defined using the previous terms for example aₙ-₁ or aₙ-₂ sometimes you see at least one of the terms aₙ=aₙ-₁+d
Sequence Recursive formula
a list of numbers, there are many types of sequences arithmetic, geometric and etc. a₁= first term, a₂= second term, etc. first term could be called a1 or a0 it does not matter
Sequences
Y over r on a terminal side
Sin
a Sin(bx-c)+d a Cos(bx-c)+d
Sin and Cos formula
opposite ÷ hypotenuse (SOH)
Sin x
sin(α+β)=sinαcosβ+cosαsinβ sin(α-β)=sinαcosβ-cosαsinβ cos(α+β)=cosαcosβ-sinαsinβ cos(α-β)=cosαcosβ+sinαsinβ tan(α+β)=(tanα+tanβ)/(1-tanαtanβ) tan(α-β)=(tanα-tanβ)/(1+tanαtanβ)
Sum and Difference Formulas
Sn = a₁(1-rⁿ) / 1-r
Sum of a finite geometric sequence
Sum of all probabilities is equal to one.
Sum of all probability is equal to 1
Sn=n/2(a₁+an)
Sum of arithmetic sequence
S = a₁ / (1-r)
Sum of infinite geometric sequence
360
Sum of rectangle
This side indicated that the angle is done going any further.
Terminal side
Use the origin. (0,0) as a test point only when your graphs go through the origin.
Testing inequality graphs
even function integral
The 2 at the end is the distance doubled because you are getting rid of the negative side
Net Change Theorem
The definite integral of the rate of change of the quantity of F'(x) gives the total change or the net change in that quantity on the interval [a.b] Its because if the function has a region below the x axis the area is negative , but if you don't think of it then you will get the total.
all of the outcomes NOT in an event, all the outcomes of an sample space that is not an event.
The compliment of an event
let f be a function and c and L are real numbers. The limit f(x) as x approaches c is L if and only if the left limit c approaches and the right limit c approaches are the same value.If the left limit equals ±∞ and the right limit equals ±∞ then the limit does not exist(DNE)
The existence of a limit
2n+1,2n-1, 2n ,2n+2
The expression to produce odd integers, 2n+1 starts with 1, and 2n-1 starts with negative 1, 2n is how to find even numbers staring from1 and 2n+2 is how to find even numbers starting from zero
f''(x)<0 for a<x<b
The graph of f is concave downward on the interval a<x<b and the graph is decreasing f(x) is also decreasing
logistic model k is the rate of change of the function, whereas L is the carrying capacity in our case with these two combined then we will get our total function the general solution is cket another one is l/1+be-kt
The logistic model is a better model as it represents population of animals and other humans more accurately dy/dt=ky(1-(y/L)) then.. y=L/1+be^(-kt)
a1=a1 a2=a2r a3=a2r^2
The nth term of a geometric seqeunce
nPr= n! / (n-r)!=n(n-1)(n-2)...(n-r+1)
The number of permutations of n objects taken r at a time
P(not E) = 1 - P(E)
The probability of complement
the rational zero test
The rational zero test shows how many rational zeros you have in your multiple polynomial equation. The way you figure it out is you put the highest polynomial over the constant in the last term. Once you do that you then use synthetic division to test if the number is a valid once we do that then we will get a polynomial factor and use those roots we produced from the synthetic division. then we then facotor thoseF roots and out and get the zeros from there on out.
Partial Fractions
The technique by which a fraction with a complicated denominator can be expressed as a sum of simpler actions whose denominators are factors of the original denominator n(x)/x^5+x^4-x-1 =n(x)/(x-1)(x+1)^2(x^2+1)=A/(x-1) +B/(x+1)+C/(x+1)^2+(Dx+e)/(x^2+1)
case 1=f(x) is continuous the limit as x approaches c is equal f(c) case 2 = if f(x) has a hole at c then the limit as x approaches c is the y value of the hole. f(c) is undefined case 3 = if f(x) has a jump it may or may not be defined at c then the limit is DNE case 4 = if f(x) has a vertical asymptote at c then f(c) = undefined and the limit is DNE
The use of limits
If the position function is given as a parametric then x(t) and then y(t) .
Then, to find the given velocity, we must derive it and then, it will be converted into the function after, after this, we cannot write our answer as a vector, what we must do instead is they should write as a derivative value and vice
What should occur or what we expect to happen in an experiment p(event)= number of favorable outcomes/Number of outcomes in sample space (aka sample size)
Theoretical Probability
absolute min or max
There is a possibility that there can be more than one min or max in your function.
sum of squares
There is no formula (a²+b²) is not factorable
integral of trig functions
This allows you to set u equal to the denominator, this allows to get a really goof integral.
Geometric series for radius of convergence
This only works id you power series can be re-written as a geometric series summation: (x-1)^n/3^n This does not loom geometric, however we can rewrote to become geometric where we set it in between the bounds of -1 and 1 which then we can solve for out radius.
Telescoping Series
This test cannot be used to show divergence first use partial fraction to decompose your fraction into two fractions. if there is no fraction then that one is wrong. once you decomposed expand the telescope and then once that is done, you basically replace the summation sign and then replace with the collasption. The last term is the second fraction then you apply limit to it. Then every single one
Are changes made to a parent function to translate or slide it reflect it and stretch or compress it.
Transformations
a(√1/b(x-h))+k
Transformations square root function a=Vertical stretch or compression factor b=horizontal stretch or compression factor h= horizontal translation K=Vertical translation
If you can tell that f(x) is a common shape such as circles, trapeziods
Using elementary school math to solve some problmes if equation is recognized
Two vectors are equal if and only if their corresponding components are equal. The magnitude (or length) of is √u₁²+u₂²+u₃² A unit vector u in the direction of is u= v/||v|| The sum for vectors is adding the corresponding terms scalar multiple is when you multiply the vector times the scalar the dot product is when you multiply the corresponding terms and then add them all up corresponding is v1 = u1 so you multiply them. convert from line way into component form it has to be terminal - initial the angle between two vectors cosθ=u×v/||u|| ||v|| If two vectors that are non-zeros have a dot product that is zero then the angle between them is 90° Recall from the definition of scalar multiplication that positive scalar multiples of a nonzero vector have the same direction, whereas negative multiples have the direction opposite. In general, two nonzero vectors are parallel when there is some scalar such that. For example, in Figure 11.12, the vectors , and are parallel because and.
Vector Operations
You have to add plus or minus
When ever your square root on the' side
Never,NEVER square root the left side if it has multiple terms while trying to solve the equation
When to square root 2 sides of an equation
absolute value inequalities
When using absolute values we use to change sign inside the number then it will never change |x-2| <1 when we rewrite this function, we will get... x-2<1 -(x-2)<1 1<x<3 as our answer
X will equal +- square root of a
When x squared to a
Absolute Convergence Test
When you absolute value a whole series and it converges then you will know that there will be an
integrals with trig odd power functions
When you have an odd power trig function, you want to split the function up and make it a even power function and then once that is completed you can re integrate it.
relationship between a function and a sequence
While they may have the same expression, you will get the same result for every positive integer. However, with a function, you will get specific dots with a line whereas with a sequence you will get points that do not have slop and they also don't have s area under the curve.
(r,θ) there are two formulas to find to find many representation of the point. (r,θ±2π) (-r,θ±(2n+1)π)
multiple representations of polar points
graphing of f prime and f double prime
always line up your y-axis and always draw your f prime in the middle and f double prime on the bottom draw dotted lines at relative mins and max and horizontal inflection points. The min and max of the f prime mean you should have inflection points at your x and normal inflection points. where your min max and your f are where the inflection points on your f prime but you want to use common logic for solving the f graph graphing from f to f prime is finding all the peaks and also the bottoms of the f and marking them on your f prime as your x-intercepts. That indicates an x-intercept. Also, draw dotted lines on the region and find the slopes either all positive or all negative. Then line up the inflection points which are to the inflection points of the graph.
Area under the curve (AUC)
area= h* w find the area for a rectangle area = f(x)(∆x) to estimate area under a curve between a and b aboce the x-axis(n all rectangles are plotted over the x-axis area of a region= ((f(x₁)+f(x₂)+(x₃)+...+f(xₙ))∆x factor ∆xout n ∑ f(xi)(∆x) i=1 ∆x= the (b-a)/(n) to fid th exacrt areazn\ area of a region= ((f(x₁)+f(x₂)+(x₃)+...+f(xₙ))∆x factor ∆xout n lim ∑ f(xi)(∆x) n→∞ i=1
Geometric sequence for calc
multiply common ratio to get second term and a,ar,ar²,ar³...
(x−h)²+(y−k)²=r² where (h,k)=center
circle equation standard form
a matrix that contains only the coefficients of a system of equations
coefficient matrix
a logarithm with base 10
common logarithm
x^2+2x=6 use factoring to make sure the x^2 is equal to one so factor out the x^2 coefficient divide the coefficient of the middle term by two then you square it and add it on both sides then you write the square as plus or minus as half of the coefficient
complete the square
two complex numbers of the form a+bi and a-bi
complex conjugates
a number of the form a+bi where a and b are real numbers and i is the square root of -1
complex number
a+bi
complex number
a+bi a is real part bi is imaginary part
complex number
(-1)(-1)(a-x) =(-1)(x-a) =-x+a
convert a - x to x-a
x^1/3
cube root of x
Multiply the number of degrees by pi radian over 180 degrees
degrees to radians
question 1: what we are trying to do is make a function which can look like our diffrential equation which leads us to D in the second equation with population all we really need to do is integrate because they are asking which equation describes linear growth and also shows how the function can be determined
diffrential equations
y=kx
direct variation
events that cannot happen at the same time
disjoint events
d=|Ax₁+By₁+C|/√A²+B²
distance between a point (x₁,y₁) and a line(Ax+By+C=0)
on a 2D plane, the distance of the hypotenuse is d=√a²+b² on a 3D plane, the distance of the hypotenuse is d=√a²+b²+c² also can be written as d=√(x₂-x₁)²+(y₂-y₁)²+(z₂-z₁)²
distance formula on a 3D vector
√a^2+b^2
distance from origin and a complex number
is undefined
division by zero (0)
e=(c/a)=(√a²+b²)/a
eccentricity of a hyperbola
monotonic sequence
either increasing or decreasing sin functions isn't mon
The behavior(what y = to)of the graph as x approaches positive infinity or negative infinity.
end behavior
horizontal:(x-h)²/a² - (y-k)²/b² = 1 vertical:(y-k)²/b²-(x-h)²/a²=1 Vertices : (h+-a,k) (h,k-+a) move a to y for vertical. focus: (h+-c,k) (h,k+-) move to y for vertical co-vertices: (h,k+-b) (h+-b,k) move to x axis for vertical. asymptote y-k=+-b/a(x-h) asymptote y-k=+-a/b(x-h)
hyperbola equation
A square matrix with ones (1s) along the main diagonal, from the upper left element to the lower right element, and zeros (0s) everywhere else.a square matrix that, when multiplied by another matrix, equals that same matrix
identity matrix
any decimal number with repeating digits
if it repeating divide by 9 with a singular base then if it two digit repeating divide it by 99 then if it three digit repeating divide it by 999 and so on.
a y value that a graph approaches as x approaches a certain value. If there has a hole at that location, limit gives you the y value of the hole. lim->f(x)= f(c) table approach graph approach algebra if your graph is osciliating at 0,c then it is DNE
limit
definition of e
ln(e)=(1,e)∫((1/t)dt =1 ln(e)-ln(1)=1 fundamental theorem of calculus.
properties of ln
ln(e)=1 ln(1)=0 ln(0)= undefined log(0)= undefined ln(a)^x=x*ln(a) ln(e)^x=x e^ln(x)=x
is equal 0
log(1) or ln(1), this is true for any base
when we want to find where these functions are increasing or decreasing in position and vleocity we need to find where the second graph is increasing and decreasing that is the key
look at def
whenever the frqs they give you two rates, and they want you to find min or max they you should set equal to zero and once that is done you can get the correct answer .
look at definition
slope in polar form
look at image
the line segment joining the vertices. (foci lie on the major axis)
major axis
A parabola is the set of all points in a plane that are equidistant from a fixed line, the directrix, and a fixed point, the focus, not on the line. (See figure.) The vertex is the midpoint between the focus and the directrix. The axis of the parabola is the line passing through the focus and the vertex.
parabola
y-y1=m(x-x1)
point slope form
position function = s(t) velocity function = s'(t) acceleration function = s''(t) s(t) be a function giving the position of an object at time t. The velocity of the object at time tt is given by v(t)=s′(t). The object's speed at time tt is given by |v(t)|. The acceleration of the object at t is given by a(t)=v′(t)=s''(t). s(t) = -a/2(t²)+v₀(t)+s₀ a(t)=v'(t) then v(t)=integram(a(t))dt
position, velocity, acceleration
Ratio test for series
ratio test is when the number is greater than 1 or equals infinity then it diverges but if it is less than one then it will converge and when it equals one it is inconclusive.
A formula that tells you how to get from one term to the next in a sequence.
recursive formula
a form of an augmented matrix in which the coefficient columns form an identity matrix. If ever column a leading one has zeros in every position above and elbow it's leading 1
reduced row echelon form
The acute angle formed by the terminal closet side of θ and the x-axis
reference angle
The tangent line to a parabola at a point makes equal angles with the following two lines
reflective property of parabola
moving particle and graphing functions with velocity graph may have x-intercepts. when it is a velocity graph and it is an x-intercept, then. The object is moving to the left or down, the velocity is negative. When the object is moving to the right or up the velocity must be positive When the direction of the object change, the velocity must be zero.
to find the points where the object changes direction set the position function equal to zero. We use an assumption that has to completely stop for it to turn around and then once you have solved, then you know where the object is, and then once you have to do∋∋ that. then you will find the total distance
total distance
total distance travled between the curve and x axis. total area = ∫(a,b) = |∫(a,c)|+|∫(b,c)|
Squeeze Theorem for Sequences
we do this when we don't know how to evaluate limit applies to Cn. Then we need to find two functions which can become possible ways to solve the function and then after that it becomes easy to solve the other functions, using the squeeze theorem for sequences.
when looking at it g''(x) is saying to us that g has a negative slope and it is getting smaller. also when graphing our f(x) we need to look when g'(x) that means g is greater than zero which tells us g is greater than zero g(x)<g(x)+1 when putting this on the graph for f(x) g(1) has to be less than g(2) then g(1) must be negative. also look at the curve and the graph. g''(x) is less than zero which makes it a hard negative and the slopes are decreasing and getting smaller and smaller and ultimately leads us to pick the option for A to be our answer.
when solving a problem similar to 88 on 2008 ap calc bc
9.8 meter /Second^2=32feet/ per second^2
Acceleration due to gravity
The product of all whole numbers except zero that are less than or equal to a number. n(n-1)(n-2)... (2)(1)
Factorial
concavity
Can be found using the 2nd derivative of f(x). For particular x value if second derrative is greater than zero it is concave up which makes it an underestimate compared to if it was concave down. To find which interval is concave up or down you find the inflection point.
nCr = n!/r!(n-r)! How many ways to select r objects from n objects.
Combination Formula
nCr = n! / (n-r)!∙r!
Combination Formula The number of combinations of n objects taken r at a time
Mean Value Theorem for Integrals
F(c) is the average height of the infinite number of heights between a and b
(x-h)² + (y-k)² = r²
Equation of a circle
Y intercept is always equal to zero unless there is vertical and horizontal shifting there is no x intercept
Exponential function
1) Avoiding division by zero that is to find all the x values that cause denominator equals zero. 2) Avoid even root of negative numbers
Finding domain of functions
solving multiple unknowns
IF there are two unknown u need n equations
to find a limit
If the gr‖aph has a hole at an f(c) then we do not look at the defined point
Replace f(x) with y and flip all the x into y and all the yinto x
Inverse function
a triangle with at least two congruent sides and angles are the same.
Isosceles triangle
Domain of a power series
It means a set of x values for which the power series converges and it could either be a single point or all real numbers The series only converges at C The second one is when you have an interval then you want to check |x-c| <R and diverges for | x-c| > R The big R stands for the radius of convergence (within this number line circle. If your radius converges at C only then it is zero If the series converges for all x then the radius of convergence is R = infinity the interval of convergence is the set of all the x-values
A sum of monomials (2 or more terms), you cannot have negative exponent, and no radicals, and no fractions in exponents. Have to be positive whole numbers
Polynomial
y=Ax^2+Bx+C a second degree polynomial
Quadratic equations
Y over x on a terminal side
Tangent
Series
Sum of terms in sequence.
180
Sum of triangle
nth Maclaurin Polynomial
This one gets rid of c by setting it equal to zero
Look at term
To find coterminal angles in radians add two pi or subtract two pi
cos(a+b)=cos(a) *cos(b)-sin(a)*sin(b) cos(a-b)=cos(a) *cos(b)+sin(a)*sin(b) sin(a+b)=sin(a)*cos(b)+cos(a)*sin(b) sin(a-b)=sin(a)*cos(b)-cos(a)*sin(b) tan(α+β)=tan(α)+tan(β)/(1−tan(α)tan(β)) tan(α-β)=tan(α)-tan(β)/(1+tan(α)tan(β))
Trigonometric Identities
An expression with three terms
Trinomial
critical points ,critical values, critical numbers
critical points are the whole point such as 2,4 and critical values are the 4 value and the critical numbers are the 2 value
ccc math rule for calclaus and determining
csc->csc->cot
derivative of exponential functions
d/dx(e^x)=e^x following are not exponential functions, therefore do not use exponential rules d/dx(x^e)=ex^e-1 d/dx(e^e)=0
finding higher order derivatives
d^2y/dx^2=d/dx(dy/dx)=(d/dt(dy/dx))/(dx/dt) d^3y/dx^3=d/dx(d^2y/dx^2)=(d/dt(d²y/dx²))/(d²x/dt²
A=P(1-r)^t
deprecation
Definition of Derivative at point x --> a
f'(a) =lim(x→a) [ f(x) - f(a) ] / [x - a]
In the rational function, compare the degree of the numerator and denominator 1. If the degree of the numerator is higher than the degree of the denominator, no horizontal asymptote 2. If the degree of the denominator is greater there is one asymptote at y=0 3. If they have the same degree then there is one horizontal asymptote at y = leading of coefficient of numerator/ leading coefficient of denominator
find horizontal asymptote
you first do final minus initial on two points and then solve for the cross product of the two. once you have the cross product form then you will find the magnitude of the the cross product form and that is the area. Then divide it by half
find the are of a triangle with vectors
you first do final minus initial on two points and then solve for the cross product of the two. once you have the cross product form then you will find the magnitude of the the cross product form and that is the area.
finding the area of a paralleg with vectors
A sequence which comes to an end, When the domain of the function consists of the first positive integers only, the sequence is a finite sequence.
finite sequence
P1f1 and p1f2 is the same distance as p3f1 and p3f2 A hyperbola is the set of all points in a plane for which the absolute value of the difference of the distances from two distinct fixed points (foci) is constant. The line through the foci intersects the hyperbola at two points (vertices). The line segment connecting the vertices is the transverse axis, and its midpoint is the center of the hyperbola. horizontal hyperbola center(h,k) vertices=(h+/-a,k) a is the dsitance from the vertex to center foci(h+/-c,k) c is the distance from the focus to center |d₂-d₁|=2a
hyperbola equation (horizontal)
Root Test
if lim(n→∞) nthroot(|a(n)|) = L < 1 then the series ∑a(n) is absolutely convergent if lim(n→∞) nthroot(|a(n)|) = L > 1 then the series ∑a(n) is divergent if lim(n→∞) nthroot(|a(n)|) = 1 then inconclusive an example would be e^n/2^n the n would cancel out by the nth root and then you can solve this series normally with ease.
2^n or n!
if the n factorial is growing at a faster rate (n≥2^n) then it allows me to believe that the function is converging or diverging based on where it located(numerator or denominator)
when ever you see your variable is the exponent and the base is not just
if you try to find the limit on an exponential function then you se the whole limit equal to y then it will get you started...
an angle whose vertex is on a circle and whose sides contain chords of the circle
inscribed angle
arc sin trick with integrals
integral trick for a function ∫x^2/√16-x^6 break x^6 into two parts= (x^2)^3 then once that part is done we then set x^2 equal to u and then after that we should be able to cancel out the numerator. then we break the constant so into a^2 so that we can rewrite it into a arctrig format and then from then on we replace the U with is respected function.
integgram of csc or cot
interacting csc(x) multiply the u to both denominator and numerator.( when you mutliply both you need to expand it and therfore provide it set u = csc(u)+cot(u) that denominator Sec(x) +cot(x)
(a,b) is the same a<x<b [a,b] [a,inf) a<=x<inf when it's a bracket it is included in the equation. If it's parentheses then it's excluded form the equation.
interval notation
Numbers that cannot be expressed as a ratio of two integers. Their decimal expansions are nonending and nonrepeating. Pi and e root2 root 3 root7 root 351
irrational numbers
If both left hand right limit equals positive infinity
it is DNE
only add n terms in a sequence
n partial sum
The value of e
lim(x-infinity) (1+1/x)^x
the chord through the center perpendicular to the focal axis.
minor axis
2 Events that cannot occur at the same time.
mutually exclusive
Infinite series tests table
nth term test: compare to infinity if it equals a finite number then we get diverging. If it equal to zero then we test again Geometric series test use the ratio they give to solve the problem telescoping series test: use partial fractions to reset our series and then solve it from a new method p-series test compare to the number on the bottom and if it greater than one it converges and then if it less than one diverges. alternating series test: There are two requirement's that need to be met. The limit as it approaches infinity needs to equal zero and then the a(n)> greater than the an+1 function which proves to converge. If one of these is met false then the series will diverge. Integral test: The integral test is simply taking an integral of our series. which can prove that our function can equal a finite number of infinity which ulitmaily tells if it converges or diverges root test: the root teste tells us to add a number over top of our sequence and then once that is done wen want to dvide an+1 over an and then get an answer. One thing to note here is that if it equals one then we must try a new test to prove our answer. ratio test: The ratio tests asks use to square root out answer by the nth term and then once that is completed we can go out and find our answer, but again if it equals 1 then our test is proven as inconclusive and then we cant use the test anymore
any set of equations using a parameter typically used to describe motion such as x=f(t) and y=f(t) t is the parameter. multiple sets of parametric equations can actually have the same graph
parametric equations
vertical inflection point
point which changes vertical concavity. The slope is always undefined.
x = rcosθ (cosθ=x/r) y = rsinθ (sinθ=y/r) r²=x²+y² tanθ=y/x
polar to rectangular coordinates
if you have 0.333333...= 3/9 which is 1/3 0.555555... = 5/9 0.66666...= 2/3 if there are repeating digits then you then you take the repeating digits divided by the correct number of 9. 0.567567567567...=567/999
repeating decimal number to fraction
Rose curve equation
rose curves r=acos(nθ) r=asin(nθ) Circle r=acos(θ) r=asin(θ) Leminscate r²=a²cos2θ r²=a²sin2θ
The position function
s(t) be a function giving the position of an object at time t. The velocity of the object at time tt is given by v(t)=s′(t). The speLeted of the object at time tt is given by |v(t)|. The acceleration of the object at t is given by a(t)=v′(t)=s''(t). s(t) = -a/2(t²)+v₀(t)+s₀
approximating arc length
s=∫(a.b)√1+[f'(x)]²dx s=∫(b,c)√1+[g'(x)]²dx
two interior angles on the same side of the transversal
same side interior angles
the set of all possible outcomes
sample space
the line through two points on a curve
secant line
Slope= Rise/Run=Change in Y/Change in X (y₂-y₁/x₁-x₂) (EX: Find slope of (5,6) and (1,3) 6-3/5-(-1)= 3/6= 1/2 slope can also be found through tan∅ which is just the rise over run
slope
(x-h)²=4p(y-k) y=k-p Direct X (h,k+p) Ex. y=3x^2 rearrange terms 3x^2=y then you isolate x^2 x^2=y/3 then x^2 should equal 4p 4p=y/3 then isolate 4 y/3 times 4 then it will give you p
vertical parabola
for evaluating limit trig functions at zero
when the trig limits approach zero you have ti mutlipy out the limit.
rules for inverse trig integrals
when using the moxy method how write as u/a
the x-coordinate of a point where a graph crosses the x-axis. X intercept equals root equals zero equals solution. If a root/zero/solution is a complex number it is not an x intercept. All X intercepts zeros/solutions/roots but no all solutions/roots/zeros are x intercepts.
x-intercept
integral rules
∫dx=x+c When the answer has c in it is a general solution, it applies to only integration.
integration rules
∫e^xdx=e^x+C ∫a^xdx=(1/ln(a))*a^x+C
Fundamental Theorem of Calculus
∫f(x)dx = F(b) - F(a)
Integration by parts formula
∫udv= uv - ∫vdu (LIATE)If it comes earlier, put it as u. Logarithmic, inverse trig, algebraic, trig, exponential You use this method to pick U then you need to find your du and v and once you found those plug them back into your formula and put them in your answer. Disclaimer, if you pick the wrong u then you will have to restart and pick a better answer.
