Trigonometry and Calculus
Convert the following to radians and give the sin, cos, tan, sec, csc, tan measurements: 0°, 30°, 45°, 60°, and 90°
0°: 0 radians; cos = 1; sin = 0; tan = 0; sec = 1; csc = undefined; cot = undefined 30°: π/6 radians; cos = sqrt(3)/2; sin = 1/2; tan = sqrt(3)/3; sec = 1*sqrt(3)/3; csc = 2; cot = sqrt(3) 45°: π/4 radians; cos = sqrt(2)/2; sin = sqrt(2)/2; tan = 1; sec = sqrt(2); csc = sqrt(2); cot = 1 60°: π/3 radians; cos = 1/2; sin = sqrt(3)/2; tan = sqrt(3); sec = 2; csc = 2*sqrt(3)/3; cot = sqrt(3)/3 90°: π/2 radians; cos = 0; sin = 1; tan = undefined; sec = undefined; csc = 1; cot = 0
Suppose that the function s(t) = t^3 represents the position of a car that begins moving in a straight line (in units of feet and seconds). Find its position, velocity, and acceleration at the end of four seconds
64 ft, 48 ft/s, 24 ft/s^2
Explain the meaning of differentiability of functions
A function is said to be differentiable at point x = a If it has a derivative at the point, that is, if f'(a) exists. For a function to be differentiable, it must be continuous because the slope cannot be defined at a point of discontinuity. Furthermore, for a function to be differentiable, it's graph must not have any sharp turn or which it is impossible to draw a pageant line. The sine function is an example of a differentiable function. It is continuous, and a tangent line can be drawn anywhere along its graph.
Explain the continuity of functions
For a function to have continuity, it's graph must be an unbroken curve. That is, it is a function that can be graphed without having to lift the pencil and move it to a different point. To say a function is continuous at Point p, you must show the function satisfies three requirements. First, f(p) must exist. if you evaluate the function at p, it must yield a real number. Second, there must exist a relationship such that lim(x-->p)f(x) = f(p). finally, the following relationship must be true: lim(x-->p+)f(x) = lim(x-->p-)f(x) = f(p). If all three of these requirements are met, a function is considered continuous at p. If any one of them is not true, the function is not continuous at p.
Discuss the properties of integrals
For each of the following properties of integrals of function f, the variables m, n, and p represent values in the domain of the given interval of f(x). The function is assumed to be integrable across all relevant intervals.
Discuss a method for calculating the area of an irregular shape
One way to calculate the area of an irregular shape is to find a formula for the width of the shape along the X Direction as a function of the y coordinate, and then integrate over y, or vice versa. Effectively, what this amounts to is dividing the area Into Thin strips and adding the area of the strips, then taking the limit as the width of the strips approaches 0. See card for example
Identify which trigonometric functions are positive in each of the four quadrants
Quad 1: sin, cos, tan, csc, cot, sec Quad 2: sin, csc Quad 3: tan and cot Quad 4: cos, sec
Discuss the difference quotient and derivative
See the card for more details, but I like the slide better
Describe accumulation of functions, otherwise known as antiderivatives
The accumulation of a function is another name for its antiderivative, or integral. We can use the relationship between a function and its antiderivative to match the corresponding graph. For example, we know that where the graph of the function is above the x-axis, the function is positive, thus the accumulation must be increasing ( it's slope is positive); where the graph of the function is below the x-axis, the accumulation must be decreasing ( it's slope is negative). it follows that where the function changes from positive to negative ( where the graph crosses the x-axis with a negative slope), it's accumulation changes from increasing to decreasing, so the accumulation has a local maximum. Where the function changes from negative to positive( where it's graph crosses the x-axis with a positive slope), the accumulation has a local minimum.
Define derivative. Describe the method for taking the derivative of a polynomial
The derivative of a function is a measure of how much that function is changing at a specific point, and it's slope of a line tangent to a curve at that specific point. The derivative of a function f(x) is written f'(x), and read, "f prime of x." The definition of the derivative is in the picture. However, this formula is rarely used because most types of functions have an established shortcut method. To find the derivative of a polynomial, use the power rule. Multiply the exponent by the constant as a given term to get the constant of the new term. To get the power of the new term, subtract 1 from the original exponent. See card for better description because math is hard to type.
Define point of inflection.
The point of inflection on the graph of a function is the point at which the concavity changes from concave downward to concave upward or from concave upward to concave downward. The easiest way to find the points of inflection is to find the second derivative of the function and then solve the equation f''(x) = 0. Remember that if f''(p) > 0, the graph is concave upward, and if f''(p) < 0, the graph is concave downward. Logically, the concavity changes at the point when f''(p) = 0.
Describe the graph of the tangent function with a period of 180° or π radians
The tangent function has a point of 360 degrees or 2π radians. This means that its graph makes one complete cycle every 360 degrees or 2π. The x-axis represents the angle measure, and the y-axis represents the tangent of the angle. The graph of the tangent function is a series of smooth curves that cross the x-axis at every 180° or π radians and have an asymptote every k*90° or kπ/2 radians, where k is an odd integer. This can be explained by the fact that the tangent is calculated by dividing the sine by the cosine, since the cosine equals zero at those asymptote points
Describe the use of the trapezoid rule
The trapezoid rule is a method of approximating the definite integral of a function by dividing the area under the function into a series of trapezoidal strips, the upper corners of the trapezoid touching the function, and adding the areas of the strips. The trapezoidal rule is related to the Riemann sum, but usually gives more accurate results than the left or right Riemann sum for the same number of intervals. In fact, it isn't hard to prove that the answer given by the trapezoidal rule is equal to the average of the left and Riemann sums using the same partition. See card for example
L'Hopital's Rule
This allows you to find the limit using derivatives. Assuming both the numerator and denominator are differentiable, and that both are equal to zero when the direct substitution method is used, take the derivative of both the numerator and the denominator and then use the direct substitution method. This rule may be applied to the function until either a limit is found or it can be determined that the limit does not exist.
Explain how to find the sine and cosine of half of a known angle
To determine the sign of your answer, you must notice the quadrant the given angle is in and apply the correct sign for the trig function you are using. Use when you have to find the sine/cosine of a weird, unknown angle (22.5 degrees, which is half of 45 degrees)
Give the properties of limits with regard to exponents, roots, and scalar multiplication
To find the limit of a power of a function or a root of a function, find the limit of the function and then raise the limit of the original Power or take the root of the limit. To find the limit of a function x a scalar, find the limit of the function and multiply the result by the scalar. See card for examples.
Explain how to find the sine, cosine, tangent, and cotangent of half of a known angle
Use one of the double angle formulas cot(2θ) = (cotθ - tanθ)/2
Explain how to find the indefinite integral of a given algebraic function
Video Code: 541913 To find the indefinite integral, reverse the process of differentiation. You will likely only need to know the most basic of integrals. Recall that in differentiation of powers of X, you multiplied the coefficient of the term by the exponent of the variable and then reduced the exponent by 1. In integration, the process is reversed: add one to the value of the exponent, and then divide the coefficient of the term by this number to get the integral. Because you do not know the value of any constant term that might have been in the original function, had C to the end of the function once you have completed this process for each term.
Explain definite integrals
While the indefinite integral has an undefined constant added at the end, the definite integral can be calculated as an exact real number. To find the definite integral of a function over a closed interval, use the formula in the picture, where F is the integral of f. Because you have been given the boundaries of a and b, no undefined constant C is needed.
Evaluate the following limits a. lim (x->4) 7 b. lim (x->4) (x-4) / (x^2 - 16) c. lim (x->4) f(x), where f(x) = x+1 when x < 4 and f(x) = x-1, when x≥4
a. 7 b. 1/8 c. DNE
Explain how to find the tangent and cotangent of half of a known angle
and cot θ/2 = sinθ / (1 - cosθ)
Using left-hand approximation with three subdivisions, calculate the area under the curve of the function f(x) = x^2 + 3 on the interval [2, 5]
approximately 38
Find f'(3) for f(x) = x^2
f'(x) = 2x, f'(3) = 6
Discuss the mean Value Theorem
if f(x) is continuous and differentiable, slope of tangent line equals slope of secant line at least once in the interval (a, b) f '(c) = [f(b) - f(a)]/(b - a)
Explain how to find the sine, cosine, or tangent of the sum or difference of two angles
practice deriving and writing them!
Suppose a farmer has 720 m of fencing, and he wants to use it to fence in a 2 by 3 block of identical rectangular pens. What dimensions of the pens will maximize their area?
see yellow notebook for work 40 ft by 45 ft
Use the first fundamental theorem of calculus to evaluate 1. ∫(x^3 + 2x) dx from 0 to 1 2. ∫(3x^3 + 2)dx from -1 to 0
1. 1.25 2. 1.25 The solution on the back of the card for 2 is for another integral
Describe the purpose of Riemann sums
A Riemann sum is a sum used to approximate the definite integral of a function over a particular interval by dividing the area under the function into vertical rectangular strips and adding the areas of the strips. The height of each strip is equal to the value of the function at some point within the interval covered by the strip. In principle, any point in the interval can be chosen, but common choices include the left endpoint of the interval ( yielding the left Riemann sum), the right endpoint ( yielding the right Riemann sum), and the midpoint of the interval ( the basis of the midpoint rule). usually it is convenient to set all the intervals to the same width, although the definition of the Riemann sum does not require this.
List rules of operations with the derivative
Many of the rules of operations with limits apply to operations with derivatives since the derivative is a limit. Addition: The derivative of a sum of two functions is equal to the sum of the individual derivatives Constant Multiplication: (cf(x)))' = c(f'(x)), where c is a constant. The derivative of a constant times a function is equal to the product of the constant times the derivative of the function Multiplication by Another Derivative: (f(x)*g(x))' = f'(x)*g(x) + g'(x)f(x), where f(x) and g(x) are both differentiable functions Division by Another Derivative: (f(x) / g(x))' = [f'(x)*g(x) + g'(x)f(x)] / (g(x))^2 where f(x) and g(x) are both differentiable functions
Explain whether the function f(x) = |x| differentiable and why
No because it has a sharp turn at x = 0 meaning we cannot draw a tangent line there.
Discuss calculating the volume of a three-dimensional shape
One way to calculate the volume of a three-dimensional shape is to find a formula for its cross-sectional area perpendicular to some axis and then integrate over that axis. effectively, what this amounts to is dividing the shape into thin, flat slices and adding the volumes of the slices, then taking the limit as the thickness of the slices approaches zero. See card for example
Define antiderivative
The antiderivative of a function is the function whose first derivative is the original function. Antiderivatives are typically represented by capital letters, while their first derivatives are represented by lowercase letters. For example, if F' = f, then F is the antiderivative of f. Antiderivatives are also known as indefinite integrals. When taking the derivative of a function, any constant terms in the function are eliminated because their derivative is zero. To account for this possibility, when you take the indefinite integral of a function, you must add an unknown constant C to the end of the function. Because there is no way to know what the value of the original constant was when looking just at the first derivative, the integral is indefinite.
Explain how to find the product of the sines and cosines of two angles
To find the product of the sines and cosines of two different angles, use one of the following formulas where u and v are two unique angles
Use the related rates method to solve the following problem: the side of a cube is increasing at a rate of 2 feet per second. Determine the rate at which the volume of the cube is increasing when the side of the cube is 4 feet long
The card has an incorrect solution. v'(t) = 3[s(t)]^2 * s'(t) v'(t) = 3*4^2*2 = 6*16 = 96 ft^3 / s
Discuss absolute value limits
To find the limit of an expression containing an absolute value sign, take the absolute value of the limit. If lim(n-->∞)a_n = L, where L is the numerical value for the limit, then lim(n-->∞) |a_n| = |L|. Also, if lim(n-->∞) |a_n| = 0, then lim(n-->∞) a_n = 0. The trip comes when you are asked to find the limit as n approaches from the left. Whenever the limit is being approached from the left, it is being approached from the negative end of the domain. The absolute value sign makes everything in the equation positive, essentially eliminating the negative side of the domain. In this case, rewrite the equation without the absolute value signs and add a negative sign in front of the expression. For example, lim(n-->0^-) |x| becomes lim(n-->0^-) (-x).
Explain differentiable functions
Video Code: 735227 and 132724 Differentiable functions are functions that have a derivative. Memorize those basic rules too
Give the properties of limits with regard to addition, subtraction, multiplication, and division
When finding the limit of the sum or difference of two functions, find the limit of each individual function and then add or subtract the results. To find the limit of the product or quotient of two functions, find the limit of each individual function and then multiply or divide the results. When finding the quotient of the limits of two functions, make sure the denominator is not equal to zero. If it is, use differentiation or L'Hôpital's rule to find the limit.
Explain how to find limits by direct substitution
To find the limit of a function by direct substitution, substitute the value of a for x in the function and solve. See card for patterns (not very helpful though). You can also use substitution for finding the limit of a trigonometric function, a polynomial function, or a rational function. Be sure that in manipulating an expression to find a limit that you do not divide terms equal to 0.
Define domain, range, and asymptotes as they relate to trigonometry
Domain: the set of all possible real number values of X on the graph of a trigonometric function. Some grass will impose limits on the values of x. Range: the set of all possible real number values of y on the graph of a trigonometric function. Some graphs will impose limits on the values of y Asymptotes: lines which the graph of a trigonometric function approaches but never reaches. Asymptotes exist for values of x in the graphs of the tangent, cotangent, secant, and cosecant. The sine and cosine graphs do not have any asymptotes.
Given that lim (x->3) f(x) = 2, lim (x->3) g(x) = 6, and k = 5, solve the following: a. lim (x->3) k*g(x) b. lim (x->3) (f(x) + g(x)) c. lim (x->3) f(x) * g(x) d. lim (x->3) g(x) ፥ f(x) e. lim (x->3) [f(x)]^n = C^n, where n = 3
a. 30 b. 8 c. 12 d. 3 e. 8
Using right-hand approximation with six subdivisions, calculate the area under the curve of the function f(x) = x^2 + 3 on the interval [2, 5]
approximately 53.375
Discuss trig differentiation and inverse trig differentiation
d/dx sin(u) = cos (u) du/dx d/dx cos(u) = -sin(u) du/dx d/dx tan(u) = sec^2(u) du/dx d/dx sec(u) = tan(u)*sec(u) du/dx d/dx csc(u) = -csc(u)*cot(u) du/dx d/dx cot(u) = -csc^2(u) du/dx Find some practice problems and use them!!
Define and discuss extrema of functions
The maximum and minimum of a function are collectively called the extrema of the function. Both Maxima and Minima can be local, also known as relative, or absolute. A local maximum or minimum refers to the value of a function near a certain value of x. An absolute maximum or minimum refers to the value of a function on a given interval. The local maximum of a function is the largest value that the function attains near a certain value of x. Function f has a local minimum at x = b is f(b) is the largest value that f attains as it approaches b. Conversely, the local minimum is the smallest value that the function attains near a certain value of x. In other words, function f has a local minimum at x = b if f(b) is the smallest value that f attains as it approaches b. The absolute maximum of a function is the largest value of the function over a certain interval. The function f has an absolute maximum at x = b if f(b) ≥ f(x) for all x in the domain of f. The absolute minimum of a function is the smallest value of the function over a certain interval. The function f has an absolute minimum x = b if f(b) ≤ f(x) for all x in the domain of f.
Explain the usefulness of the second derivative
The second derivative, designated by f''(x), is helpful in determining whether the relative extrema of a function are relative maximum or relative minimum. If the second derivative of the critical point is greater than zero, the critical point is a relative minimum. If the second derivative at the critical point is less than zero, the critical point is a relative maximum. If the second derivative at the critical point is equal to 0, you must use the first derivative test to determine whether the point is a relative minimum or a relative maximum.
Explain how to use the derivative tests to draw a rough sketch of a function's graph
Begin by solving the equation f(x) = 0 to find all the zeros of the function, if they exist. Plot these points on the graph. Then, find the first derivative of the function and solve the equation f'(x) = 0 to find the critical points. Remember the numbers obtained here are the x portions of the coordinates. Substitute these values for x in the original function and solve for y to get the full coordinates of the points. Plot these points on the graph. Take the second derivative of the function and solve the equation f''(x) = 0 to find the points of inflection. Substitute in the original function to get the coordinates and graph these points. Test points on both sides of the critical points to test for concavity and draw the curve
Define integral and differential calculus
Calculus, also called analysis, is the branch of mathematics that studies the link, area, and volume of objects, and the rate of change of quantities ( which can be expressed as slopes of curves). The two principal branches of calculus are differential and integral. Differential calculus is based on derivatives and takes the form d/dx[f(x)]. integral calculus is based on integrals and takes the form ∫f(x)dx. Some of the basic ideas of calculus were utilized as far back in history as Archimedes. However, its modern forms were developed by Newton and Leibniz.
De Moivre's Theorem
De Moivre's TheoremIs used to find the powers of complex numbers ( numbers that contain the imaginary number i) written in polar form. Given a trigonometric expression that contains i, use the bottom equation in the picture attached. R and n are real numbers, θ is the angle measure in POLAR form, and i is sqrt(-1). Note that De Moivre's Theorem is only for angles in polar form. If you are given an angle in degrees, you must convert to polar form before using the formula.
Explain how to find the indefinite integral of a given trigonometric or logarithmic function
Finding the integral of a function is the opposite of finding the derivative of the function. Where possible, you can use the trigonometric or logarithmic differentiation formula in reverse, and add sea to the end to compensate for the unknown term. In instances where a negative sign appears in the differentiation formula, move the negative sign to the opposite side (multiply both sides by -1) to reverse the integration formula. integral of 1/x is ln|x| +C and integral of e^x is e^xx + C Study these integrals!!
Give the formula for integration by substitution and explain its usefulness
Integration by substitution is the integration version of the chain rule for differentiation. The formula for integration by substitution is in the picture attached. When a function is in a format that is the plural or impossible to integrate using traditional integration methods and formulas due to multiple functions being combined, use the formula shown above to convert the fraction to a simpler form that can be integrated directly. FIND PRACTICE PROBLEMS
Discuss Rolle's Theorem
Rolle's Theorem states that if a differentiable function has two different values in the domain that correspond to a single value in the range, then the function must have a point between them where the slope of the tangent to the graph is zero. This point will be a maximum or minimum value of the function between those two points. The maximum or minimum point is the point at which f'(c) = 0, where c is within the appropriate interval of the function's domain.
Explain how to determine if the graph of a function is concave upward or concave downward
There are a couple of ways to determine the concavity of the graph of a function. To test a portion of the graph that contains a point with domain p, find the second derivative of the function and evaluate it for p. If f''(p)>0, then the graph is concave upward at that point. If f''(p)<0, then the graph is concave downward at that point.
Suppose that the function s(t) = t^2 + 4t + 5 represents the position of a train that begins moving in a straight line (in units of meters and second).Find its position, velocity, and acceleration at the end of three seconds
26 m, 10 m/s, 2 m/s^2
Explain left and right Riemann sums and their effects
A Riemann sum is an approximation to the definite integral of a function over a particular interval performed by dividing it into smaller intervals and summing the products of the width of each interval and the value of the function evaluated at some point within the interval. The left Riemann sum is a Riemann sum in which the function is evaluated at the left endpoint of each interval. In the right Riemann sum, the function is evaluated at the right end of each interval. When the function is increasing, the left Riemann sum will always underestimate the function. This is because we are evaluating the function at the minimum Point within each interval; the integral of the function in the interval will be larger than the estimate. Conversely, the right Riemann sum is evaluating the function at the maximum Point within each interval, that's it will always overestimate the function. For a decreasing function these considerations are reversed: a left Riemann sum will overestimate the interval, and a right Riemann sum will underestimate it.
Define critical point and sign diagram
A critical point is a point (x, f(x)) that is part of the domain of a function such that either f'(x) = 0 or f'(x) does not exist. If either of these conditions is true, then X is either an inflection point or a point at which the slope of the curve changes sign. If the slope changes sign, then a relative minimum or maximum occurs. In graphing an equation with relative extrema, use a sign diagram to approximate the shape of the graph. Once you have determined the relative extrema, calculate the sign of a point on either side of each critical point. This will give a general shape of the graph, and he will know whether each critical point is a relative minimum, a relative maximum, or a point of inflection.
Explain continuity of functions
A function can be either continuous or discontinuous. A conceptual way to describe continuity is this: a function is continuous if it's graph can be traced with a pen without lifting the pen from the page. In other words, there are no breaks or gaps in the graph of the function. However, this is only a description, not a technical definition. A function is continuous at the point x = a if the three following conditions are met: f(a) is defined, lim(x-->a)f(x) exists and = f(a). If any of these conditions are not met, the function is discontinuous at the point x = a. A function can be continuous at a point, Continuous over an interval, or continuous everywhere. The above rules define continuity at a point. A function that is continuous over an interval [a, b] is continuous at the points a and b and at every point between them. A function that is continuous everywhere is continuous or every real number, that is, for all points in its domain.
Discuss limits to Riemann Sums
One way to calculate the area of an irregular shape is to find a formula for the width of the shape along the X Direction as a function of the y coordinate, and then integrate over y, or vice versa. Effectively, what this amounts to is dividing the area Into Thin strips and adding the areas of the strips, then taking the limit as the width of the strips approaches zero. See card for example
Describe use of the first and second derivative
Video Code: 787269 The first derivative of a function is equal to the rate of change of the function. The sign of the rate of change shows whether the value of the function is increasing or decreasing. A positive rate of change, and therefore a positive first derivative, represents that the function is increasing at that point. A negative rate of change represents that the function is decreasing. If the rate of change is zero, the function is not changing, i.e. it is constant.
Find f'(x) for the function f(x) = 3x and f'(x) for the function f(x) = x^2
f'(x) = 3 f'(x) = 2x
Define related rates problems and explain how to approach them
A related rate problem is one in which one variable has a relation with another variable, and the rate of change of one of the variables is known. With that information, the rate of change of the other variable can be determined. The first step in solving related rates problems is defining the known rate of change. Then, determine the relationship between the two variables, then the derivatives (the rate of change closing), and finally substitute the problem's specific values.
State the derivative relationship between the general equations for motion (position, velocity, and acceleration) and use this relationship to formulate the velocity and acceleration equations for the position equation s(t) = t^2 + 4t - 7
The relationship among the three functions is one of rate of change. Velocity is the rate of change of position, and acceleration is the rate of change of velocity. Therefore one is the derivative of another. For position, the function is s(t) = t^2 + 4t - 7. For velocity, the function is s'(t) = 2t + 4. For acceleration, the function is s''(t) = 2.
Describe the graph of the sine function with a period of 360° or 2π radians
The sine function has a point of 360 degrees or 2π radians. This means that its graph makes one complete cycle every 360 degrees or 2π. Because sin(0) = 0, the graph of y = sin(x) begins at the origin, with the x-axis representing the angle measure, and the y-axis representing the sine of the angel. The graph of the sine function is a smooth curve that begins at the origin, peaks at the point (π/2, 1), crosses the x-axis at (π,0), has its lowest point at (3π/2, -1) and returns to the x-axis to complete one cycle at (2π,0)
Explain the trapezoidal rule
The trapezoidal rule, also called the trapezoidal rule or the trapezium rule, is another way to approximate the area under a curve. In this case, a trapezoid is drawn such that one side of the trapezoid is on the x-axis and the parallel sides terminate on points of the curve so that part of the curve is above the trapezoid and part of the curve is below the trapezoid. While this does not give an exact area or the region under the curve, it does provide a close approximation.
Discuss how to find the limit of a composite function. Discuss intermediate form
To find the limit of a composite function, begin by finding the limit of the innermost function. For example, to find lim(x-->a)f(g(x)), first find lim(x-->a)g(x) = H. Then substitute H in for x in f(x) and solve. The result is the limit of the original problem. Sometimes solving a limit of the form lim(x-->a)[f(x) / g(x)] by the direct substitution method will result in the numerator and denominator both being equal to zero, or both being equal to infinity. This outcome is called an indeterminate form. The limit cannot be directly found by substitution in these cases. L'Hopital's rule should be used to find the limit of indeterminate forms.
Explain the difference between solving trig equations and algebraic equations
Trigonometric and algebraic equations are solved following the same rules, but why algebraic expressions have one unique solution, trigonometric equations could have multiple Solutions, and you must find them all. When solving for an angle with a known trigonometric value, you must consider the sign and include all angles with that value. Your calculator will probably only give one value as an answer, typically in the following ranges: For the inverse sine function, [-π/2, π/2] or [-90°, 90°] For the inverse cosine function, [0,π] or [0°, 180°] For the inverse tangent function, [-π/2, π/2] or [-90°, 90°] It is important to determine if there is another angle in a different quadrant that also satisfies the problem. To do this, find the other quadrant(s) with the same sign for that trigonometric function and find the angle that has the same reference angle. Then check whether this angle is also a solution.
Explain unit circles and standard position
Video Code: 333922 A unit circle is a circle with a radius of one that has its Center at the origin. The equation of the unit circle is x^2 + y^2 = 1. notice that this is an abbreviated version of the standard equation of a circle. Because the center is the point (0,0), the values of h and k in the general equation are equal to 0 and the equation simplifies to this form. Standard position is the position of an angle of measure θ whose vertex is at the origin, the initial side crosses the unit circle at the point (1, 0), and the terminal side crosses the unit circle at some other point (a, b). In the standard position, sin θ = c, cos θ = a, and tan θ = b/a
Discuss the rectangular and polar coordinate system and explain how to convert between the two
Video Code: 694585 Rectangular coordinates are those that lie on the square grids of the Cartesian plane. The polar coordinate system are based on a circular graph. Points in the polar coordinate system are written as (r, θ), where r is the distance from the origin (think radius of the circle) and θ is the smallest positive angle (moving counterclockwise around the circle) made with the positive horizontal axis. See picture to go from rectangular to polar To go from polar to rectangular use x = r*cosθ and y = r*sinθ When in doubt for θ, look at the unit circle
Define the six inverse trig functions
Video Code: 996431 Each of the trigonometric functions except an angular measure, either degrees or radians, and gives a numerical value as the output. The inverse functions do the opposite; they accept a numerical value and give an angular measure of the output. The inverse sine, or arcsine, commonly written as either sin^-1(x) or arcsin(x), gives the angle whose sine is x. Similarly: The inverse of cos(x) is written as cos^-1(x) or arccos(x) and means the angle whose cosine is x. The inverse of tan(x) is written as tan^-1(x) or arctan(x) and means the angle whose tangent is x. The inverse of csc(x) is written as csc^-1(x) or arccsc(x) and means the angle whose cosecant is x. The inverse of sec(x) is written as sec^-1(x) or arcsec(x) and means the angle whose secant is x. The inverse of cot(x) is written as cot^-1(x) or arccot(x) and means the angle whose cotangent is x.
Describe the graph of the cosine function with a period of 360° or 2π radians
The cosine function has a point of 360 degrees or 2π radians. This means that its graph makes one complete cycle every 360 degrees or 2π. Because cos(0) = 1, the graph of y = cos(x) begins at (0,1), with the x-axis representing the angle measure, and the y-axis representing the sine of the angel. The graph of the cosine function is a smooth curve that begins at (0,1), crosses the x-axis at (π/2,0), has its lowest point at (π, -1) and returns to the x-axis to complete one cycle at (2π,1)
Give the domain, range, and asymptotes of each of the six trig functions
Cosine (x/r): domain = ℝ ; range = -1 ≤ y ≤ 1; no asymptotes Sine (y/r): domain = ℝ ; range = -1 ≤ y ≤ 1; no asymptotes Tangent (y/x): domain = ℝ exlcuding x = π/2 + kπ ; range = ℝ; asymptote at x = π/2 + kπ Secant (r/x): domain = ℝ exlcuding x = π/2 + kπ; range = (∞, -1) ⋃ (1, ∞); asymptote at x = π/2 + kπ Cosecant (r/y): domain = ℝ exlcuding y = kπ; range = (∞, -1) ⋃ (1, ∞); asymptote at y= kπ Cotangent (x/y): domain = ℝ exlcuding y = kπ; range = ℝ; asymptote at y= kπ
Describe the process of approximating a derivative from a table of values
The derivative of a function at a particular point is equal to the slope of the graph of the function at that point. For a nonlinear function, it can be thought of as the limit of the slope of a line drawn between two other points on the function as those points become closer to the point in question. Such a line drawn through two points on the function is called a secant of the function. This definition of derivative in terms of the second allows us to approximate the derivative of a function at a point from a table of values: we take the slope of the line through the points on either side. That is, if the points lie between (x_1, y_1) and (x_2, y_2), the slope of the secant ( the approximate derivative) is (y_2 - y_1) / (x_2 - x_1). This is also equal to the average slope over the interval [x_1, x_2].
List the trig reciprocal, ratio, trig and cofunction identities
Trig reciprocal identities: csc θ = 1 / (sin θ); sec θ = 1 / (cos θ); cot θ = 1 / (tan θ) Trig ratio identities: tanθ = (sinθ)/(cosθ); cotθ = (cosθ)/(sinθ) Trig Identities: sin^2 θ + cos^2 θ = 1; tan^2 θ + 1 = sec^2 θ; cot^2 θ + 1 = csc^2 θ Cofunction Identities: see picture
Discuss rectilinear motion
Rectilinear motion is motion along a straight line rather than a curved path. This concept is generally used in problems involving distance, velocity, and acceleration. Average velocity over a period of time is found by Δd / Δt. Instantaneous velocity at a specific time is found by using the formula lim(h-->0) [s(t + h) - s(t)] / h, or v = s'(t). Remember that velocity at a given point is found using the first derivative, and acceleration at a given point is found using the second derivative. Therefore, the formula for acceleration at a given point in time is found using the formula a(t) = v'(t) = s''(t), where a is acceleration, v is velocity, and s is distance or location.
Explain the squeeze theorem
The Squeeze theorem is known by many names, including the sandwich theorem, the sandwich Rule, The Squeeze Mama, the squeezing your room, and the pinching theorem. No matter what you call it, the principle is the same. To prove the limit of a difficult function exists, find the limits of two functions one on either side of the unknown, that are easy to compute. If the limits of these functions are equal, then that is also the limit of the unknown function.
Discuss the relationship between a derivative and the graph of a function
We can use what we know about the meaning of a derivative to match the graph of a function with a graph of its derivative. For one thing, we know that where the function has a critical point, the derivative is zero. Therefore, at every x value at which the graph of a function has a maximum or minimum, the derivative must cross the x axis. And conversely, everywhere the graph of the derivative crosses the x-axis, the function must have a critical point: either a maximum, a minimum, or an inflection point. If this is still not enough to identify the correct match, we can also use the fact that the sign of the derivative corresponds to whether the function is increasing or decreasing: everywhere the graph of the derivative is above the x-axis, the function must be increasing ( it's slope is positive), and everywhere the graph of the derivative is below the x-axis, the function must be decreasing ( it's slope is negative).
Discuss implicit differentiation
An implicit function is one where it is impossible, or very difficult, to express one variable in terms of another by normal algebraic method. This would include functions that have both variables raised to a power greater than one, functions that have two variables x each other, or a combination of the two. To differentiate such a function with respect to x, take the derivative of each term that contains a variable, either x or y. When differentiating a term with y, use the chain rule, first taking the derivative with respect to y, and then multiplying by dy/dx. if a term contains both x and y, you will have to use the product rule as well as the chain rule. Once the derivative of each individual term has been found, use the rules of algebra to solve for dy/dx to get the final answer.
Describe optimization problems
An optimization problem is a problem in which we are asked to find the value of a variable that maximizes or minimizes a particular value. Because the maximum or minimum occurs at a critical point, and because the critical point occurs when the derivative of the function is zero, we can solve an optimization Problem by setting the derivative of the function to zero and solving the desired variable.
Describe discontinuous functions
Discontinuous functions are categorized according to the type of cause of discontinuity. Three examples are Point, infinite, and jump discontinuity. a function with a point discontinuity has one value of x for which it is not continuous. A function with infinite discontinuity has a vertical asymptote at x = a and f(a) is undefined. it is said to have an infinite discontinuity at x = a. a function with jump discontinuity has one sided limits from the left and from the right, but they are not equal to one another. It is said to have a jump discontinuity at x = a.
Describe the mean value theorem of integrals, and give an example of its usage
Discrete function with finitely many points, the average value of a function is simply the sum of all the values of the function, divided by the number of values. In the case of a continuous function, the definition is analogous: the average value of a function over an interval is equal to the definite integral of the function over that interval, divided by the width of the interval ( that is, the difference between the endpoints of the interval). See card for example.
Discuss exponential and logarithmic differentiation
Exponential functions in the form e^x, which has itself as its derivative d/dx e^x = e^x. For functions that have a function as the exponent rather than just an x, use the formula d/dx e^u = e^u du/dx. The inverse of the exponential function is the natural logarithm, To find the derivative if the natural logarithm, use the formula d/dx ln(u) = 1/u du/dx. If you are trying to solve an expression with a variable in the exponent, use the formula a^x = e^(x*ln(a)), where a is a positive real number and x is any real number. To find the derivative of a function in this format, use the formula d/dx a^x = a^x * ln(a). If the exponent is a function rather than a single variable x, use the formula d/dx a^u = a^u * ln(a) du/dx. If you are trying to solve an expression involving a logarithm, use the formula d/dx(log_a x) = 1 / x*ln(a) or d/dx (log_a |u|) = 1 / (u*ln(a)) du/dx; u≠0
Explain the fundamental theorem of calculus
Fundamental theorem of calculus shows that the process of indefinite integration can be reversed by finding the first derivative of the resulting function. It also gives the relationship between differentiation and integration over a closed interval of the function. For example, assuming a function is continuous over the interval [m, n], you can find the definite integral by using the formula (m to n) ∫f(x) dx = F(n) - F(m). Many times the notation ∫f(x) dx = F(x) | m to n = F(n) - F(m) is also used to represent the Fundamental Theorem of Calculus. To find the average value of the function over the given interval, use the formula 1/(n-m) * ∫f(x) dx from m to n. The second fundamental theorem of calculus is related to the first. This theorem states that, assuming the function is continuous over the interval you are considering, taking the derivative of the integral of a function will yield the original function. The general format for this theorem is d/dx ∫f(x) dx from c to x from any point having a domain value equal to c in the given interval.
Discuss cases when a limit does not exist
If the limit as x approaches a differs depending on the direction from which it approaches, then the limit does not exist at a. In other words if the left-handed limit is not equal to the right-handed limit, then the limit at that point does not exist. Situations in which the limit does not exist include a function that jumps from one value to another at a, one that oscillates between two different values as x approaches a, or one that increases or decreases without bound as x approaches a. If the limit you calculate has a value of c/0. where c is any constant, this means the function goes to infinity and the limit does not exist. It is possible for two functions that do not have limits to be multiplied to get a new function that does have a limit. Just because two functions do not have limits, do not assume that the product will not have a limit.
Explain integration by parts
Integration by parts is the integration version of the product rule for differentiation. Whenever you are asked to find the integral of the product of two different functions or parts, integration by parts can make the process simpler. When using integration by parts, the key is selecting the best function to substitute for u and v so that you make the integral easier to solve and not harder. See card for more detail or the formula. FIND PRACTICE PROBLEMS!!
Identify the discontinuity in the graph of the function f(x) = x for x < 0 and f(x) = x + 1 for x ≥ 0 (see card for graph)
Jump discontinuity because the left and right limits exits, but are not equal to one another
Discuss the relative minimum, relative maximum, and relative extremum of functions
Remember Rolle's Theorem, which stated that if two points have the same value in the range that there must be a point between them where the slope of the graph is zero. This point is located at a peak or Valley on the graph. A peak is a maximum point, and a valley is a minimum point. The relative minimum is the lowest point on a graph for a given section of the graph. It may or may not be the same as the absolute minimum, which is the lowest point on the entire graph. The relative maximum is the highest point on one section of the graph. Again, it may or may not be the same as the absolute maximum. A relative extremum (plural extrema) is a relative minimum or relative maximum point on a graph.
Explain the first derivative test
Remember that critical points occur where the slope of the curve is zero. Also remember that the first derivative of a function gives the slope of the curve at a particular point on the curve. Because of this property of the first derivative, the first derivative test can be used to determine if a critical point is a minimum or maximum. If f'(x) is negative at a point to the left of a critical number and f'(x) is positive at a point to the right of a critical number, then the critical number is a relative minimum. If f'(x) is positive to the left of a critical number and f'(x) is -2 the right of a critical number, then the critical number is a relative maximum. If f'(x) has the same sign on both sides, then the critical number is a point of inflection.
Explain the power rule of differentiation and its usage and use it to differentiate f(x) = 4x^2, f(x) = x^4, and f(x) = 3x^2 - 5x + 6
The power rule is useful for finding the derivative of polynomial functions. f'(x) = 8x, 4x^3, 6x - 5
Explain tangents and normals as they relate to functions
Tangents are lines that touch a curve in exactly one point and have the same slope as the curve at that point. To find the slope of a curve at a given point and a slope of its tangent line at that point, find the derivative of the function of the curve. If the slope is undefined, the tangent is a vertical line. If the slope is 0, the tangent is a horizontal line. A line that is normal to a curve at a given point is perpendicular to the tangent at that point. Assuming f'(x) ≠ 0, the equation for the normal line at point (a, b) is: y - b = (-1 / [f'(a)] ) * (x-a).The easiest way to find the slope of the normal is to take the negative reciprocal of the slope of the tangent. If the slope of the tangent is zero, the slope of the normal is undefined. If the slope of the tangent is undefined the slope of the normal is 0.
Describe methods for computing the area under a curve
The common methods for computing the area under a curve include Riemann sums and the trapezoidal rule. Riemann sums is a method in which the area under a curve is divided into narrow rectangles and then the individual areas of the rectangles are added together to obtain the total area. There are several variations on this method, including the left hand approximation, the midpoint approximation, and the right hand approximation. In left hand approximation, the value of the function at the left endpoint of each equal width rectangle is used as the height of the rectangle. In the midpoint approximation, the value of the function at the midpoint of each equal width rectangle is used as the height of the rectangle. In right hand approximation, the value of the function at the right endpoint of each equal width rectangle is used as the height of the rectangle. The trapezoid rule divides the area under a curve into narrow trapezoids, using the value of the function at the right and left endpoints of each section to determine the height of the two uneven corners of the trapezoid. Their areas are then summed to approximate the total area under the curve. For all of the above methods, the greater the number of subdivisions into which the area is divided, the greater the accuracy of the approximation.
Discuss the process of accumulation
The definite integral of a function represents the accumulated value of the function over an interval. Therefore, given a function representing a process that has a cumulative value, we can find that cumulative value by taking the definite integral of the function and adding the initial value. (This last step is important and often forgotten. Since the definite integral is the change in the accumulated value over the interval, it is necessary to add the initial value to find the final value of the accumulation.) See card for example (we find the change from day 1 to day 2, but to find the value for day 2, we must add the change just calculated to the given day 1 value)
Define derivative problems and explain what a derivative represents in the following two instances: 1. Temperature as a function of time 2. Profit as a function of units sold
The derivative represents the rate of change of a function, thus derivatives are a useful tool for solving any problem that involves finding the rate at which a function is changing. At its simplest form, such a problem might provide a formula for a quantity as a function of time and ask for its rate of change at a particular time. The derivative of a function of time in reference to temperature gives you the rate of change of the temperature over time. The derivative of profit as a function of units sold is the additional profit for each additional unit sold, a quantity known as the marginal profit.
Suppose that the function s(t) = 4t^3 represents the position of a rocket that has been fired in a straight line. The position is measures in feet, and t is the time in seconds that has elapsed since its motion started. Determine the instantaneous rate of change of s(t) at the time of t = 3
The instantaneous rate of change measures the slope of a function at a certain point. Therefore, the instantaneous rate of change is expressed by the derivative. Use s'(t) = 12t^2 s'(3) = 108 ft/s
Discuss the limits of functions
The limit of a function is represented by the notation lim(x-->a)f(x). it is read as " the limit of f of x as X approaches a." in many cases, lim(x-->a)f(x) will simply be equal to f(a), but not always. Limits are important because some functions are not defined or are not easy to evaluate at certain values of x. The limit at the point is said to exist only if the limit is the same when approached from the right side as from the left. Notice the symbol by the a in each case. When x approaches a from the right, it approaches from the positive end of the number line. When x approaches a from the left, it approaches from the negative end of the number line.
Explain the midpoint rule
The midpoint rule is a way of approximating the definite integral of a function over an interval by dividing the interval into smaller subintervals, multiplying the width of each sub interval by the value of the function at the midpoint of the sub interval, and then selling these products. This is a special case of the Riemann sum, specifying the midpoint of the interval as the point at which the function is to be evaluated. The approximation found using the midpoint rule is usually more accurate than that found using the left or right Riemann sum, though as the number of intervals becomes very large the difference becomes negligible.
Describe finding distance when given velocity of an object over time
When given the velocity of an object over time, it's possible to find a Distance by integration. The velocity is the rate of change of the position; therefore, the distance is the accumulation of the Velocity: that is, the integral of the Velocity is the distance. However, if asked to find the total distance traveled ( as opposed to the displacement), it's important to take the sign into account: we must integrate not just the velocity, but the absolute value of the Velocity, which essentially means integrating separately over each interval in which the velocity has a different sign. See card for example