AP Calculus AB: Practice Midterm Exam
This set of flashcards covers key concepts in calculus and algebra, including average rates of change, conditions for intersections between lines and parabolas, limits of functions, and slopes of tangent lines at specific points. It’s designed to help reinforce understanding of fundamental mathematical principles and problem-solving skills.
An object is dropped from the top of a tall building. At 2 seconds, it is 64 feet from the top of the building. At 4 seconds, it is 256 feet from the top of the building. What is the average rate the object was traveling in the interval between 2 and 4 seconds?
96 ft / s
Key Terms
An object is dropped from the top of a tall building. At 2 seconds, it is 64 feet from the top of the building. At 4 seconds, it is 256 feet from the top of the building. What is the average rate the object was traveling in the interval between 2 and 4 seconds?
96 ft / s
For which values of k will the line y = x + k meet the parabola of the equation y = −x ^2 + 4x − 8 in two distinct points?
k < −23/4
What is the limit of the function in the graph at x = 4?
6
Determine, if it exists, limx→1 x^2−2x+1/√x+3 −2
0
What is the slope of the tangent line of the function f (x) = 4x ^2 − 2x + 1 at x = 3?
22
Consider the function y = x^ 2 − 2x + 1. What is the slope of the tangent line at x = 2?
2
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| Term | Definition |
|---|---|
An object is dropped from the top of a tall building. At 2 seconds, it is 64 feet from the top of the building. At 4 seconds, it is 256 feet from the top of the building. What is the average rate the object was traveling in the interval between 2 and 4 seconds? | 96 ft / s |
For which values of k will the line y = x + k meet the parabola of the equation y = −x ^2 + 4x − 8 in two distinct points? | k < −23/4 |
What is the limit of the function in the graph at x = 4? | 6 |
Determine, if it exists, limx→1 x^2−2x+1/√x+3 −2 | 0 |
What is the slope of the tangent line of the function f (x) = 4x ^2 − 2x + 1 at x = 3? | 22 |
Consider the function y = x^ 2 − 2x + 1. What is the slope of the tangent line at x = 2? | 2 |
The instantaneous rate of change of a ball (in ft/sec) is given by f′(x)=1/√x. When was the ball traveling at a rate of 1/4 ft/sec? | 16 sec |
What is f ’ (x) if f (x) = x^64? | 64x^63 |
Compute the derivative of the function f(x)=x−√x / (x^3−x+3). | (1−1/2x^−1/2)(x^3−x+3)−(x−√x)(3x^2−1)/(x^3−x+3)^2 |
Find the derivative of: P(t)=(3t^2/3−6t^1/3)^3⋅(3t^2−6t)^1/3 | P′(t)=2(3t^2/3−6t^1/3)^3(3t^2−6t)^−2/3(t−1)+6(3t^2−6t)^ 1/3(3t^2/3−6t^1/3)^2(t^−1/3−t^−2/3) |
What is the value of sin (π / 4)? | √2/2 |
What is the derivative of the function f(x) = e^x/2−tanx/x? | (e^x/2 /2−sec^2x)x−(e^x/2−tanx)/x^2 |
Evaluate the following as true or false. (ln(−x))′=1/x | true |
If dy / dx is undefined for a given value of x, then the line tangent to the curve y = f (x) at that value does not exist. | false |
Find an equation of the tangent line to the curve x^2/a^2−y^2/b^2=1, where a and b are constants, at the point (x0,y0). | x0x/a^2−y0y/b^2=1 |
Below is the graph of a function f (x). Which graph could be the graph of its inverse f −1 (x)? | The graph of f −1 (x) looks like the reflection of the graph of f (x) across the line y = x. Another way to think of this is that whatever is true of the y-coordinates in the graph of f (x) must be true of the x-coordinates in the graph of f −1 (x). Because the graph of f (x) has no negative y-coordinates, the graph of f −1 (x) must not have any negative x-coordinates. Also in the original f (x) graph the y-coordinates decrease as the x-coordinates increase. |
What is the value of d/dx[f−1(x)] when x=0, given that f(x)=x1−x, and f−1(0)=0? | 1 |
Suppose I want to find an inverse to the function |cos x|. I intend to restrict the domain of the function to an interval whose left endpoint is x = 0. On which of the following intervals is |cos x| one-to one? | [0, π / 2] |
Find the derivative d/dx[arcsec2x]. | 2/|2x|√4x^2−1 |
What does sech (0) equal? | 1 |