AP Calculus AB: Chapter 3 Test
This content focuses on calculating the average rate of change and the slopes of secant and tangent lines for quadratic and cubic functions. It also includes finding derivatives and applying them to evaluate function behavior at specific points.
What is the slope of the secant line of the function y = −2x ^2 + 3x − 1 between x = 2 and x = 6?
-13
Key Terms
What is the slope of the secant line of the function y = −2x ^2 + 3x − 1 between x = 2 and x = 6?
-13
What is the average rate of change of the function y = 2x^ 2 + 3 between x = x1 and x = x2?
2(x2+x1)
What is the slope of the secant line of the function y = 4x^ 2 − 2x + 1 between x = x1 and x = x2?
4x2 + 4x1 − 2
What is the average rate of change of the function y = 4x^ 3 − 2 between x = x1 and x = x2?
4 (x2^2 + x2x1 + x1^2 )
What is the derivative of the function f (x) = 2x^ 2 + 3 at x ?
4x
What is the slope of the tangent line of the function f (x) = 4x ^2 − 2x + 1 at x ?
8x − 2
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| Term | Definition |
|---|---|
What is the slope of the secant line of the function y = −2x ^2 + 3x − 1 between x = 2 and x = 6? | -13 |
What is the average rate of change of the function y = 2x^ 2 + 3 between x = x1 and x = x2? | 2(x2+x1) |
What is the slope of the secant line of the function y = 4x^ 2 − 2x + 1 between x = x1 and x = x2? | 4x2 + 4x1 − 2 |
What is the average rate of change of the function y = 4x^ 3 − 2 between x = x1 and x = x2? | 4 (x2^2 + x2x1 + x1^2 ) |
What is the derivative of the function f (x) = 2x^ 2 + 3 at x ? | 4x |
What is the slope of the tangent line of the function f (x) = 4x ^2 − 2x + 1 at x ? | 8x − 2 |
What is the slope of the tangent line of the function f (x) = −4x ^2 − 5x + 2 at x ? | −8x − 5 |
Consider the function y = x ^2 − 3x + 2. What is the slope of the tangent line at x = 2? | 1 |
Consider the function y = x ^2 − 3x + 5. At what value of x is the slope of the tangent line equal to 7? | 5 |
Consider the function y = x ^2 + x + 9. At what value of y is the slope of the tangent line equal to 5? | 15 |
The position of a car at time t is given by the function p (t) = t ^2 + 2t − 4. What is the velocity at t = 2? Assume t ≥ 0. | 6 |
The position of a car at time t is given by the function p (t) = t ^2 − t − 7. What is the velocity when p (t) = 5? Assume t ≥ 0. | 7 |
The position of a car at time t is given by the function p (t) = t ^2 − 3t − 26. At what time will the velocity of the car be 11? Assume t ≥ 0. | 7 |
The position of a car at time t is given by the function p (t) = t ^2 + 4t − 17. Where will the car be when it moves at a velocity of 10? Assume t ≥ 0. | 4 |
Apply the definition of the derivative to differentiate the function f (x) = x | 1 |
Evaluate the derivative of the function: f (x) = (x − 2)^−1 | −1/(x−2)^2 |
Given that f(x)=√6x−3, evaluate f′(x) | 3 / √6x−3 |
Find the slope of the line tangent to f(x)=x^2−x at the point x=2. | 3 |
True or false? The derivative of a function at a point x represents the instantaneous rate of change at x. | true |
A ball is rolling along the x-axis. Its position (in feet) at time x (in seconds) is given by f(x)=2√x. Find its instantaneous rate of change when x=9 seconds. | 1/3 ft/sec |