Question 1

A car's position at time $$t (in $$seconds) is given by $$s(t) = 3t^2$ + 2t + 1. $$What is the instantaneous velocity of the car at $$t = 4$$ seconds?

Accepted Answer

$$v(4) = 3(2)(4) + 2 = 26$$

Question 2

Given $$f(x) = x^3$, use the limit definition of the derivative to find f$$'(x)$$. Which of the following is correct?

Accepted Answer

$$f'(x) = 3x^2$

Question 3

Which of the following best describes the meaning of $$f'(a)$$ for a differentiable function $$f$$?

Accepted Answer

$$f'(a) is $$the slope of the tangent line to $$f at$$ $$x = a$$

Question 4

If $$f(x) = 5x + 7$$, what is $$f'(x)$$ using the limit definition of the derivative?

Accepted Answer

$$f'(x) = 5$$

Question 5

Suppose the velocity of a particle is given by $$v(t) = 4t^3$ - 2t. $$What is the acceleration $$a(t) at$$ $$t = 2$$?

Accepted Answer

$$a(2) = 48 - 2 = 46$$

Question 6

Which of the following is the correct application of the product rule for differentiation?

Accepted Answer

$$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$

Question 7

Let $$f(x) = \sin(x^2$). $$Using the chain rule, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 2x \cos(x^2$)$$

Question 8

A medication's concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.5t}$$. What is the rate of change of concentration at t = 2$ hours?

Accepted Answer

$$C'(2) = -5e$$^{-1}$$

Question 9

Which of the following is the correct derivative of $$f(x) = 	an(x)$$?

Accepted Answer

$$f'(x) = \sec^2$(x)$$

Question 10

A company’s profit function is $$P(x) = 100x - 0.5x^2$, where x$ is the number of units sold. What is the marginal profit when x = 20$?

Accepted Answer

$$P'(20) = 100 - 0.5(2)(20) = 80$$

Question 11

Given $$f(x) = \frac{2x^2$ + 3x}{x}$$, find $$f'(x)$$ using the quotient rule.

Accepted Answer

$$f'(x) = \frac{(4x + 3)x - (2x^2$ + 3x) \cdot 1}{x^2$}$$

Question 12

If $$f(x) = \cos(x)$$, what is $$f''(x)$$?

Accepted Answer

$$f''(x) = -\cos(x)$$

Question 13

Which of the following is the correct derivative of $$f(x) = e$$^{2x}$$?

Accepted Answer

$$f'(x) = 2e$$^{2x}$$

Question 14

A ball is thrown upward and its height at time $$t is$$ $$s(t) = 20t - 5t^2$. What is the average velocity between t = 1$ and t = 3$ seconds?

Accepted Answer

$$v_{avg} = \frac{s(3) - s(1)}{3 - 1} = \frac{15 - 15}{2} = 0$$

Question 15

Which of the following is the correct derivative of $$f(x) = \sec(x)$$?

Accepted Answer

$$f'(x) = \sec(x)	an(x)$$

Question 16

If $$f(x) = x^2$ + 3x$$, what is the slope of the secant line between $$x = 1$$ and $$x = 4$$?

Accepted Answer

$$\frac{f(4) - f(1)}{4 - 1} = \frac{28 - 4}{3} = 8$$

Question 17

Which of the following is the correct derivative of $$f(x) = \cot(x)$$?

Accepted Answer

$$f'(x) = -\csc^2$(x)$$

Question 18

Suppose $$f(x) = 7. $$What is $$f'(x)$$?

Accepted Answer

$$f'(x) = 0$$

Question 19

If $$f(x) = x^5$ - 3x^2$ + 2$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = 5x^4$ - 6x$$

Calculus Midterm 2 Study Guide: Limits, Derivatives, and Derivative Rules

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