Q: Which of the following is the correct definition of the derivative using limits?

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Question 1

Which of the following best describes the derivative of a function $$f(x) at a $$point $$x = a$$?

Accepted Answer

The slope of the tangent line to the graph of $$f(x) at$$ $$x = a.$$

Question 2

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

Accepted Answer

$$f'(x) = 3x^2$ - 4x + 1$$

Question 3

A medication is administered to a patient and the amount in the bloodstream is modeled by $$A(t) = 100e$$^{-0.2t}$$, where t$ is in hours. What is the rate of change of the medication amount at t = 5$ hours?

Accepted Answer

$$A'(5) = -20e$$^{-1}$$

Question 4

Which of the following is the correct application of the chain rule for $$y = \sin(3x^2$)$$?

Accepted Answer

$$y' = \cos(3x^2$) \cdot 6x$$

Question 5

A company’s profit $$P(x) in $$thousands of dollars is given by $$P(x) = -2x^2$ + 40x - 100$$, where $$x is $$the number of units sold in thousands. What is the number of units sold that maximizes profit?

Accepted Answer

$$x = 10$$

Question 6

Evaluate the following limit: $$\lim_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

1

Question 7

Which of the following statements about the Mean Value Theorem is correct?

Accepted Answer

If $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b)$$, then there exists $$c \in (a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}.$$

Question 8

Find the derivative of $$y = \ln(x^2$ + 1).$$

Accepted Answer

$$y' = \frac{2x}{x^2$ + 1}$$

Question 9

A population of bacteria grows according to $$P(t) = 500e$$^{0.3t}$$. What is the instantaneous rate of growth at t = 2$ hours?

Accepted Answer

$$P'(2) = 1500e$$^{0.6}$$

Question 10

Which of the following is the correct derivative of $$y = e$$^{2x}$$ \cos x$$?

Accepted Answer

$$y' = e$$^{2x}$$(2\cos x - \sin x)$$

Question 11

Suppose $$f(x) is $$continuous and differentiable everywhere. If $$f'(x) = 0$$ for all $$x$$, what can you conclude about $$f(x)$$?

Accepted Answer

$$f(x) is a $$constant function.

Question 12

A ball is thrown upward with velocity $$v(t) = 20 - 9.8t. At $$what time does the ball reach its maximum height?

Accepted Answer

$$t = 2.04$$ seconds

Question 13

Which of the following is the correct definition of the derivative using limits?

Accepted Answer

$$f'(x) = \lim_{h 	o 0} \frac{f(x+h) - f(x)}{h}$$

Question 14

If $$y = 	an$$^{-1}$$(x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{1}{1 + x^2$}$$

Question 15

A tank is being filled with water such that the volume $$V(t) in $$liters at time $$t (in $$minutes) is $$V(t) = 5t^2$ + 3t. $$What is the rate at which water is entering the tank at $$t = 4$$ minutes?

Accepted Answer

$$V'(4) = 43$$ liters/minute

Question 16

Which of the following is the correct derivative of $$y = \ln(\sin x)$$?

Accepted Answer

$$y' = \cot x$$

Question 17

If $$f(x) = x^2$ e^x$$, what is $$f''(x)$$?

Accepted Answer

$$f''(x) = x^2$ e^x + 4x e^x + 2e^x$$

Question 18

A car’s position is given by $$s(t) = 4t^3$ - 3t^2$ + 2t + 5 (in $$meters, $$t in $$seconds). What is the car’s acceleration at $$t = 2$$ seconds?

Accepted Answer

$$a(2) = 36 m/s$^2$$

Question 19

Which of the following is the correct application of the product rule for $$y = x^2$ \sin x$$?

Accepted Answer

$$y' = 2x \sin x + x^2$ \cos x$$

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