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

$$\displaystyle \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Q: Which of the following limits does NOT exist?

$$\displaystyle \lim_{x \to 0} \frac{1}{x}$$

Question 1

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

Accepted Answer

$$\displaystyle \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 2

A pharmaceutical company models the concentration $$C(t) of a $$drug in the bloodstream (in mg/L) as $$C(t) = 10e$$^{-0.5t}$$, where t$ is in hours. What is the rate of change of the drug concentration at t = 2$ hours?

Accepted Answer

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

Question 3

Which of the following statements about the derivative is TRUE?

Accepted Answer

The derivative of a function at a point gives the slope of the tangent line at that point.

Question 4

Given $$f(x) = x^3$ - 6x^2$ + 9x + 1$$, at what value of $$x$$ does $$f(x)$$ have a local minimum?

Accepted Answer

$$x = 2$$

Question 5

A company’s profit function is $$P(x) = -2x^2$ + 40x - 100$$, where $$x is $$the number of units sold. What is the maximum profit, and how many units should be sold to achieve it?

Accepted Answer

Maximum profit is $$300 at$$ $$x = 10$$

Question 6

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

Accepted Answer

$$f''(x) = -\frac{1}{x^2$}$$

Question 7

Which of the following is the correct derivative of $$f(x) = \sin(x) \cdot e^x$$?

Accepted Answer

$$f'(x) = \cos(x) \cdot e^x + \sin(x) \cdot e^x$$

Question 8

A population of bacteria grows according to $$P(t) = 500e$$^{0.2t}$$, where t$ is in hours. What is the instantaneous rate of growth at t = 5$ hours?

Accepted Answer

$$P'(5) = 100e$$^{1}$$

Question 9

Which of the following limits does NOT exist?

Accepted Answer

$$\displaystyle \lim_{x 	o 0} \frac{1}{x}$$

Question 10

A company wants to minimize the cost $$C(x) = x^2$ - 12x + 36$$ for producing $$x$$ units. What is the minimum cost and at how many units is it achieved?

Accepted Answer

Minimum cost is $$0 at$$ $$x = 6$$

Question 11

If $$y = \frac{2x + 1}{x - 3}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{7}{(x-3)^2$}$$

Question 12

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

Accepted Answer

$$\frac{dy}{dx} = \cos(3x^2$) \cdot 6x$$

Question 13

A company’s revenue function is $$R(x) = 50x - x^2$, and its cost function is C(x) = 10x + 20$. What is the marginal profit when x = 10$ units?

Accepted Answer

Marginal profit is $$30$$

Question 14

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

Accepted Answer

$$f''(x) = 4e$$^{2x}$$

Question 15

Suppose the position of a car is given by $$s(t) = 4t^3$ - 3t^2$ + 2t$$, where $$t is in $$hours and $$s in $$miles. What is the car’s acceleration at $$t = 2$$ hours?

Accepted Answer

Acceleration is $$36$$ miles/hour$$^2$$

Question 16

Which of the following is the correct derivative of $$f(x) = \ln(x^2$ + 1)$$?

Accepted Answer

$$f'(x) = \frac{2x}{x^2$ + 1}$$

Question 17

A company’s cost to produce $$x$$ items is $$C(x) = 100 + 5x + 0.1x^2$. What is the marginal cost when x = 20$?

Accepted Answer

Marginal cost is $$9$$

Question 18

Which of the following best describes the meaning of a critical point for a function $$f(x)$$?

Accepted Answer

A point where $$f'(x) = 0 or$$ $$f'(x)$$ does not exist

Question 19

A company wants to maximize its revenue, given by $$R(x) = 100x - 2x^2$. What is the number of units x$ that maximizes revenue?

Accepted Answer

$$x = 25$$

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