Q: A function $$f(x)$$ has a local maximum at $$x = c if$$:

$$f'(c) = 0$$ \,\text{and}\, f''(c) \u003c 0$

Q: A company wants to minimize the cost $$C(x) of $$producing $$x$$ units. If $$C(x) is $$differentiable, what must be true at a local minimum?

$$C'(x) = 0$$ \,\text{and}\, C''(x) \u003e 0$

Question 1

Which of the following is the correct derivative of $$y = 	an$$^{-1}$$(x)$$?

Accepted Answer

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

Question 2

Suppose $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). $$According to the Mean Value Theorem, which statement must be true?

Accepted Answer

There exists $$c in$$ $$(a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}$$

Question 3

Given $$y = \ln(x^2$ + 1)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 4

A function $$f(x)$$ has a local maximum at $$x = c if$$:

Accepted Answer

$$f'(c) = 0$$ \,	ext{and}\, f''(c) \u003c 0$

Question 5

If $$y = \sec$$^{-1}$$(x)$$, what is $$\frac{dy}{dx}$$ for $$|x| \u003e 1$$?

Accepted Answer

$$\frac{1}{|x| \sqrt{x^2$ - 1}}$$

Question 6

Which of the following is the correct derivative of $$y = \ln|x|$$?

Accepted Answer

$$\frac{1}{x}$$

Question 7

A pharmaceutical company wants to maximize the profit $$P(x)$$ from selling $$x$$ units of a drug. If $$P(x) is $$continuous and differentiable, which method should they use to find the maximum profit on $$[a, b]$$?

Accepted Answer

Evaluate $$P(x) at $$the endpoints and at any critical points in $$(a, b)$$, then choose the largest value.

Question 8

If $$y = 	an$$^{-1}$$(x)$$, what is $$\frac{d^2$y}{dx^2$}$$?

Accepted Answer

$$\frac{-2x}{(1 + x^2$)^2$$}$

Question 9

Which statement is true about the function $$f(x) if$$ $$f'(x) \u003e 0$$ for all $$x in$$ $$(a, b)$$?

Accepted Answer

$$f(x) is $$increasing on $$(a, b)$$

Question 10

A biologist models the population of a species by $$P(t) = 1000e$$^{0.05t}$$. What is the instantaneous rate of change of the population at t = 10$?

Accepted Answer

$$P'(10) = 50e$$^{0.5}$$

Question 11

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

Accepted Answer

$$\cot x$$

Question 12

If $$f(x) is $$continuous on $$[a, b]$$, which theorem guarantees that $$f(x)$$ attains both an absolute maximum and minimum on $$[a, b]$$?

Accepted Answer

Extreme Value Theorem

Question 13

A company wants to minimize the cost $$C(x) of $$producing $$x$$ units. If $$C(x) is $$differentiable, what must be true at a local minimum?

Accepted Answer

$$C'(x) = 0$$ \,	ext{and}\, C''(x) \u003e 0$

Question 14

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

Accepted Answer

$$\frac{1}{\sin x \cos x}$$

Question 15

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

Accepted Answer

$$\frac{1}{\sqrt{1 - x^2$}}$$

Question 16

A medical researcher models the concentration of a drug in the bloodstream by $$C(t) = 5\ln(t + 1). $$What is the rate of change of concentration at $$t = 4$$?

Accepted Answer

$$C'(4) = 1$$

Question 17

Which of the following is the correct derivative of $$y = 	an$$^{-1}$$(2x)$$?

Accepted Answer

$$\frac{2}{1 + 4x^2$}$$

Question 18

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

Accepted Answer

$$\frac{2}{x}$$

Question 19

If $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) = f(b)$$, which theorem guarantees there is $$c in$$ $$(a, b)$$ such that $$f'(c) = 0$$?

Accepted Answer

Rolle's Theorem

Calculus I: Differentiation, Applications, and Extrema – Exam Review Study Notes

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