Question 1

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

Accepted Answer

The instantaneous rate of change of $$f(x)$$ with respect to $$x at$$ $$x = a.$$

Question 2

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

Accepted Answer

$$1$$

Question 3

If $$f(x) = x^3$ - 3x^2$ + 2x$$, at which $$x-$$value does $$f(x)$$ have a local minimum?

Accepted Answer

$$x = 2$$

Question 4

Which of the following is the correct expression for the Fundamental Theorem of Calculus?

Accepted Answer

$$\int_a^b f'(x) dx = f(b) - f(a)$$

Question 5

A car's velocity (in m/s) at time $$t (in $$seconds) is given by $$v(t) = 3t^2$. What is the total distance traveled from t = 0$ to t = 2$?

Accepted Answer

$$12$$

Question 6

Which of the following functions is continuous for all real numbers?

Accepted Answer

$$f(x) = e^x$$

Question 7

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

Accepted Answer

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

Question 8

A medication is administered at a rate of $$R(t) = 5e$$^{-0.2t}$$ mg/hour. What is the total amount administered from t = 0$ to t = 5$ hours?

Accepted Answer

$$\$$int_0$$^5 5e$$^{-0.2t}$$ dt$$

Question 9

Which of the following statements about the curve $$y = x^3$ - 3x is $$true?

Accepted Answer

The curve has two points of inflection.

Question 10

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

Accepted Answer

$$2x \cos(x^2$)$$

Question 11

A spherical balloon is being inflated so that its volume increases at a rate of $$10 	ext{ cm}^3/	ext{min}. $$What is the rate of change of the radius when the radius is $$5 cm$$? (Recall $$V = \frac{4}{3}\pi r^3$)

Accepted Answer

$$\frac{3}{100\pi} cm/$$min

Question 12

Which of the following integrals gives the area between $$y = x^2$ and y = x$ from x = 0$ to x = 1$?

Accepted Answer

$$\$$int_0$$^1 (x - x^2$) dx$$

Question 13

If $$f(x) = e$$^{2x}$$, what is $$\int f(x) dx$$?

Accepted Answer

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

Question 14

Which of the following is the best interpretation of $$\$$int_0$$^T v(t) dt in a $$real-world context where $$v(t) is $$velocity?

Accepted Answer

The total distance traveled over $$[0, T].$$

Question 15

Let $$f(x) = x^4$ - 4x^2$. How many relative extrema does f(x$$)$$ have?

Accepted Answer

2

Question 16

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

Accepted Answer

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

Question 17

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

Accepted Answer

Mean Value Theorem

Question 18

If $$\$$int_0$$^2 f(x) dx = 5$$ and $$\$$int_2$$^5 f(x) dx = -3$$, what is $$\$$int_0$$^5 f(x) dx$$?

Accepted Answer

$$2$$

Question 19

Which of the following is the best definition of the definite integral $$\int_a^b f(x) dx$$?

Accepted Answer

The area under the curve $$y = f(x)$$ from $$x = a to$$ $$x = b$$, accounting for sign.

CLEP Calculus Official Study Guide: Key Concepts and Practice Questions

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