Question 1

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

Accepted Answer

The instantaneous rate of change of $$f(x) at$$ $$x=a$$, given by $$f'(a) = \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 2

Given $$f(x) = x^3$ \ln(x)$$, what is $$f'(x)$$?

Accepted Answer

$$3x^2$$ \ln(x) + $$x^2$$

Question 3

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

Accepted Answer

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

Question 4

Evaluate $$\int (2x^3$ - 5x^2$ + 4) dx.$$

Accepted Answer

$$\frac{1}{4}x^4$ - \frac{5}{3}x^3$ + 4x + C$$

Question 5

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

Accepted Answer

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

Question 6

Which of the following integrals requires $$u-$$substitution to solve?

Accepted Answer

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

Question 7

Find the area between the curves $$f(x) = x^2$ and g(x) = x$ from x=0$ to x=1$.

Accepted Answer

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

Question 8

Which of the following is the correct statement of the Fundamental Theorem of Calculus?

Accepted Answer

$$\int_a^b f(x) dx = F(b) - F(a)$$, where $$F(x) is $$any antiderivative of $$f(x)$$

Question 9

A medication's concentration in the bloodstream is modeled by $$C(t) = 5e$$^{-0.2t}$$. What is the instantaneous rate of change of concentration at t=3$ hours?

Accepted Answer

Question 10

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

Accepted Answer

$$\frac{\cos(x)}{\sin(x)}$$

Question 11

Evaluate $$\int \frac{1}{x \ln(x)} dx$$ using $$u-$$substitution.

Accepted Answer

$$\ln|\ln(x)| + C$$

Question 12

A tank is being filled with water at a rate given by $$R(t) = 3t^2$ liters per minute. How much water is added between t=1$ and t=4$ minutes?

Accepted Answer

$$[t^3$]_1^4$ = 64 - 1 = 63$$ liters

Question 13

Which of the following is the correct formula for the area between two curves $$f(x)$$ and $$g(x)$$ from $$x=a to$$ $$x=b$$?

Accepted Answer

$$\int_a^b [f(x) - g(x)] dx$$

Question 14

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

Accepted Answer

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

Question 15

Evaluate $$\$$int_0$$^\pi \sin(x) dx.$$

Accepted Answer

$$0$$

Question 16

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

Accepted Answer

$$2e^{2x}$$

Question 17

A population of bacteria grows according to $$P(t) = 100e$$^{0.05t}$$. What is the rate of growth at t=10$ hours?

Accepted Answer

$$0.05 \cdot 100e$$^{0.5}$$

Calculus Review: Differentiation, Integration, and Area Between Curves

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