Question 1

Let $$f(x) = x^2$ e$$^{-x}$$. Which of the following is the correct expression for f$$'(x)$$?

Accepted Answer

$$2x e$$^{-x}$$ - x^2$ e$$^{-x}$$

Question 2

Evaluate the definite integral $$\$$int_0$$^1 x^2$ e$$^{-x}$$ dx$$ using integration by parts.

Accepted Answer

$$2 - 3e$$^{-1}$$

Question 3

Which of the following is the correct formula for integration by parts?

Accepted Answer

$$\int u dv = uv - \int v du$$

Question 4

Suppose $$f(x) is $$continuous on $$[a, b]. $$Which theorem guarantees that $$f(x)$$ attains a maximum and a minimum on $$[a, b]$$?

Accepted Answer

Extreme Value Theorem

Question 5

Given $$\$$int_0$$^1 x^2$ e$$^{-x}$$ dx$$, which substitution is most appropriate for integration by parts?

Accepted Answer

$$u = x^2$, dv$$ = $$e^{-x} dx$

Question 6

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

Accepted Answer

$$2e^{-x} - 4x$$ $$e^{-x}$$ + $$x^2$$ $$e^{-x}$$

Question 7

Which of the following best describes the Fundamental Theorem of Calculus?

Accepted Answer

It relates differentiation and integration, stating that if $$F(x) is an $$antiderivative of $$f(x)$$, then $$\int_a^b f(x) dx = F(b) - F(a).$$

Question 8

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 = 0$ and t = 2$ minutes?

Accepted Answer

$$8$$ liters

Question 9

Which of the following is the correct evaluation of $$\$$int_0$$^1 x^2$ e$$^{-x}$$ dx$$ using integration by parts twice?

Accepted Answer

$$2 - 3e$$^{-1}$$

Question 10

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

Accepted Answer

Mean Value Theorem

Question 11

Let $$f(x) = x^2$ e$$^{-x}$$. What is the value of f(0$$)$$?

Accepted Answer

$$0$$

Question 12

Which of the following is a correct application of the chain rule?

Accepted Answer

If $$y = (x^2$ + 1)^3$, then $$\frac{dy}{dx} = $$3(x^2$$ + $$1)^2$$ \cdot 2x$$

Question 13

A medication is administered at a rate $$R(t) = 5e$$^{-0.5t}$$ mg/hour. How much medication is administered in the first 2 hours?

Accepted Answer

$$5(1 - e$$^{-1}$$) mg$$

Question 14

Which of the following statements is true about the function $$f(x) = x^2$ e$$^{-x}$$?

Accepted Answer

It has a maximum at $$x = 2$$

Calculus Exam Study Guidance: Integrals, Vector Calculus, and Theorems

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