Question 1

Given the function $$f(x) = x^3$ - 3x^2$ + 4x - 7$$, which of the following is the correct set of critical points?

Accepted Answer

$$x = 1, x = 2$$

Question 2

The population $$P(t) (in $$thousands) of a fish species in a lake is modeled by $$P(t) = 200 \left(1 - e$$^{-0.1t}$$\$$right)^3$$. What is the population after 5 years?

Accepted Answer

$$P(5) \approx 38.7$$

Question 3

Which of the following best describes the Average Rate of Change (AROC) of a function $$f(x)$$ over the interval $$[a, b]$$?

Accepted Answer

$$\frac{f(b) - f(a)}{b - a}$$

Question 4

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

Accepted Answer

$$e^{x}(x^2$ + 2x)$

Question 5

Suppose the population of Texas (in millions) increased from 11.2 in 1970 to 14.2 in 1990. What is the average rate of change in population per year over this interval?

Accepted Answer

$$0.15$$ million/year

Question 6

Given $$f(x) = x^3$ - 3x^2$ + 4x - 7$$, on which interval is $$f(x)$$ increasing?

Accepted Answer

$$(2, \infty)$$

Question 7

Find the equation of the tangent line to $$f(x) = 4x^2$ + 7x at$$ $$x = 2.$$

Accepted Answer

$$y = 23x - 22$$

Question 8

Which of the following is the correct derivative of $$g(x) = 3x$$^{5/3}$$?

Accepted Answer

Question 9

Evaluate the definite integral $$\int_{0}^{1} (4x^2$ - 3\sqrt{x}) dx.$$

Accepted Answer

$$\frac{4}{3} - 2$$

Question 10

A temperature function $$T(t) = 40e$$^{-0.5t}$$ + 98$$ models the temperature (in $$^\circF) of a $$body after an injection. What is the temperature after 2 hours?

Accepted Answer

$$T(2) = 40e$$^{-1}$$ + 98 \approx 112.7$$

Question 11

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

Accepted Answer

$$\int_{a}^{b} f'(x) dx = f(b) - f(a)$$

Question 12

For the function $$f(x) = x^2$ e$$^{-x}$$, which of the following is the correct expression for f$$'(x)$$?

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

A company finds that the marginal cost $$C'(x) of $$producing $$x$$ units is $$C'(x) = 2x + 5. $$What is the total cost function $$C(x)$$, assuming $$C(0) = 0$$?

Accepted Answer

$$C(x) = x^2$ + 5x$$

Question 15

Which of the following intervals describes where the function $$f(x) = x^3$ - 3x^2$ + 4x - 7 is $$concave up?

Accepted Answer

$$x \u003c 1$$

Question 16

Evaluate $$\int_{1}^{2} x^2$ dx.$$

Accepted Answer

$$\frac{7}{3}$$

Question 17

If $$f(x) = x^2$ + 2x$$, what is $$\int_{0}^{1} f(x) dx$$?

Accepted Answer

$$\frac{5}{3}$$

Question 18

A fish population $$P(t) is $$modeled by $$P(t) = 200(1 - e$$^{-0.1t}$$)^3. $$What happens to the population as $$t 	o \infty$?

Accepted Answer

It approaches 200

Question 19

Which of the following is the correct expression for the instantaneous rate of change of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$f'(a)$$

Business Calculus Final Review: Key Concepts and Applications

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