Question 1

Suppose the population of Texas (in millions) is given in the table below for selected years. What is the average rate of change of the population between 1980 and 2000?

Year: 1970, 1980, 1990, 2000, 2010
Population: 11.2, 14.2, 17.0, 20.9, 25.1

Accepted Answer

$\frac{20.9 - 14.2}{2000 - 1980} = 0.335$ million per year

Question 2

Given $f(x) = x^3 - 3x^2 + 4x - 7$, which of the following is a critical point of $f(x)$?

Accepted Answer

$x = 2$

Question 3

For the function $f(x) = e^x \sqrt{x} - x^2$, what is $f'(x)$?

Accepted Answer

$e^x \sqrt{x} + \frac{1}{2} e^x x^{-1/2} - 2x$

Question 4

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

Accepted Answer

The population approaches 200.

Question 5

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

Accepted Answer

$y = 23x - 14$

Question 6

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

Accepted Answer

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

Question 7

Let $f(x) = x^3 - 3x^2 + 4x - 7$. On which interval is $f(x)$ increasing?

Accepted Answer

$(2, \infty)$

Question 8

Evaluate the definite integral $\int_0^1 (4x^3 - 3\sqrt{x}) dx$.

Accepted Answer

$2.5$

Question 9

A temperature function is given by $T(t) = 40e^{-0.5t} + 60$ for $t \geq 0$. What is the long-term temperature as $t 	o \infty$?

Accepted Answer

$60$

Question 10

Suppose $C(x) = x \ln(x^2 + 1)$. What is $C'(x)$?

Accepted Answer

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

Question 11

If $f(x)$ is continuous on $[a, b]$ 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 12

A company’s profit function is $P(x) = -2x^2 + 8x + 10$. What is the maximum profit, and at what value of $x$ does it occur?

Accepted Answer

Maximum profit is $18$ at $x = 2$

Question 13

Evaluate $\int x e^{x^2} dx$.

Accepted Answer

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

Question 14

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

Accepted Answer

It states that differentiation and integration are inverse processes.

Question 15

Suppose $f(x) = x^4 - 4x^3 + 6x^2$. Find all inflection points.

Accepted Answer

$x = 2$

Question 16

A medication is injected into a patient and its concentration in the bloodstream is modeled by $C(t) = 100e^{-0.2t}$. What is the instantaneous rate of change at $t = 5$?

Accepted Answer

$-20e^{-1}$ mg/unit$

Question 17

If $f(x) = 3x^5 + x^3$, what is $f''(x)$?

Accepted Answer

$60x^3 + 6x$

Question 18

Evaluate $\int_1^e \frac{1}{x} dx$.

Accepted Answer

$1$

Question 19

Suppose the cost function for producing $x$ units is $C(x) = 5x^2 + 2x + 7$. What is the marginal cost when $x = 3$?

Accepted Answer

$32$

Business Calculus Final Review: Key Concepts, Applications, and Practice Problems

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