Q: Given the piecewise graph below, which formula correctly represents the function for $-1 < x \\leq 2$ and $1 < x \\leq 3$?

$y = \\begin{cases} x + 3, \\ -1 < x \\leq 2 \\ 3x - 1, \\ 1 < x \\leq 3 \\end{cases}$

Q: Apply the transformations on the graph of $p(x) = \\sqrt{x}$ into the graph of $h(x) = 4\\sqrt{x-3} - 6$. Which sequence of transformations is correct?

Vertical stretch by a factor of $4$, right shift by $3$ units and shift down by $6$ units.

Q: Write the logarithm $\\log_5{120}$ in terms of the natural logarithm and evaluate its value to four decimal places.

$\\frac{\\ln{120}}{\\ln{5}} \\approx 2.9746$

Q: Use a graphing calculator to decide if the functions $f(x) = \\ln{\\left(\\frac{x+5}{x-5}\\right)}$ and $g(x) = \\ln{\\left(\\frac{x-5}{x+5}\\right)}$ differ by a constant.

The functions differ by a constant.

Q: A coffee shop charges for Wi-Fi access based on the time used. The graph of the charge function $C$ for $0 \\leq t \\leq 40$ is given below. Write the intervals of continuity of $C$.

$(0, 10) \\ (10, 20) \\ (20, 30) \\ (30, 40)$

Q: Given a function $H(t)$ that models the height (in centimeters) of a plant $t$ days after it was planted, estimate $H'(90)$ from the provided graph. What does this derivative indicate?

$H'(90) = \\frac{1}{5}$; At $t = 90$ days, the plant is growing at a rate of $\\frac{1}{5}$ cm/day.

Q: Determine the derivative of $h(x) = \\frac{x^2}{x^2 + 5}$.

$h'(x) = \\frac{10x}{(x^2 + 5)^2}$

Question 1

A company manufactures a product with cost function $C(x) = -0.03x^2 + 70x + 75$ and sells each unit at a price $p(x) = 80$. What is the profit function $P(x)$?

Accepted Answer

$P(x) = -0.03x^2 + 10x + 5$

Question 2

Given the piecewise graph below, which formula correctly represents the function for $-1 < x \leq 2$ and $1 < x \leq 3$?

Accepted Answer

$y = \begin{cases} x + 3, \ -1 < x \leq 2 \ 3x - 1, \ 1 < x \leq 3 \end{cases}$

Question 3

For the function $g(x) = 3x - 6 + 2$, what is the largest interval on which $g(x)$ is increasing?

Accepted Answer

$[2, \infty)$

Question 4

A telecommunications company must install a fiber optic cable from a central office to a new development 3 miles away on the opposite side of a lake that is 1200 ft wide. The cost of laying cable under the lake is $250/ft$ and $150/ft$ along the road. What is the least expensive point for the cable to reach the development if it is less than or greater than 3000 ft from the starting point along the road?

Accepted Answer

$1200$ ft

Question 5

Apply the transformations on the graph of $p(x) = \sqrt{x}$ into the graph of $h(x) = 4\sqrt{x-3} - 6$. Which sequence of transformations is correct?

Accepted Answer

Vertical stretch by a factor of $4$, right shift by $3$ units and shift down by $6$ units.

Question 6

Simplify the composite function $a(b(c(x)))$ given $a(x) = -x^3$, $b(x) = \cos(x)$, $c(x) = \sqrt{x}$.

Accepted Answer

$-\cos^3{\sqrt{x}}$

Question 7

Write the logarithm $\log_5{120}$ in terms of the natural logarithm and evaluate its value to four decimal places.

Accepted Answer

$\frac{\ln{120}}{\ln{5}} \approx 2.9746$

Question 8

Use a graphing calculator to decide if the functions $f(x) = \ln{\left(\frac{x+5}{x-5}\right)}$ and $g(x) = \ln{\left(\frac{x-5}{x+5}\right)}$ differ by a constant.

Accepted Answer

The functions differ by a constant.

Question 9

Given $\log_{10}{p} = 0.42$ and $\log_{10}{r} = 0.91$, find the value of $\log_{10}{\sqrt{\frac{p}{r^3}}}$.

Accepted Answer

$-0.1075$

Question 10

For the function $f(x) = -5x + 3$, find the inverse function $f^{-1}(x)$.

Accepted Answer

$f^{-1}(x) = \frac{x-3}{-5}$

Question 11

A runner's position along a track is recorded at various times during a race. The data is presented in a table. Find the runner's average velocity between the time interval of $0$ to $2$ seconds if $s(0) = 0$, $s(1) = 20$, $s(2) = 28$.

Accepted Answer

$4$ m/s

Question 12

Find the value of $\lim_{x \to 0} f(x)$ given that $\lim_{x \to 3} 2x \lim_{x \to 0} f(x) = -12$.

Accepted Answer

$-2$

Question 13

A coffee shop charges for Wi-Fi access based on the time used. The graph of the charge function $C$ for $0 \leq t \leq 40$ is given below. Write the intervals of continuity of $C$.

Accepted Answer

$(0, 10) \ (10, 20) \ (20, 30) \ (30, 40)$

Question 14

The cost function $C(x)$ and the selling price $p(x)$ are given below. Determine the profit function $P(x)$. $C(x) = -0.03x^2 + 70x + 75$, $p(x) = 80 - 0.02x$.

Accepted Answer

$P(x) = -0.03x^2 + 10x - 75$

Question 15

Determine the derivative of the function $f(x) = 2x^3$ using the definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$.

Accepted Answer

$f'(x) = 6x^2$

Question 16

Given a function $H(t)$ that models the height (in centimeters) of a plant $t$ days after it was planted, estimate $H'(90)$ from the provided graph. What does this derivative indicate?

Accepted Answer

$H'(90) = \frac{1}{5}$; At $t = 90$ days, the plant is growing at a rate of $\frac{1}{5}$ cm/day.

Question 17

Determine whether the following piecewise function is differentiable at $x = 0$: $g(x) = \begin{cases} 4x + \sin{x}, \ x < 0 \ 2x^2, \ x \geq 0 \end{cases}$.

Accepted Answer

No

Question 18

The graphs of $f$ and $g$ are shown. Find $K'(6)$ if $K = 5f - 2g$.

Accepted Answer

$\frac{5}{4}$

Question 19

Calculate the fourth derivative of the function $f(x) = 5\cos{x}$.

Accepted Answer

$5\cos{x}$

Question 20

Determine the derivative of $h(x) = \frac{x^2}{x^2 + 5}$.

Accepted Answer

$h'(x) = \frac{10x}{(x^2 + 5)^2}$

Question 21

Given that $f(-1) = -1$, $f'(-1) = -2$, $g(-1) = -1$, and $g'(-1) = 5$, find the derivative of $f^4(x)g(x)$ at $x = -1$.

Accepted Answer

$-8$

Business Calculus Study Guide: Functions, Derivatives, Applications, and Problem Solving

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