Question 1

A company's total cost, in millions of dollars, is given by $C(t) = 220 + 8t^2 - t^3$ where $t$ is time in years. What is the marginal cost when $t = 5$? Round to the nearest hundredth.

Accepted Answer

$1.48$ million dollars per year

Question 2

The nationwide attendance per day for a certain motion picture can be approximated using the equation $A(y) = 15y^2 - y^3$, where $A$ is the attendance per day in thousands of persons, and $y$ is the number of months since the release of the film. Find and interpret the rate of change of the daily attendance after 4 months. Round to the nearest thousand.

Accepted Answer

$2,196$ thousand persons/day-month; the change in daily attendance is decreasing.

Question 3

The population of a small country increases according to the function $B(t) = 2,500,000e^{0.08t}$, where $t$ is measured in years. How many people will the country have after 3 years? Round to the nearest whole number.

Accepted Answer

$3,149,684$ people

Question 4

Five hundred grams of a particular radioactive substance decays according to the function $m(t) = 500e^{-0.0004t}$, where $t$ measures the time in years. When does the mass of the substance reach $10$ grams? Round to the nearest year.

Accepted Answer

$4608$ years

Question 5

Differentiate the function $f(x) = 8x^9$.

Accepted Answer

$72x^8$

Question 6

Differentiate $y = 5x^{4/3}$.

Accepted Answer

$\frac{20}{3}x^{1/3}$

Question 7

Differentiate $f(x) = -5x^4$.

Accepted Answer

$-20x^3$

Question 8

Differentiate $f(x) = 3x^4 - e^x$.

Accepted Answer

$12x^3 - e^x$

Question 9

Differentiate $y = 9x^4 - 3x^2$.

Accepted Answer

$36x^3 - 6x$

Question 10

Differentiate $y = \frac{x^2}{2x^2 + 1}$.

Accepted Answer

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

Question 11

The sales in thousands of a new type of product are given by $S(t) = 270 - 50t - 0.5t^2$, where $t$ represents time in years. Find the rate of change of sales at the time when $t = 7$. Round to the nearest tenth.

Accepted Answer

$-8.0$ thousand per year

Question 12

Find the tangent line to the graph of $f(x) = 3x^2 + 2x$ at the point $(0,3)$.

Accepted Answer

$y = 2x + 3$

Question 13

Find the derivative of the function $y = \ln(2x^2 - x)$.

Accepted Answer

$\frac{4x - 1}{2x^2 - x}$

Question 14

At what interest rate would a deposit of $20,000$ grow to $74,868$ in $30$ years with continuous compounding?

Accepted Answer

$4.5\%$

Question 15

Determine where the given function is concave up and where it is concave down. $g(x) = \frac{1}{3}x^3 - x^2 + 6$

Accepted Answer

Concave up on $(2, \infty)$, concave down on $(-\infty, 2)$

Question 16

The annual revenue and cost functions for a manufacturer of precision gauges are approximately $R(x) = 480x - 0.01x^2$ and $C(x) = 150x + 1,000,000$, where $x$ denotes the number of gauges made. What is the maximum annual profit, rounding to the nearest dollar?

Accepted Answer

$2,250,000$

Question 17

The equation below is an approximation to the total profit (in thousands of dollars) from the sale of $x$ hundred thousand tires. Find the number of tires that must be sold to maximize profit. $P(x) = -\frac{1}{2}x^2 + 65x - 100, x \geq 5$

Accepted Answer

$45$ hundred thousand

Question 18

For a dosage of cubic centimeters (cc) of a certain drug, assume that the resulting blood pressure $B$ is approximated by $B(x) = 0.3x^3 - 0.2x^2$. Find the dosage at which the resulting blood pressure is maximized. Round to two decimal places.

Accepted Answer

$0.67$ cc

Question 19

Find the absolute maximum and absolute minimum values of the function, if they exist, on the indicated interval. $f(x) = x^2 - 12x + 33$, $[2, 8]$

Accepted Answer

Absolute maximum is $5$; absolute minimum is $1$

Business Calculus: Key Concepts, Applications, and Problem-Solving Strategies

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