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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