Q: If $$f(x) = \sec(3 \ln x)$$, what is $$f'(x)$$?

$$3 \sec(3 \ln x) \tan(3 \ln x) / x$$

Question 1

Given the function $$f(x) = -4x^4$ - 3x^3$ + 36x^2$ + 12$$, which of the following is the $$x-$$coordinate of a relative maximum?

Accepted Answer

$$x = \frac{3}{4}$$

Question 2

Suppose a company wants to build an open-top box with a square base and a volume of $$32$$ cubic inches. If the cost of the base is $$0.50$$ per square inch and the sides cost $$0.30$$ per square inch, what is the minimum cost of materials (rounded to two decimal places)?

Accepted Answer

$$30.72$$

Question 3

Given $$f(x) = \frac{2x^2$ - 3x}{x^2$}$$, which of the following is the correct domain of $$f(x)$$?

Accepted Answer

$$x 
eq 0$$

Question 4

A vertical fence is to be built to enclose a rectangular garden of $$400 m$^2$$ using two types of fencing: vertical fencing at $$5.50$$ per meter and horizontal fencing at $$12.00$$ per meter. What dimensions minimize the cost?

Accepted Answer

Width $$= 20 m$$, Length $$= 20 m$$

Question 5

If $$y = 5x^2$ - 8x - x^3$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$10x - 8 - 3x^2$

Question 6

A population of bacteria grows according to $$P(t) = 1840e$$^{0.02t}$$. How many bacteria will there be after 40$ hours? (Round to the nearest whole number.)

Accepted Answer

$$4081$$

Question 7

Given $$f(x) = \ln(5x - 8) + \frac{1}{4} \ln(x + 4)$$, express $$f(x) as a $$single logarithm.

Accepted Answer

$$\ln\$$left((5x-8)(x+4)^{1/4}$$\right)$$

Question 8

If the surface area of a cylinder is increasing at a constant $$50 cm$^2/$$min, while the radius is increasing at $$3 cm/$$min, what is happening to the height when the height is $$20 cm $$and the radius is $$4 cm$$?

Accepted Answer

The height is decreasing at a rate of about $$1.99 cm $$per second.

Question 9

Given $$f(x) = e$$^{4x^2 + 3x}$$, find f$$'(x)$$.

Accepted Answer

$$(8x + 3)e$$^{4x^2 + 3x}$$

Question 10

A sample of Carbon-14 has $$73.8\% of $$its original content. About how many years old is the sample? (The half-life of Carbon-14 is $$5730$$ years.)

Accepted Answer

$$2130$$ years

Question 11

Given $$y = x \cos y - 4y^2$ + 9x + 5$$, find $$\frac{dy}{dx}$$ using implicit differentiation.

Accepted Answer

$$\frac{dy}{dx} = \frac{\cos y + 9}{1 - x \sin y + 8y}$$

Question 12

If $$f(x) = \frac{5x}{x+4}$$, what is the absolute maximum value of $$f(x) on $$the interval $$[1, 5]$$?

Accepted Answer

$$\frac{25}{9}$$

Question 13

Given $$f(x) = \ln(e$$^{4x^2 + 3x}$$)$$, find $$f'(x).$$

Accepted Answer

$$8x + 3$$

Question 14

A population is modeled by $$P(t) = 1840e$$^{0.02t}$$. How long will it take for the population to double?

Accepted Answer

$$69.3$$ hours

Question 15

Given $$y = 3x^2$ - 4x + x \ln x$$, find $$\frac{dy}{dx}.$$

Accepted Answer

$$6x - 4 + \ln x + 1$$

Question 16

If $$f(x) = \frac{2x^2$ - 3x}{x^2$}$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{4x^2$ - 3x - 4x^2$ + 6x}{x^4$}$$

Question 17

If $$f(x) = \sec(3 \ln x)$$, what is $$f'(x)$$?

Accepted Answer

$$3 \sec(3 \ln x) 	an(3 \ln x) / x$$

Question 18

A sample of element X decays exponentially. 20 years ago, a sample weighed 65 g; now it weighs 43 g. What is the half-life of X (rounded to the nearest year)?

Accepted Answer

$$27$$ years

Question 19

Given $$\int \frac{3x^2$ - 7}{4x + x^2$} dx$$, which substitution is most appropriate to simplify the integral?

Accepted Answer

$$u = 4x + x^2$

Question 20

If $$\log_3$ \left( \frac{x^2$ + y^3$}{81z^5$} \right) is to be $$expanded using properties of logarithms, which of the following is correct?

Accepted Answer

$$2 \log_3$ x + 3 \log_3$ y - 4 - 5 \log_3$ z$$

Business Calculus: Applications, Optimization, and Differentiation Study Guide

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