Question 1

Evaluate the following limit: $$\lim_{x 	o 0^-} \frac{\ln(1 + 2x)}{x^2$}$$

Accepted Answer

$$+\infty$$

Question 2

Find the local maximum and minimum values of $$f(x) = 2x^3$ - 3x^2$ - 12x + 24 on $$the interval $$[-3, 4].$$

Accepted Answer

Local max at $$x = -2$$, local min at $$x = 2$$

Question 3

Given $$f(x) = \frac{x^2$}{x^2$ + 1}$$ for $$x \in (-\infty, \infty)$$, what is the absolute maximum value of $$f(x)$$?

Accepted Answer

$$1 is $$approached but never reached$

Question 4

A company’s weekly price–demand equation is $$p = 500 - 0.4x$$ and cost equation is $$C(x) = 20000 + 20x. $$What is the number of smartphones $$x$$ that maximizes weekly revenue?

Accepted Answer

$$x = 625$$

Question 5

For the same company, what is the maximum weekly profit, and at what price should the phones be sold to achieve this profit?

Accepted Answer

Maximum profit is $$\ 78$$,125$$ at price p$$ = \$$250$$

Question 6

A parcel delivery service requires that the length plus girth of a box does not exceed 108 inches. For a box with square ends, what is the maximum possible volume?

Accepted Answer

$$\approx 13,824$$ cubic inches

Question 7

A fence is to be built to enclose a rectangular area of 800 square feet. Three sides use material costing $$\6$ per foot, and the fourth side uses material costing $$\$$18$$ per foot. What are the dimensions of the rectangle that minimize the total cost?

Accepted Answer

Width $$= 20 ft$$, Length $$= 40 ft ($$with the expensive side as length)

Question 8

Given $$C(t) = \frac{0.16t}{t^2$ + 4t + 4}$$, where $$t is $$hours after a drug is taken, at what time is the concentration maximized?

Accepted Answer

$$t = 2$$ hours

Question 9

Evaluate $$\lim_{x 	o \infty} \frac{x^2$}{e$$^{3x}$$}.$$

Accepted Answer

$$0$$

Question 10

For $$f(x) = 4x \ln(x) - 7x on$$ $$(0, \infty)$$, at what value of $$x$$ does $$f(x)$$ attain a local extremum?

Accepted Answer

$$x = e$$^{7/4}$$

Question 11

Find the limit: $$\lim_{x 	o -2} \frac{x^2$ + 2x}{x^2$ + x - 2}.$$

Accepted Answer

$$0$$

Question 12

For $$f(x) = x^3$ - 6x^2$ on $$[0, \infty)$$, what is the absolute minimum value?

Accepted Answer

$$-108$$

Question 13

A company’s revenue function is $$R(x) = x(500 - 0.4x). $$What is the maximum revenue?

Accepted Answer

$$\ 78$$,125$

Question 14

For the drug concentration $$C(t) = \frac{0.16t}{t^2$ + 4t + 4}$$, what is the maximum concentration?

Accepted Answer

$$0.04 mg/cc$$

Question 15

If $$f(x) = 2x^3$ - 3x^2$ - 12x + 24$$, which of the following is a critical point?

Accepted Answer

$$x = 2$$

Question 16

For the box with square ends and length plus girth not exceeding 108 inches, what is the relationship between the side length $$x$$ and the length $$L of $$the box?

Accepted Answer

$$L + 4x = 108$$

Question 17

A rectangular area of 800 square feet is to be enclosed with fencing. If three sides use material costing $$\6$ per foot and the fourth side uses $$\$$18$$ per foot, which side should be the expensive one to minimize cost?

Accepted Answer

The shorter side

Question 18

For $$f(x) = \frac{x^2$}{x^2$ + 1}$$, what is $$\lim_{x 	o \infty} f(x)$$?

Accepted Answer

$$1$$

Question 19

If $$f(x) = 4x \ln(x) - 7x$$, what is $$f'(x)$$?

Accepted Answer

$$4 \ln(x) + 4 - 7$$

Question 20

A company’s cost function is $$C(x) = 20000 + 20x. $$What is the marginal cost?

Accepted Answer

$$20$$

Business Calculus Assignment Guidance: Limits, Extrema, Optimization, and Applications

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