Q: A couple wants to invest so that their money grows to $$10,000 in$$ $$10$$ years at $$9\%$$ interest compounded continuously. What should the initial investment be?

$$10,000 e$$^{-0.09 \times 10}$$

Q: The monthly cost for producing $$x$$ units is $$C(x) = -0.035x^2$ + 40x + 25. $$What is the marginal cost when $$600$$ units are produced?

$$-0.035 \times 2 \times 600 + 40 = -42 + 40 = -2$$

Q: A continuous money flow of $$3000$$ per year is invested for $$20$$ years at $$8\%$$ interest compounded continuously. What is the accumulated present value?

$$\frac{3000}{0.08} (1 - e$$^{-0.08 \times 20}$$)$$

Question 1

Evaluate the limit $$\lim_{x 	o -5} (x^2$ - 25x + 5). $$Which of the following is correct?

Accepted Answer

$$155$$

Question 2

A manufacturer’s monthly revenue is given by $$R(x) = x^3$ + 4x^2$ + 160x. $$What is the marginal revenue when 70 units are sold?

Accepted Answer

$$15,660$$

Question 3

If the position of an object is $$s(t) = t^4$ + t^2$ + 3t$$, what is the average velocity from $$t = 0 to$$ $$t = 2$$ minutes?

Accepted Answer

$$14$$

Question 4

Differentiate $$f(x) = 3x^2$ - 7x + 9. $$What is $$f'(x)$$?

Accepted Answer

$$6x - 7$$

Question 5

Find the equation of the tangent line to $$y = x^3$ at the point (-2$$, -8)$$.

Accepted Answer

$$y = -12x - 32$$

Question 6

A certain radioactive element decays at a rate of $$7\%$$ per day. If the initial amount is $$800 g$$, what is the amount remaining after $$20$$ days?

Accepted Answer

$$800 e$$^{-0.07 	imes 20}$$

Question 7

For the function $$f(x) = -x^3$ + 3x - 2$$, which of the following is a relative maximum point?

Accepted Answer

$$(1, 0)$$

Question 8

A couple wants to invest so that their money grows to $$10,000 in$$ $$10$$ years at $$9\%$$ interest compounded continuously. What should the initial investment be?

Accepted Answer

$$10,000 e$$^{-0.09 	imes 10}$$

Question 9

Given $$R(x) = -0.4x^2$ + 40x + 60$$ and $$C(x) = 0.1x^2$ + 25x + 70$$, for $$x$$ units of a product, what is the number of units that must be produced and sold to maximize profit?

Accepted Answer

$$20$$

Question 10

Find the area of the region bounded by $$y = 4x^2$ and y = 8x$.

Accepted Answer

$$\frac{32}{3}$$

Question 11

If $$f(x, y) = 3xy^2$ - 4x^3$ + e$$^{2y}$$, what is f$$_{xy}$$?

Accepted Answer

$$6y$$

Question 12

For the demand function $$D(x) = x - 92$$ and supply function $$S(x) = x^2$ - 7x + 4$$, what is the equilibrium quantity?

Accepted Answer

$$89$$

Question 13

The monthly cost for producing $$x$$ units is $$C(x) = -0.035x^2$ + 40x + 25. $$What is the marginal cost when $$600$$ units are produced?

Accepted Answer

$$-0.035 	imes 2 	imes 600 + 40 = -42 + 40 = -2$$

Question 14

If $$y = e$$^{x^2}$$, what is $$\frac{$$d^2y$$}{$$dx^2$$}$$?

Accepted Answer

$$2e^{x^2}$$ + $$4x^2$$ $$e^{x^2}$$

Question 15

A gourmet coffee chain starts with 20 franchises in 1980 and grows at 8% per year. After how many years will the number of franchises double?

Accepted Answer

$$\frac{\ln 2}{0.08}$$

Question 16

Find the integral $$\int x^2$ \ln x \, dx.$$

Accepted Answer

$$\frac{x^3$}{3} \ln x - \frac{x^3$}{9} + C$$

Question 17

For the function $$f(x) = x^4$ - 2x^2$ + 1$$, which of the following is a point of inflection?

Accepted Answer

$$(0, 1)$$

Question 18

A continuous money flow of $$3000$$ per year is invested for $$20$$ years at $$8\%$$ interest compounded continuously. What is the accumulated present value?

Accepted Answer

$$\frac{3000}{0.08} (1 - e$$^{-0.08 	imes 20}$$)$$

Question 19

Find the area under the graph of $$y = 2x - 13$$ from $$x = 2 to$$ $$x = 3.$$

Accepted Answer

$$-10.5$$

Question 20

If the sales growth rate is $$S'(t) = 30e$$^{1.3t}$$, what is the accumulated sales from the beginning of the 3rd day to the end of the 6th day?

Accepted Answer

$$\int_{3}^{6} 30e$$^{1.3t}$$ dt$$

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