Q: Which of the following is the correct application of L'Hospital's Rule for $$\lim\limits_{x \to 0} \frac{\sin x}{x}$$?

$$\lim\limits_{x \to 0} \frac{\cos x}{1} = 1$$

Question 1

Evaluate the following limit: $$\lim\limits_{x 	o 2} \frac{x^2$ - 4}{x - 2}$$

Accepted Answer

$$4$$

Question 2

Which of the following best describes the derivative $$f'(x) at a $$point $$x = a$$?

Accepted Answer

The instantaneous rate of change of $$f(x) at$$ $$x = a$$

Question 3

Given the price-demand equation $$p = 600 - 0.04x$$, where $$p is $$price and $$x is $$the number of TVs sold, what is the revenue function $$R(x)$$?

Accepted Answer

$$R(x) = 600x - 0.04x^2$

Question 4

Find the derivative of $$f(x) = x^3$ + 2x^2$ - 5x + 1.$$

Accepted Answer

$$f'(x) = 3x^2$ + 4x - 5$$

Question 5

Suppose the marginal profit function for a product is $$P'(x) = 100 - 0.2x. $$What is the total profit function $$P(x)$$, assuming fixed costs are zero?

Accepted Answer

$$P(x) = 100x - 0.2x^2$ + C$$

Question 6

If $$f(x) = e$$^{2x}$$, what is f$$'(x)$$?

Accepted Answer

$$2e^{2x}$$

Question 7

A company’s cost function is $$C(x) = 5000 + 20x. $$What is the marginal cost?

Accepted Answer

$$20$$

Question 8

Which of the following is the correct formula for the present value $$PV of an $$amount $$A to be $$received in $$t$$ years at interest rate $$r$$, compounded continuously?

Accepted Answer

$$PV = Ae$$^{-rt}$$

Question 9

If $$\$$int_0$$^2 (3x^2$ + 2x) dx is $$evaluated, what is the result?

Accepted Answer

$$16$$

Question 10

A business invests $$2000$$ per year into an account earning $$5\%$$ interest, compounded annually, for $$10$$ years. Which formula gives the future value $$FV$$?

Accepted Answer

$$FV = 2000 \left( \frac{(1.05$$)^{10}$$ - 1}{0.05} \right)$$

Question 11

Which of the following best describes the Fundamental Theorem of Calculus?

Accepted Answer

It relates differentiation and integration as inverse processes.

Question 12

If $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

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

Question 13

A loan of $$P$$ dollars is repaid at an annual interest rate $$r$$ compounded continuously for $$t$$ years. What is the formula for the amount owed $$A$$?

Accepted Answer

$$A = Pe$$^{rt}$$

Question 14

If $$f(x) = \sqrt{x}$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{1}{2\sqrt{x}}$$

Question 15

Suppose the demand function for a product is $$p = 100 - 2x. $$What is the marginal revenue when $$x = 10$$?

Accepted Answer

$$100 - 4x$$

Question 16

Which of the following is the correct formula for the future value $$FV of a $$continuous income stream $$R(t)$$ invested at rate $$r$$ for $$T$$ years?

Accepted Answer

$$FV = \$$int_0$$^T R(t) e$$^{r(T-t)}$$ dt$$

Question 17

If $$f(x) = \frac{1}{x}$$, what is $$f'(x)$$?

Accepted Answer

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

Question 18

A company’s revenue function is $$R(x) = 50x - x^2$. What production level maximizes revenue?

Accepted Answer

$$x = 25$$

Question 19

If $$\$$int_1$$^e \frac{1}{x} dx is $$evaluated, what is the result?

Accepted Answer

$$1$$

Question 20

Which of the following is the correct application of L'Hospital's Rule for $$\lim\limits_{x 	o 0} \frac{\sin x}{x}$$?

Accepted Answer

$$\lim\limits_{x 	o 0} \frac{\cos x}{1} = 1$$

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