Q: Suppose you invest $$1,000 at an annual interest rate of 6% compounded quarterly. What is the amount after 5 years? Use the compound interest formula A = P$$\left(1 + \frac{r}{n}\$$right)^{nt}$.

$$A = 1000\left(1 + \frac{0.06}{4}\$$right)^{4 \times 5}$$

Q: Given $$C(x) = 100 + 5x + 0.1x^2$, what is the marginal cost when x = 10$?

$$C'(10) = 5 + 0.2 \times 10 = 7$$

Question 1

A company’s revenue in thousands of dollars is given by $$R(t) = 50e$$^{0.04t}$$, where t$ is the number of years since 2020. What is the average rate of change of revenue between t = 2$ and t = 5$?

Accepted Answer

$$\frac{R(5) - R(2)}{5 - 2} = \frac{50e$$^{0.2}$$ - 50e$$^{0.08}$$}{3}$$

Question 2

A medication degrades in the bloodstream according to $$C(t) = 100e$$^{-0.3t}$$, where C$ is the concentration in mg/L and t$ is hours after administration. How long will it take for the concentration to drop to 20 mg/L?

Accepted Answer

$$t = \frac{\ln(0.2)}{-0.3}$$

Question 3

Given the table below, which function is linear?

$$x$$: 1, 2, 3, 4
$$y$$: 5, 8, 11, 14

Accepted Answer

The function is linear because the difference between consecutive $$y-$$values is constant.

Question 4

Suppose you invest $$1,000 at an annual interest rate of 6% compounded quarterly. What is the amount after 5 years? Use the compound interest formula A = P$$\left(1 + \frac{r}{n}\$$right)^{nt}$.

Accepted Answer

$$A = 1000\left(1 + \frac{0.06}{4}\$$right)^{4 	imes 5}$$

Question 5

If $$A(10) = 3222$$ for an investment, what does this mean in context?

Accepted Answer

After 10 years, the investment is worth $3,222.

Question 6

Solve for $$x$$: $$2e^{3x} = 10$.

Accepted Answer

$$x = \frac{\ln(5)}{3}$$

Question 7

Given $$f(x) = x^2$ + 1$$ and $$g(x) = 2x - 3$$, what is $$(f \circ g)(2)$$?

Accepted Answer

$$(f \circ g)(2) = f(g(2)) = f(1) = 1^2$ + 1 = 2$$

Question 8

At a point $$x = a on $$the graph of $$f(x)$$, the tangent line is horizontal and the graph is concave down. What can you say about $$f'(a)$$ and $$f''(a)$$?

Accepted Answer

$$f'(a) = 0$$ and $$f''(a) \u003c 0$$

Question 9

Find the equation of the tangent line to $$f(x) = 3x^2$ - 2x + 1 at$$ $$x = 1.$$

Accepted Answer

$$y = 4x - 2$$

Question 10

Find the derivative of $$f(x) = (2x^3$ - x) e$$^{x}$$ using the product rule.

Accepted Answer

$$f'(x) = (6x^2$ - 1) e$$^{x}$$ + (2x^3$ - x) e$$^{x}$$

Question 11

Given $$C(x) = 100 + 5x + 0.1x^2$, what is the marginal cost when x = 10$?

Accepted Answer

$$C'(10) = 5 + 0.2 	imes 10 = 7$$

Question 12

If $$f(x) = \sqrt{3x^2$ + 1}$$, what is $$f'(x)$$ using the chain rule?

Accepted Answer

$$f'(x) = \frac{1}{2\sqrt{3x^2$ + 1}} \cdot 6x = \frac{3x}{\sqrt{3x^2$ + 1}}$$

Question 13

Given the marginal revenue function $$MR(x) = 20 - 0.2x$$ and $$R(0) = 0$$, what is the total revenue function $$R(x)$$?

Accepted Answer

$$R(x) = 20x - 0.1x^2$

Question 14

Find the critical points of $$f(x) = x^3$ - 6x^2$ + 9x + 1$$ and determine their nature using the second derivative test.

Accepted Answer

Critical points at $$x = 1 ($$local max), $$x = 3 ($$local min)

Question 15

Estimate the area under $$f(x) = x^2$ from x = 0$ to x = 2$ using a left-hand Riemann sum with two subintervals. Is this an underestimate or overestimate?

Accepted Answer

Left-hand sum: $$0^2$$ 	imes 1 + $$1^2$$ 	imes 1 = 1$$. It is an underestimate.

Question 16

Find the antiderivative of $$f(x) = 4x^3$ - 2x.$$

Accepted Answer

$$F(x) = x^4$ - x^2$ + C$$

Question 17

Use the Fundamental Theorem of Calculus to evaluate $$\int_{1}^{3} (2x) dx.$$

Accepted Answer

$$[x^2$]_{1}^{3} = 9 - 1 = 8$$

Question 18

A company’s cost function is $$C(x) = 500 + 10x$$, and its revenue function is $$R(x) = 25x. $$What is the production level $$x$$ that maximizes profit?

Accepted Answer

Profit is maximized for all $$x$$ since $$R(x) - C(x)$$ increases with $$x.$$

Question 19

Find the area between the curves $$y = x^2$ and y = 2x$ from x = 0$ to x = 2$.

Accepted Answer

$$\int_{0}^{2} (2x - x^2$) dx = [x^2$ - \frac{1}{3}x^3$]_{0}^{2} = 4 - \frac{8}{3} = \frac{4}{3}$$

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