Q: A biologist models the population of bacteria by $$P(t) = 1000e$$^{0.05t}$$, where t$ is in hours. What is the instantaneous rate of change of the population at t = 10$ hours?

$$P'(10) = 1000 \times 0.05 \times e$$^{0.5}$$ \approx 82$$

Q: Which of the following statements about the function $$f(x) = \frac{1}{x} is $$true?

$$f(x) is $$continuous for all $$x \neq 0$$

Q: Which of the following is a correct statement of L'Hospital's Rule?

If $$\lim_{x \to a} \frac{f(x)}{g(x)}$$ yields $$\frac{0}{0} or$$ $$\frac{\infty}{\infty}$$, then $$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} if $$the latter limit exists.

Question 1

A hospital monitors the rate at which a drug leaves a patient's bloodstream, modeled by $$R(t) = 5e$$^{-0.2t}$$ mg/hour, where t$ is in hours. What is the total amount of drug that leaves the bloodstream in the first 5 hours?

Accepted Answer

$$\$$int_0$$^5 5e$$^{-0.2t}$$ \, dt = 5 \left[ -5e$$^{-0.2t}$$ \right]_0^5$ = 5(1 - e$$^{-1}$$)$$

Question 2

A car's position along a straight road is given by $$s(t) = t^3$ - 6t^2$ + 9t$$ meters, where $$t is in $$seconds. What is the car's acceleration at $$t = 2$$ seconds?

Accepted Answer

$$s''(2) = 6t - 12 = 0$$

Question 3

Which of the following is the correct application of the Mean Value Theorem for the function $$f(x) = x^2$ on the interval $$[1, 3]$$?

Accepted Answer

$$f'(c) = \frac{f(3) - f(1)}{3 - 1}$$ for some $$c \in (1, 3)$$

Question 4

A biologist models the population of bacteria by $$P(t) = 1000e$$^{0.05t}$$, where t$ is in hours. What is the instantaneous rate of change of the population at t = 10$ hours?

Accepted Answer

$$P'(10) = 1000 	imes 0.05 	imes e$$^{0.5}$$ \approx 82$$

Question 5

Evaluate the limit: $$\lim_{x 	o 0} \frac{\sin(3x)}{x}$$

Accepted Answer

$$3$$

Question 6

A medication is administered at a rate $$r(t) = 2 + \sin(t) mg/$$hour for $$0 \leq t \leq \pi. $$What is the total amount of medication administered over this interval?

Accepted Answer

$$\$$int_0$$^\pi (2 + \sin t) \, dt = 2\pi$$

Question 7

Find the derivative of $$f(x) = x^2$ e^x.$$

Accepted Answer

$$f'(x) = 2x e^x + x^2$ e^x$$

Question 8

A tank is being filled with water at a rate $$R(t) = 4t$$ liters per hour. How much water is added to the tank from $$t = 0 to$$ $$t = 3$$ hours?

Accepted Answer

$$\$$int_0$$^3 4t \, dt = 18$$

Question 9

Which of the following statements about the function $$f(x) = \frac{1}{x} is $$true?

Accepted Answer

$$f(x) is $$continuous for all $$x 
eq 0$$

Question 10

A company’s profit $$P(x) in $$thousands of dollars is given by $$P(x) = -2x^2$ + 8x + 10$$, where $$x is $$the number of units sold. What is the maximum profit?

Accepted Answer

$$P(2) = 18$$

Question 11

Find $$\frac{d}{dx} \left( \$$arctan(x^2$$) \right).$$

Accepted Answer

$$\frac{2x}{1 + x^4$}$$

Question 12

A spherical balloon is being inflated so that its radius increases at a rate of $$2 cm/$$min. What is the rate of change of its volume when the radius is $$5 cm$$?

Accepted Answer

$$\frac{dV}{dt} = 4\pi r^2$ \frac{dr}{dt} = 4\pi (5)^2$ (2) = 200\pi$$

Question 13

Evaluate $$\$$int_0$$^1 x^2$ \, dx.$$

Accepted Answer

$$\frac{1}{3}$$

Question 14

Which of the following is a correct statement of L'Hospital's Rule?

Accepted Answer

If $$\lim_{x 	o a} \frac{f(x)}{g(x)}$$ yields $$\frac{0}{0} or$$ $$\frac{\infty}{\infty}$$, then $$\lim_{x 	o a} \frac{f(x)}{g(x)} = \lim_{x 	o a} \frac{f'(x)}{g'(x)} if $$the latter limit exists.

Question 15

A patient’s heart rate is modeled by $$H(t) = 70 + 10\sin(\frac{\pi t}{6})$$ beats per minute, where $$t is in $$minutes. What is the average heart rate over the interval $$[0, 6]$$?

Accepted Answer

$$70$$

Question 16

Find the area under the curve $$y = x^2$ from x = 0$ to x = 2$.

Accepted Answer

$$\$$int_0$$^2 x^2$ \, dx = \frac{8}{3}$$

Question 17

Which of the following is the correct derivative of $$f(x) = \ln(x^2$ + 1)$$?

Accepted Answer

$$f'(x) = \frac{2x}{x^2$ + 1}$$

Question 18

A patient’s temperature $$T(t) in $$degrees Celsius is modeled by $$T(t) = 37 + 2e$$^{-0.1t}$$, where t$ is in hours. What is the rate of change of temperature at t = 0$?

Accepted Answer

$$T'(0) = -0.2$$

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