Q: A population of bacteria grows at a rate proportional to its current size. If $$P(0) = 100$$ and $$P(2) = 400$$, what is the population at $$t = 3$$ hours?

$$P(3) = 100 \times 2$$^{3/2}$$ = 100 \times 2.828 = 282.8$$

Q: A ball is thrown upward with an initial velocity of $$20 m/s $$from a height of $$50 m. $$The height at time $$t is $$given by $$h(t) = -4.9t^2$ + 20t + 50. At $$what time does the ball reach its maximum height?

$$t = \frac{20}{2 \times 4.9}$$

Question 1

A pharmaceutical company is studying the rate at which a drug is absorbed into the bloodstream. If the concentration of the drug in the blood at time $$t$$ hours is given by $$C(t) = 5e$$^{-0.4t}$$ mg/L, what is the total amount of drug absorbed in the first 3 hours?

Accepted Answer

$$\$$int_0$$^3 5e$$^{-0.4t}$$ \, dt = \frac{5}{0.4}(1 - e$$^{-1.2}$$)$$

Question 2

Which of the following statements about the Mean Value Theorem for Integrals is correct?

Accepted Answer

If $$f is $$continuous on $$[a, b]$$, then there exists $$c in$$ $$[a, b]$$ such that $$\int_a^b f(x) \, dx = f(c)(b-a).$$

Question 3

Evaluate the following limit using L'Hospital's Rule: $$\lim_{x 	o 0} \frac{\sin(3x)}{x}.$$

Accepted Answer

3

Question 4

A tank is being filled with water at a rate given by $$R(t) = 4t^2$ + 2$$ liters per minute. How much water enters the tank between $$t = 1$$ and $$t = 3$$ minutes?

Accepted Answer

$$\$$int_1$$^3 (4t^2$ + 2) \, dt = [\frac{4}{3}t^3$ + 2t]_1^3$

Question 5

Which of the following is the correct substitution to evaluate $$\int x \sqrt{x^2$ + 1} \, dx$$?

Accepted Answer

Let $$u = x^2$ + 1$$, then $$du = 2x \, dx$$

Question 6

A population of bacteria grows at a rate proportional to its current size. If $$P(0) = 100$$ and $$P(2) = 400$$, what is the population at $$t = 3$$ hours?

Accepted Answer

$$P(3) = 100 	imes 2$$^{3/2}$$ = 100 	imes 2.828 = 282.8$$

Question 7

Which of the following integrals can be evaluated using integration by parts?

Accepted Answer

$$\int x e^x \, dx$$

Question 8

Evaluate the improper integral $$\$$int_1$$^\infty \frac{1}{x^2$} \, dx.$$

Accepted Answer

1

Question 9

Which of the following series converges?

Accepted Answer

$$\sum_{n=1}^\infty \frac{1}{n^2$}$$

Question 10

A ball is thrown upward with an initial velocity of $$20 m/s $$from a height of $$50 m. $$The height at time $$t is $$given by $$h(t) = -4.9t^2$ + 20t + 50. At $$what time does the ball reach its maximum height?

Accepted Answer

$$t = \frac{20}{2 	imes 4.9}$$

Question 11

Which of the following is the correct formula for the arc length of the curve $$y = f(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$L = \int_a^b \sqrt{1 + (f'(x))^2$} \, dx$$

Question 12

A radioactive substance decays at a rate proportional to its mass. If the half-life is 5 years, what is the decay constant $$k in $$the equation $$m(t) = m_0$ e$$^{-kt}$$?

Accepted Answer

$$k = \frac{\ln 2}{5}$$

Question 13

Which of the following is the correct Maclaurin series for $$e^x$$?

Accepted Answer

$$e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$$

Question 14

Evaluate $$\$$int_0$$^{\pi/2} \sin x \, dx.$$

Accepted Answer

1

Question 15

Which of the following is the correct formula for the volume of a solid of revolution generated by rotating $$y = f(x)$$ about the $$x-$$axis from $$x = a to$$ $$x = b$$?

Accepted Answer

$$V = \pi \int_a^b [f(x)]^2 \, dx$$

Question 16

Which of the following is the correct partial fraction decomposition for $$\frac{3x+5}{(x+1)(x-2)}$$?

Accepted Answer

$$\frac{A}{x+1} + \frac{B}{x-2}$$

Question 17

A curve is defined parametrically by $$x = t^2$, y$$ = $$t^3. $$What is $$\frac{dy}{dx} at$$ $$t = 2$$?

Accepted Answer

$$\frac{dy}{dx} = \frac{3t^2$}{2t} = \frac{3t}{2}$$, so at $$t=2$$, $$\frac{dy}{dx} = 3$$

Question 18

Which of the following is the correct formula for the area enclosed by one loop of the polar curve $$r = \sin 	heta$$?

Accepted Answer

$$A = \frac{1}{2} \$$int_0$$^{\pi} (\sin \$$theta)^2$$ \, d	heta$$

Calculus I: Comprehensive Study Notes (Course Syllabus Overview)

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