Question 1

A bacteria culture starts with 200 cells and grows at a rate proportional to its size. After half an hour, the population has increased to 360 cells. What is the population after 4 hours?

Accepted Answer

$$22,040$$

Question 2

Cobalt-60 has a half-life of $$5.24$$ years. How much of a $$100 mg $$sample remains after $$20$$ years?

Accepted Answer

$$7 mg$$

Question 3

Solve the differential equation $$\frac{dy}{dx} = 2y$$, $$y(0) = \frac{1}{5}.$$

Accepted Answer

$$y = \frac{1}{5}e$$^{2x}$$

Question 4

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

Accepted Answer

$$36$$

Question 5

A tank in the shape of an inverted circular cone with height $$10 m $$and base radius $$4 m is $$filled with water to a height of $$8 m. $$Find the work required to pump all the water to the top of the tank. (Use $$\rho = 1000 kg/m$^3$$, $$g = 9.8 m/s$^2.)$$

Accepted Answer

$$3.4 	imes 10^6$ Joules

Question 6

Evaluate the integral $$\$$int_0$$^1 \frac{x^3$}{\sqrt{16 - x^2$}} dx.$$

Accepted Answer

$$\frac{13}{3}$$

Question 7

Determine whether the improper integral $$\$$int_2$$^\infty \frac{1}{2x - 5} dx$$ converges or diverges.

Accepted Answer

Diverges

Question 8

Find the sum of the series $$\sum_{n=1}^\infty \frac{2^n + (-4)^n}{6^n} or $$show that it is divergent.

Accepted Answer

$$\frac{1}{10}$$

Question 9

Find the interval of convergence of the series $$\sum_{n=0}^\infty \frac{(x+1)^n (2n+1)!}{10^n n!}.$$

Accepted Answer

$$R = 0; I = \{-1\}$$

Question 10

Find the centroid of the region bounded by the graphs of $$y = x$$ and $$y = x^2$.

Accepted Answer

$$\left( \frac{1}{2}, \frac{1}{6} \right)$$

Question 11

Evaluate the integral $$\int \frac{\ln x}{x^2$} dx.$$

Accepted Answer

$$\frac{1 - \ln x}{x} + C$$

Question 12

Find the area of the region bounded by $$x + y = 0$$ and $$x = y^4$ + 3y.$$

Accepted Answer

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

Question 13

Find the surface area obtained by rotating the curve $$y = \frac{x}{6} + \frac{1}{2x}$$, $$1 \leq x \leq 2$$ around the x-axis.

Accepted Answer

$$\frac{47\pi}{10}$$

Question 14

Find the Maclaurin series representation for $$f(x) = \sin\left(\frac{x}{7}\right).$$

Accepted Answer

$$\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)!} \frac{x$$^{2n+1}$$}{7$$^{2n+1}$$}$$

Question 15

Convert the parametric equations $$x = 6 \cos 	heta$$, $$y = 1 - 3 \sin 	heta to $$rectangular form.

Accepted Answer

$$\frac{x^2$}{36} + \frac{(y-1)^2$}{9} = 1$$

Question 16

Find the area of one petal of the rose $$r = 3 \cos 5	heta.$$

Accepted Answer

$$\frac{9\pi}{10}$$

Question 17

Evaluate $$\int \frac{3x\sqrt{x}}{\sqrt{2x - 6}} dx.$$

Accepted Answer

$$6x^{3/2} - 6c$

Question 18

Find the area of the surface formed by revolving the curve $$r = 2 \sin 	heta$$ about the polar axis.

Accepted Answer

$$4\pi^2$

Question 19

A steel tank has the shape of a circular cylinder oriented vertically with diameter $$4 m $$and height $$5 m. $$The tank is currently filled to a level of $$3 m $$with cooking oil that has a density of $$920 kg/m$^3. $$Find the work required to pump the oil out through a $$1-m $$spout at the top of the tank.

Accepted Answer

$$1.53 	imes 10^6$ J

Comprehensive Calculus II Study Guide: Differential Equations, Applications of Integration, Series, and Polar Coordinates

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