Q: Rewrite $$\ln \sin \theta - \ln \left(\sin \$$theta^5$$\right) as a $$single logarithm.

$$\ln(\sin \theta) - \ln(\sin^5$ \theta) = \ln(\sin \theta / \sin^5$ \theta) = \ln(\sin$$^{-4}$$ \theta)$$

Q: A cable weighing $$4.5 lb/ft is $$attached to an elevator. When the elevator is at the first floor, $$180 ft of $$cable are paid out. How much work does the motor do on the cable as the elevator rises to the top floor (where no cable is paid out)?

$$4.5 \times \frac{180^2$}{2}$$

Q: Given $$y = \ln\left(\frac{e$$^{\theta}$$}{1 + e$$^{\theta}$$}\right)$$, what is $$\frac{dy}{d\theta}$$?

$$\frac{1}{1 + e$$^{\theta}$$}$$

Q: Evaluate $$\int_{0}^{\pi/4} (1 + e$$^{\tan \theta}$$) \sec^2$ \theta d\theta.$$

$$\tan \theta + e$$^{\tan \theta}$$ \Big|_{0}^{\pi/4}$$

Question 1

A solid is formed by rotating the region bounded by $$0 \leq x \leq 	an\left(\frac{\pi}{4}yight)$$ from $$x=0 to$$ $$x=1$$ about the $$y-$$axis. Which of the following integrals correctly represents the volume of the solid?

Accepted Answer

Question 2

A bag of sand weighing 144 lb is lifted 18 ft, but sand leaks out at a constant rate so that only half the sand remains at the top. Neglecting the weight of the bag, what is the total work done in lifting the sand?

Accepted Answer

$$1944\ 	ext{ft-lb}$$

Question 3

Find the arclength of the curve $$y = \frac{1}{3}(x^2$ + 2$$)^{3/2}$$ from x=0$ to x=3$.

Accepted Answer

$$\int_{0}^{3} \sqrt{1 + x^2$(x^2+2$$)} dx$

Question 4

Given $$y = (x^2$ \ln x)^4$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$4(x^2$$ \ln $$x)^3 (2x$$ \ln x + x)$$

Question 5

Rewrite $$\ln \sin 	heta - \ln \left(\sin \$$theta^5$$ight) as a $$single logarithm.

Accepted Answer

$$\ln(\sin 	heta) - \ln(\sin^5$ 	heta) = \ln(\sin 	heta / \sin^5$ 	heta) = \ln(\sin$$^{-4}$$ 	heta)$$

Question 6

Evaluate $$\int_{0}^{\pi/12} 6 	an(3x) dx.$$

Accepted Answer

$$2 \ln|\sec(3x)| \Big|_{0}^{\pi/12}$$

Question 7

Given $$\cosh(x) = \frac{12}{5}$$ and $$x \u003e 0$$, what is $$\sinh(x)$$?

Accepted Answer

$$\frac{\sqrt{119}}{5}$$

Question 8

Simplify $$\sinh(2 \ln x).$$

Accepted Answer

$$x^2$$ - $$x^{-2}$$

Question 9

Order the following functions from slowest to fastest growth as $$x 	o \infty$$: $$x$$, $$x^2$$, $$x^{10}$$, $$5^x$$, $$2^x$$, $$x!$$, $$x^x$$, $$\ln x$$, $$\ln \ln x.$$

Accepted Answer

$$\ln \ln x \u003c \ln x \u003c x \u003c x^2$ \u003c x$$^{10}$$ \u003c 2^x \u003c 5^x \u003c x! \u003c x^x$$

Question 10

A cable weighing $$4.5 lb/ft is $$attached to an elevator. When the elevator is at the first floor, $$180 ft of $$cable are paid out. How much work does the motor do on the cable as the elevator rises to the top floor (where no cable is paid out)?

Accepted Answer

$$4.5 	imes \frac{180^2$}{2}$$

Question 11

Find the surface area of the surface generated by revolving $$y = \sqrt{x}$$, $$\frac{3}{4} \leq x \leq \frac{15}{4}$$ about the $$x-$$axis.

Accepted Answer

$$2\pi \int_{3/4}^{15/4} \sqrt{x} \sqrt{1 + (\frac{1}{2\sqrt{x}})^2$} dx$$

Question 12

Evaluate $$\int_{1}^{2} \frac{\cosh(\ln t)}{t} dt.$$

Accepted Answer

$$\frac{1}{2}(t + \frac{1}{t}) \Big|_{1}^{2}$$

Question 13

Given $$y = \ln\left(\frac{e$$^{	heta}$$}{1 + e$$^{	heta}$$}ight)$$, what is $$\frac{dy}{d	heta}$$?

Accepted Answer

$$\frac{1}{1 + e$$^{	heta}$$}$$

Question 14

Evaluate $$\int_{0}^{\pi/4} (1 + e$$^{	an 	heta}$$) \sec^2$ 	heta d	heta.$$

Accepted Answer

$$	an 	heta + e$$^{	an 	heta}$$ \Big|_{0}^{\pi/4}$$

Question 15

Simplify $$\ln(\cosh x + \sinh x) + \ln(\cosh x - \sinh x).$$

Accepted Answer

$$\ln(2)$$

Question 16

If $$y = 	heta (\sin(\ln 	heta) + \cos(\ln 	heta))$$, what is $$\frac{dy}{d	heta}$$?

Accepted Answer

$$\sin(\ln 	heta) + \cos(\ln 	heta) + 	heta (\cos(\ln 	heta) - \sin(\ln 	heta)) \frac{1}{	heta}$$

Question 17

Evaluate $$\int_{0}^{\pi/2} 2 \sinh(\sin 	heta) \cos 	heta d	heta.$$

Accepted Answer

$$2 (\cosh(\sin 	heta)) \Big|_{0}^{\pi/2}$$

Question 18

Given $$\cosh(x) = \frac{12}{5}$$ and $$x \u003e 0$$, what is $$	anh(x)$$?

Accepted Answer

$$\frac{\sqrt{119}}{12}$$

Comprehensive Calculus Practice Exam Guidance

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