Q: Find the derivative of $$y = \ln\left(\frac{e$$^{\theta}$$}{1 + e$$^{\theta}$$}\right)$$ with respect to $$\theta.$$

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

Question 1

A region is bounded by $$0 \leq x \leq 	an\left(\frac{\pi}{4}yight)$$ for $$x = 0 to$$ $$x = 1. $$What is the volume of the solid obtained by rotating this region about the $$y-$$axis?

Accepted Answer

$$V = \frac{\pi}{2}$$

Question 2

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

Accepted Answer

$$W = 1944 ft-lb$$

Question 3

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

Accepted Answer

$$L = 11$$

Question 4

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

Accepted Answer

$$\frac{13}{5}$$

Question 5

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

Accepted Answer

$$-4\ln(\sin 	heta)$$

Question 6

Calculate the derivative of $$y = (x^2$ \ln x)^4$ with respect to x$.

Accepted Answer

$$y' = 4(x^2$ \ln x)^3$ (2x \ln x + x)$$

Question 7

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

Accepted Answer

$$2 \ln(2)$$

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

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

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

Question 11

A force of 2 N stretches a rubber band 2 cm. Assuming Hooke's law applies, how far will a 4 N force stretch the band, and what is the work required to do so?

Accepted Answer

4 cm, $$0.04 J$$

Question 12

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

Accepted Answer

$$\frac{3}{8}$$

Question 13

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

Accepted Answer

$$0$$

Question 14

Find the derivative of $$y = \ln\left(\frac{e$$^{	heta}$$}{1 + e$$^{	heta}$$}ight)$$ with respect to $$	heta.$$

Accepted Answer

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

Question 15

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

Accepted Answer

$$2(\cosh(1) - 1)$$

Question 16

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

Accepted Answer

Comprehensive Calculus Practice Exam Guidance

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