Q: Find $$\frac{dy}{dx} if$$ $$y = x \tan(2\sqrt{x}) + 4.$$

$$\tan(2\sqrt{x}) + \frac{1}{\cos^2$(2\sqrt{x})} \cdot \frac{1}{\sqrt{x}}$$

Question 1

Evaluate the following limit: 
$$
\lim_{x 	o 0} \frac{\sin x^2$}{\cos 2x - 1}
$$

Accepted Answer

$$0$$

Question 2

Find $$\frac{dy}{dx} if$$ $$y = e$$^{x+y}$$ - x.$$

Accepted Answer

$$\frac{dy}{dx} = \frac{e$$^{x+y}$$ - 1}{1 - e$$^{x+y}$$}$$

Question 3

Evaluate the integral: 
$$
\int_{-1}^{1} \frac{x}{1 + x^2$} dx
$$

Accepted Answer

$$0$$

Question 4

A rectangular plot of land is to be fenced on three sides (the fourth side is a stream). If $$800 ft$^2 is to be $$enclosed, what dimensions minimize the amount of fencing used?

Accepted Answer

Width $$= 20 ft$$, Length $$= 40 ft$$

Question 5

Find $$\frac{dy}{dx} if$$ $$y = x \arcsin(2x).$$

Accepted Answer

$$\arcsin(2x) + \frac{2}{\sqrt{1 - 4x^2$}}$$

Question 6

Evaluate the following limit: 
$$
\lim_{x 	o 0} \$$left(e^{2x}$$ + x\$$right)^{1/x}$$
$$

Accepted Answer

$$e^2$$

Question 7

Find the average value of $$f(t) = 3t^2$ - 2t - 4 on$$ $$[1, 3].$$

Accepted Answer

$$7$$

Question 8

Air is being pumped into a spherical balloon at a rate of $$20 in$^3/s. $$How fast is the radius increasing when the surface area is $$4 in$^2$$? (Recall $$V = \frac{4}{3}\pi r^3$, A = 4$$\pi $$r^2)$$

Accepted Answer

$$5 in/s$$

Question 9

Find $$\frac{dy}{dx} if$$ $$y = \ln(\cos e^x).$$

Accepted Answer

$$-	an(e^x) \cdot e^x$$

Question 10

Evaluate the integral: 
$$
\int 3x^2$ \sqrt{5x^3$ + 9} \, dx
$$

Accepted Answer

$$2(5x^3$$ + $$9)^{1/2} + C$

Question 11

Find all vertical asymptotes of the function $$y = \frac{4 + x^2$}{x}.$$

Accepted Answer

$$x = 0$$

Question 12

For the function $$y = \frac{4 + x^2$}{x}$$, on which intervals is the function increasing?

Accepted Answer

$$(0, \infty)$$

Question 13

Find $$\frac{dy}{dx} if$$ $$y = x 	an(2\sqrt{x}) + 4.$$

Accepted Answer

$$	an(2\sqrt{x}) + \frac{1}{\cos^2$(2\sqrt{x})} \cdot \frac{1}{\sqrt{x}}$$

Question 14

Evaluate the integral: 
$$
\int 3x^2$ (1 - x$$^{-6}$$) dx
$$

Accepted Answer

$$x^3$$ - \frac{1}{$$x^3$$} + C$$

Question 15

Find $$\frac{dy}{dx} if$$ $$y = \$$int_0$$^{\sqrt{x}} \cos t^2$ dt.$$

Accepted Answer

$$\cos(x) \cdot \frac{1}{2\sqrt{x}}$$

Question 16

Find the intervals where the function $$y = \frac{4 + x^2$}{x} is $$concave up.

Accepted Answer

$$(-\infty, -2) \cup (2, \infty)$$

Question 17

Find the slant (oblique) asymptote of $$y = \frac{4 + x^2$}{x}.$$

Accepted Answer

$$y = x$$

Question 18

Find the area of the region bounded by $$y = x^2$ - 6$$ and $$y = 2x + 2.$$

Accepted Answer

$$\frac{64}{3}$$

Math 111 Final Exam Calculus Study Guidance

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