Q: Evaluate the definite integral $$\$$int_0$$^4 \frac{3}{9x^2$ + 16} dx$$ using a change of variables.

$$\frac{3}{4} \left[ \tan$$^{-1}$$ \left( \frac{3x}{4} \right) \right]_0^4$

Q: Which of the following is the correct general formula for $$\int \frac{1}{a^2$ + x^2$} dx$$?

$$\frac{1}{a} \tan$$^{-1}$$ \left( \frac{x}{a} \right) + C$$

Question 1

A cylindrical container with a volume of $$50\pi ft$^3 is to be $$made from two different materials. The material used for the ends costs twice as much as the material for the side. What are the dimensions of the container that minimize the cost of the materials?

Accepted Answer

$$r = 5$$, $$h = 2$$

Question 2

A ladder is leaning against a wall. If the bottom of the ladder slides away from the wall at a rate of $$2 ft/s$$, and the ladder is $$10 ft $$long, how fast is the top of the ladder descending when the bottom is $$6 ft $$from the wall?

Accepted Answer

$$-1.6 ft/s$$

Question 3

A conical tank (as shown in the image) is being filled with water at a rate of $$3 ft$^3/$$min. If the tank has a height of $$10 ft $$and a radius of $$5 ft at $$the top, how fast is the water level rising when the water is $$6 ft $$deep?

Accepted Answer

$$\frac{1}{4\pi} ft/$$min

Question 4

Given the function $$f(x) = \frac{1}{x}$$, which of the following graphs correctly represents $$f(x)$$?

Accepted Answer

Graph C

Question 5

A piece of wire of length $$87 cm is $$cut, and the resulting two pieces are formed to make a circle and a square. Which value of $$x ($$the length of the piece used for the circle) will minimize the total area enclosed by the circle and the square?

Accepted Answer

$$x = 43.5 cm$$

Question 6

Evaluate the definite integral $$\$$int_0$$^4 \frac{3}{9x^2$ + 16} dx$$ using a change of variables.

Accepted Answer

$$\frac{3}{4} \left[ 	an$$^{-1}$$ \left( \frac{3x}{4} ight) ight]_0^4$

Question 7

Find the indefinite integral $$\int \frac{1}{1 - 25x^2$} dx.$$

Accepted Answer

$$\frac{1}{10} \sin$$^{-1}$$(5x) + C$$

Question 8

Use linear approximation to estimate $$f(0.03)$$ for $$f(x) = 4 + 4x at$$ $$a = 0. $$What is the percent error if the exact value is $$4.12$$?

Accepted Answer

Percent error $$= 0.0441\%$$

Question 9

Which of the following is the correct general formula for $$\int \frac{1}{a^2$ + x^2$} dx$$?

Accepted Answer

$$\frac{1}{a} 	an$$^{-1}$$ \left( \frac{x}{a} ight) + C$$

Question 10

A tank in the shape of an inverted cone is being drained at a rate of $$2 ft$^3/$$min. If the tank has a height of $$8 ft $$and a radius of $$4 ft at $$the top, how fast is the water level dropping when the water is $$5 ft $$deep?

Accepted Answer

$$\frac{1}{8\pi} ft/$$min

Question 11

A ladder $$12 ft $$long is leaning against a wall. If the bottom slides away from the wall at $$1 ft/s$$, how fast is the top descending when the bottom is $$9 ft $$from the wall?

Accepted Answer

$$-0.6 ft/s$$

Question 12

Which of the following is the correct formula for the derivative of $$f(x) = \frac{1}{x}$$?

Accepted Answer

$$f'(x) = -\frac{1}{x^2$}$$

Question 13

If $$f(x) = \sin(ax)$$, what is $$\int \cos(ax) dx$$?

Accepted Answer

$$\frac{1}{a} \sin(ax) + C$$

Question 14

A wire of length $$L is $$cut into two pieces, one bent into a circle and the other into a square. If $$x is $$the length used for the circle, what value of $$x$$ minimizes the total area?

Accepted Answer

$$x = \frac{L}{2}$$

Question 15

Which of the following is the correct formula for $$\int \sec(ax) dx$$?

Accepted Answer

$$\frac{1}{a} 	an(ax) + C$$

Question 16

Which of the following is the correct formula for $$\int \frac{1}{\sqrt{a^2$ - x^2$}} dx$$?

Accepted Answer

$$\sin$$^{-1}$$ \left( \frac{x}{a} \right) + C$$

Calculus Study Guidance: Related Rates, Optimization, and Graphing

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