Question 1

A company wants to design an open-top cylindrical can with a volume of $$1000\pi cm$^3. $$What is the radius $$r$$ that minimizes the surface area of the can?

Accepted Answer

$$r = 10 cm$$

Question 2

Given the equation $$x^3y - x$$ $$y^3 = 6$, find $$\frac{dy}{dx}$$ using implicit differentiation.

Accepted Answer

$$\frac{dy}{dx} = \frac{y^3$ - 3x^2$y}{x^3$ - 3xy^2$}$$

Question 3

A rock is thrown into a pond, creating ripples. If the radius of the outermost ripple increases at $$6 ft/sec$$, at what rate is the area inside the ripple changing when the radius is $$4 ft$$?

Accepted Answer

$$48\pi ft$^2/sec$$

Question 4

Find the most general antiderivative of $$f(x) = x$$^{3/4}$$ + 8x - 6.$$

Accepted Answer

$$\frac{4}{7}x$$^{7/4}$$ + 4x^2$ - 6x + C$$

Question 5

A closed box is to be made from $$108$$ square feet of material. The base is a rectangle with length twice its width. What is the maximum volume that can be enclosed?

Accepted Answer

$$72 ft$^3$$

Question 6

Find the particular solution to the differential equation $$\frac{d^2$y}{dx^2$} = 24x^2$ + \cos x$$ given $$y'(0) = -3$$ and $$y(0) = 5.$$

Accepted Answer

$$y = 2x^4$ - \cos x - 3x + 6$$

Question 7

A chlorine tablet is a cylinder with diameter three times its height. When the diameter is $$1.8 in $$and shrinking at $$0.2 in/$$min, at what rate is the volume changing?

Accepted Answer

$$-0.509 in$^3/$$min

Question 8

Evaluate $$\int 4 \csc^2$ x\, dx.$$

Accepted Answer

$$-4\cot x + C$$

Question 9

A large cubic block of ice melts so that its sides are $$2 ft $$and the volume is decreasing at $$1.5 ft$^3/$$min. At what rate are the sides changing?

Accepted Answer

$$-0.125 ft/$$min

Question 10

Find the general solution $$F(x) to$$ $$F'(x) = 6e$$^{2x}$$ and the particular solution if F(0) = 10$.

Accepted Answer

$$F(x) = 3e$$^{2x}$$ + 7$$

Question 11

Use logarithmic differentiation to find $$\frac{dy}{dx}$$ for $$y = (5x)^x.$$

Accepted Answer

$$(5x)^x (\ln(5x) + 1)$$

Question 12

A rocket is launched vertically and observed from a point $$12 km $$away. When the rocket is $$5 km $$high and rising at $$7.9 km/sec$$, how fast is the distance between the observer and the rocket changing?

Accepted Answer

$$3.038 km/sec$$

Question 13

Find the most general antiderivative of $$f(x) = \frac{3}{x} + \frac{3}{x^2$}.$$

Accepted Answer

$$3\ln|x| - 3x$$^{-1}$$ + C$$

Question 14

A rectangle is inscribed in the region between the $$x-$$axis, $$y-$$axis, and the curve $$y = -x^2$ + 6 in $$the first quadrant. What is the maximal area of such a rectangle?

Accepted Answer

$$4\sqrt{2}$$

Question 15

Evaluate $$\int 15e$$^{5x}$$ dx.$$

Accepted Answer

$$3e^{5x} + C$

Question 16

Which of the following best describes the difference between the First Derivative Test and the Second Derivative Test for finding extrema?

Accepted Answer

The First Derivative Test uses sign changes of $$f'$$ around critical points, while the Second Derivative Test uses the value of $$f'' at $$the critical point.

Question 17

If $$f is $$the antiderivative of $$g$$, and $$g is $$the antiderivative of $$h$$, what is the relationship between $$f$$ and $$h$$?

Accepted Answer

$$f''(x) = h(x)$$

Question 18

Evaluate $$\int \sec(2x)	an(2x) dx.$$

Accepted Answer

$$\frac{1}{2}\sec(2x) + C$$

Question 19

Suppose $$y = \frac{e$$^{4x}$$}{3x^2$ + x + 4}^5. $$Using logarithmic differentiation, which of the following is $$\frac{dy}{dx}$$?

Accepted Answer

$$\left(\frac{e$$^{4x}$$}{3x^2$ + x + 4}\$$right)^5$$ \left[5\left(4 - \frac{6x + 1}{3x^2$ + x + 4}\right)\right]$$

Calculus Exam III Review: Step-by-Step Guidance

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