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

$$r = 10$$

Question 2

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

Accepted Answer

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

Question 3

A rocket is launched vertically and observed by a person 12 km away. When the rocket is 5 km above the ground 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 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 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 value of $$x$$ that gives the rectangle the maximal area?

Accepted Answer

$$x = \sqrt{2}$$

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 $$1.8$$ inches and shrinking at $$0.2 in/$$min, at what rate is the volume changing?

Accepted Answer

$$-0.509$$ cubic in/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 $$2 ft $$long and the volume is decreasing at $$1.5 ft$^3/$$min. At what rate are the sides of the cube changing?

Accepted Answer

$$-0.125 ft/$$min

Question 10

Find the general solution $$F(x) to $$the differential equation $$F'(x) = 6e$$^{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 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

$$48\pi sq ft/sec$$

Question 13

Evaluate $$\int \frac{4}{3\sqrt{x}} dx.$$

Accepted Answer

$$6x^{2/3} + C$

Question 14

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

Accepted Answer

The First Derivative Test uses sign changes in $$f' to $$determine local extrema, while the Second Derivative Test uses the value of $$f'' at $$critical points.

Question 15

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

Accepted Answer

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

Question 16

Given $$y = -\frac{11}{2}x + 12 is $$the tangent line to $$x^3y$$ - $$xy^3 = 6$ at (2$$,1)$$, what is the slope of the tangent line at that point?

Accepted Answer

$$-\frac{11}{2}$$

Question 17

Find the most general antiderivative of $$f(x) = 15e$$^{5x}$$.

Accepted Answer

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

Question 18

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

Accepted Answer

$$72 ft$^3$$

Question 19

Evaluate $$\int 	an t \cos t\, dt.$$

Accepted Answer

$$\sec t + C$$

Calculus Exam Review: Applications of Derivatives & Antiderivatives

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