MATH 133, section 1 – Calculus With Fundamentals 1 Exam 3 Practice Problems November 4, 2015

I. Do not use the differentiation rules from Chapter 3 in this question.

A) State the limit definition of the derivative f′(x). B) Use the definition to compute the derivative function of f(x) = \frac{1}{3x}. C) Find the equation of the line tangent to the graph y = \frac{1}{3x} at x = 2.

II. Use the sum, product, quotient, and/or chain rules to compute the following derivatives. You may use any correct method, but must show work for full credit.

A)

\frac{d}{dx} (5x\sqrt{x} − \frac{2}{x^{3}} + 11x − 4)

B)

\frac{d}{dt} (\frac{t^{2}e^{3t}}{t^{4} + 1})

C)

\frac{d}{dz} \frac{z^{2} − 2z + 4}{z^{2} + 1}

D)

\frac{d}{dx} (\frac{π^{2} + tan(e^{π}) − 2x^{e}}{4} )

E)

\frac{d}{dx} (sin(x) (x^{7} − \frac{4}{\sqrt{x}}))

F) Find y′ (note this is just another way of asking the same question!)

y = (e^{2x} + 2)^{3}

G) Find y′

y = \frac{x + 1}{3x^{4} − 1}

H) Find y′

y = \frac{sin(x)}{1 + cos(x)} + x^{2} cos(x^{3} + 3)

III. The total cost (in $) of repaying a car loan at interest rate of r% per year is C = f(r).

A) What is the meaning of the statement f(7) = 20000?

B) What is the meaning of the statement f′(7) = 3000? What are the units of f′(7)?

IV. The quantity of a reagent present in a chemical reaction is given by Q(t) = t^{3} − 3t^{2} + t + 30 grams at time t seconds for all t ≥ 0. (Note: For a question like this, I could also give you the plot of the function and ask questions like those below. In this case you need to start from the formula and compute Q′(t); if you were given the graph, you need to make the connection between slopes of tangent lines and signs of Q′(t) visually.)

(a) Over which intervals with t ≥ 0 is the amount increasing? (i.e. Q′(t) > 0) decreasing (i.e. Q′(t) < 0)?

(b) Over which intervals is the rate of change of Q increasing? decreasing? decreasing?

V. A spherical balloon is being inflated at 20 cubic inches per minute. When the radius is 6 inches, at what rate is the radius of the balloon increasing? At what rate is the surface area increasing? (The volume of a sphere of radius r is V = \frac{4πr^{3}}{3} and the surface area is A = 4πr^{2}.)

VI. Review problems 2 and 3 from Quiz 5.