MATH 110, SECTION 003, WRITTEN ASSIGNMENT 5
Hand in full solutions to the questions below. Make sure to justify all your work, and include complete arguments and
explanations. Your answers must be legible (no tiny drawings or micro-handwriting!). Your answers must be stapled, with
your name and student number at the top of each page.
Please print and write your answers in the space provided.
(1) For each of the following functions, sketch the curves y = f (x) and y = f 0 (x) on the same graph.
(a) f (x) = 2x + 1.
(b) f (x) = −x2 + 4.
(c) f (x) = 3 sin(2t) + 3.
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MATH 110, SECTION 003, WRITTEN ASSIGNMENT 5
(2) Suppose that f is a differentiable function, defined everywhere, such that f 0 (x) = cos2 (x), and f (4) = π/4. Find the
equation of the tangent line to the curve
y = f (ln(x)) + x2
at x = e4 .
MATH 110, SECTION 003, WRITTEN ASSIGNMENT 5
(3) You have a debt of $700 on a credit card with a monthly interest rate of 10%. After t months, your debt is
d(t) = 700(1.1)t
dollars.
(a) Write d(t) in the form
d(t) = Cekt
for some appropriate constants C and k.
(b) How much do you owe after one year?
(c) What is the rate of change of your debt after one year? (Remember to include units.)
(d) How many months will it be until you owe one million dollars?
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MATH 110, SECTION 003, WRITTEN ASSIGNMENT 5
(4) Since the derivative of ln(x) is 1/x, we know that, by the definition of the derivative,
lim
h→0
ln(2 + h) − ln(2)
1
= .
h
2
Use this fact to show carefully, using the definition of the derivative, that the function
f (x) = |ln(x) − ln(2)|
is not differentiable at x = 2.