(10 points) Give an example of two functions f and g
such that f °g = g °f. Be sure to verify that the
domains are the same.
Common Mistake: Not verifying the domains
4.
(10 points) Give an example of a function f(x) defined
for all real numbers such that limx → 0f(x)
exists but does not equal f(0).
Common Mistake: Not verifying the domain or not
redefining the function at x=0 so that it is discontinuous at
0.
5.
(20 points) Let
F(x)
=
√
x2 − 1
,
G(x)
=
√
1 − x2
.
Describe:
•
domain(F) and domain(G).
Common Mistake: Not identifying correctly domain(F) = {|x| ≥ 1}.
•
domain(G °F) must be a subset of
domainF. Answer: {1 ≤ |x| ≤ √2}
III. Proofs
6.
(20 points) Show, using only P1 - P9:
For all a, a ·0 = 0.
You may abbreviate (distributive, … ).
Common Mistakes:
•
Starting by assuming the result and manipulating:
a ·0
= 0
(1)
a−1 ·a ·0
= 0 ·a−1
(2)
1 ·0
= 0 ·a−1
(3)
0
= 0
RHSassumesresult!(4)
If all the steps were justified, one would have to "reverse" to obtain the result.
Going from (4) to (3) assumes the result. All that has been shown
is: a ·0 = 0 implies 0 · a−1 = 0.
•
Not using P9 (distributive) somewhere.
7.
(20 points) Show by mathematical induction or otherwise:
(Bernoulli's Inequality) For all natural numbers n = 1, 2, …, for x > −1,
(1 + x)n ≥ 1 + n x.
Common Mistakes:
•
(Clarity) P(n) is not stated clearly.
•
(Minor) Not being explicit as to how the condition x > −1 is used in passing from P(n) to P(n+1)
•
Writing P(n) = (1 + x)n ≥ 1 + n x
•
Writing P(n+1) and manipulating to try to obtain
P(n).
•
(We won't do this again!) (1 + x)n+1 = 1n+1 + xn+1.
IV. Qualitative Properties of Functions
8.
(30 points) The graph below shows how the height of a liquid
in beaker X varies as water is steadily dripped into it.
Copy the graph, and on the same diagram
show the height-volume relationship for the Ink Bottle.
Describe the features of the graph you have drawn. Your
description should include …
Common Mistakes
•
Identifying the domain
•
Identifying the maximum height (same as for Beaker
X)
•
Interchanging the the roles of Height and Volume when
saying "faster", etc.
•
Incomplete word descriptions
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