Study Guide
Test 1
Fall 2012
Carter
1. Definitions and proofs (between 15-20 points on these items):
(a) Be able to state the definition of a limit. See also
http://www.youtube.com/watch?v=wSTJLUAu5ZI
(b) Be able to state the definition of a continuous function. See page 83.
(c) Be able to prove that limh→0+
sin (h)
h
= 1. See
http://www.youtube.com/watch?v=o6S6RbfhRTU
(d) Be able to prove that limh→0+
cos (h)−1
h
= 0. See
http://www.youtube.com/watch?v=encQhh2JeYc
2. The following questions are the quiz questions to date:
(a) State the definition of a function.
(b) A box with an open top is to be constructed from a rectangular piece of cardboard
that is 14 inches by 22 inches by cutting out equal squares of side length x from
each corner and folding up the sides as indicated. Express the volume as a function
of x.
x
x
x
x
x
x
14”
x
22”
1
x
(c) Sketch the graph of the function y = sin (2x).
(d) The half-life of phosphorus-32 is about 14 days. There are 6.6 grams present
initially.
i. Express the amount of phosphorus-32 that will remain as a function of time,
t, which is measured in days.
ii. When will there be 1 gram remaining? Leave your answer in the form
of a logarithm.
(e) Determine the inverse function, y = f −1 (x), for the function
y = f (x) =
x+3
.
x−2
(f) For the function above verify that f (f −1 (x)) = x and that f −1 (f (x)) = x.
(g)
x2 + 3x − 10
x→−5
x+5
lim
(h)
u4 − 1
u→1 u3 − 1
lim
√
√
(i) According to my calculator, 2 − 3 = 0.26795 while 5 − 2 = 0.23607. Consider
the function y = x2 in a neighborhood of c = 2. Find a δ > 0 so that |x2 − 4| < 1
provided |x − 2| < δ. See the blackboard for an illustration.
(j) Demonstrate your awareness of the difference between frequency and amplitude
by showing detailed work while computing the following limit:
sin (3y)
y→0
4y
lim
(k) Give the -δ definition of a limit. That is define precisely what is meant by the
sentence,
“ lim f (x) = L.”
x→c
(l) At what points is the function g(x) that is indicated below continuous?
g(x) =
x2 −x−6
x−3
5
if x 6= 3,
if x = 3.
(m) Use the definition of the derivative ( f 0 (x) = limh→0
equation of the tangent line for
y = x3
f (x+h)−f (x)
h
) to determine the
at the point(−2, −8).
3. The next list of problems are closely related to the quiz problems and deserve your
close attention. p. 13 # 64,68; p. 27 # 13-20; p. 50 # 45-48; p. 68 #23-42; p. 95 #
1-10; p. 109 # 63-68; p. 119 # 11-18; p. 126 # 23-26.
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4. Test Outline
(a) Questions on definitions and proofs (15-20 points).
(b) Questions elementary functions — graphing trig, quadratic, rational functions
(20-30 points).
(c) Questions on logs and/or exponentials (10-15 points).
(d) Questions on limits (20-40 points).
(e) Questions on the definition of derivative (20-40 points).
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