Study Guide Test 1 Fall 2013 Carter
This study guide consists of two parts: The first part is a suggested problem list to study.
Note, these problems are compiled from this semester’s quizzes, the study guide from Fall
2012, and some other homework problems that I find amusing. The second part of the study guide consists of the study guide and the first test from Fall 2012. As it happens, the break in the material is almost identical to last fall’s class. The difference here is that I will include a few problems from section 3.3 — rules of differentiation.
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 lim h → 0 + sin ( h ) h
= 1. See http://www.youtube.com/watch?v=o6S6RbfhRTU
(d) Be able to prove that lim h → 0
+ cos ( h ) − 1 h
= 0. See http://www.youtube.com/watch?v=encQhh2JeYc
(e) Be able to prove the product rule: See http://www.youtube.com/watch?v=RALGAHo9Ll4
2. The following questions are the quizzes to date:
(a) (First Quiz) Write a formula for f ◦ g ◦ h where f ( x ) = x + 1 g ( x ) = 3 x h ( x ) = 4 − x
(b) (2nd Quiz) x x x x
14” x x x x
22”
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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 .
(c) (3rd Quiz) i. Sketch the graph of the function y = cos ( πx ) over two full periods.
ii. Solve for t in the equation e
− 0 .
3 t
= 27 .
(d) (4th Quiz) i. Write the expression ln 3
√
2 in terms of ln (2) and ln (3) .
ii. Find the equation of the line that is tangent to the curve y = 5 − x 2 f (1+ h ) − f (1) point P = (1 , 4). (Hint: First compute .) h at the
(e) (5th Quiz) Use the limit laws ( LK , L Σ, L Π, and LQ ) and elementary algebra to evaluate the following limits.
i.
lim x → 0
1 x − 1
+ x
1 x +1 ii.
u
4 − 1 lim u → 1 u 3 − 1
(f) (6th Quiz) Use the limit laws ( LK , L Σ, L Π, and LQ ) and elementary algebra to evaluate the following limits.
i.
ii.
lim t → 1 t
2 t
+ t − 2
2 − 1 lim x → 9
√ x − 3 x − 9
(g) (7th Quiz) i. Give the , δ -definition of a limit. That is, give the rigorous meaning of the equation: lim x → c f ( x ) = L.
ii. Compute lim y → 0 sin (3 y )
.
4 y
(h) (8th Quiz) i. Determine the points at which the function is continuous. Remember to exclude certain values of x if necessary.
y = x tan ( x ) x 2 + 1
2

ii. Evaluate the function f ( x ) = x
3 − 15 x + 1 at the points x = − 4 , x = 0, x = 1, and x = 4. Give a clear, concise, explanation that the equation x 3 − 15 x + 1 = 0 has three solutions in the closed interval [ − 4 , 4].
(i) (9th Quiz) i. Compute the limit lim y →∞ sin (2 y ) y ii. Compute the limits
7 x
3 lim x →±∞ x 3 − 3 x 2 + 6 x
(j) (10th Quiz) Use the Newton quotient to compute the equation of the line tangent to the curve y = 4 − x
2 at the point ( − 1 , 3) .
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; p. 137 #1-18.
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).
(f) Routine calculations of the derivative (10-20 points).
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 lim h → 0
+ sin ( h ) h
= 1. See http://www.youtube.com/watch?v=o6S6RbfhRTU
3

(d) Be able to prove that lim h → 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
14” x x x x
22”
(c) Sketch the graph of the function y = sin (2 x ) .
(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) lim x →− 5 x 2 + 3 x − 10 x + 5
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(h)
(i) According to my calculator, 2 − the function y = x 2
√ lim u → 1
3 = 0 u 4
.
in a neighborhood of
− 1 u 3 − 1
26795 while c
√
= 2. Find a
5 −
δ >
2 = 0 .
23607. Consider
0 so that | x 2 − 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: lim y → 0 sin (3 y )
4 y
(k) Give the δ definition of a limit. That is define precisely what is meant by the sentence,
“ lim x → c f ( x ) = L.
”
(l) At what points is the function g ( x ) that is indicated below continuous?
g ( x ) = x
2 − x − 6 x − 3
5 if x = 3 , if x = 3 .
(m) Use the definition of the derivative ( f
0
( x ) = lim h → 0 equation of the tangent line for f ( x + h ) − f ( x ) h
) to determine the y = x
3 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.
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).
General Instructions: Write your name on only the outside of your blue book. Put your test paper inside your blue book as you leave. Solve each of the following problems (105 total points). Point values are indicated on the problems. Remember to use your turn signals when changing lanes.
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1.
(5 points) What does the expression lim x → c f ( x ) = L mean?
State the definition of a limit.
2.
(10 points) Give a proof that lim h → 0 + sin ( h )
= 1 .
h
3.
(5 points) Use the fact that lim h → 0 algebra to prove that sin ( h ) h lim h → 0
= 1, the Pythagorean identity, and elementary cos ( h ) − 1
= 0 .
h
4.
(10 points) Sketch the graph of f ( x ) = 3 cos (2 x ) over two full periods.
5.
(10 points) Sketch the graph of the linear fractional transformation f ( x ) =
2 x − 3
.
x + 4
6.
(5 points) Use the properties of exponentials and logarithms to simplify the expression ln ( e x
2
+ y
2
) .
7. Use the rules for computing the limit for each of the following problems (10 points each) .
(a) x 2 + x − 6 lim x → 2 x 2 − 3 x + 2
(b) lim x → 0 sin (2 x ) sin (3 x )
(c) lim x → 16
√ x − 4 x − 16
8.
(10 points) In the figure below, a rectangle is inscribed under the parabola y = 12 − x
2 with its base along the x -axis and its two upper vertices on the parabola. Express the area of the rectangle as a function of x which is the horizontal coordinate of the lower right vertex.
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y x x
9.
(10 points) Use the definition of the derivative, f
0
( x ) = lim h → 0 lim z → x f ( z ) − f ( x ) z − x
), to compute the derivative of the function f ( x + h ) − f ( x ) h
(or f
0
( x ) = f ( x ) = 2 x
2
+ 3 x − 1 .
10.
(10 points) Use the definition of the derivative (as given above) to compute the equation of the line tangent to the curve
1 y = x at the point (2 ,
1
2
) .
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