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math21_2007f_final.pdf
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Applied Mathematics  21  Calculus of a Single Va...
Applied Mathematics  21  Calculus of a Single Variable I  exam UC Merced: MATH 21 — Final Exam — 15 December 2007 On the front of your bluebook print (1) your name, (2) your student ID number, (3) your discussion section number, (4) your room and seat number, and (5) a grading table. Show all work in your bluebook and BOX IN YOUR FINAL ANSWERS where appropriate. A correct answer with no supporting work may receive no credit wh
Applied Mathematics  21  Calculus...
Applied Mathematics  21  Calculus of a Single Variable I  exam UC Merced: MATH 21 — Final Exam — 15 December 2007 On the front of your bluebook print (1) your name, (2) your student ID number, (3) your discussion section number, (4) your room and seat number, and (5) a grading table. Show all work in your bluebook and BOX IN YOUR FINAL ANSWERS where appropriate. A correct answer with no supporting work may receive no credit wh
Page 2
4. (15 points: 5 each) Find the value of the following expressions.
(a)
π/
2
0
d
dx
sin
x
2
cos
x
2
dx
(b)
d
dx
π/
2
0
sin
x
2
cos
x
2
dx
(c)
d
dx
π/
2
x
sin
t
2
cos
t
2
dt
5. (20 points total) Answer the following unrelated integral problems.
(a) (5 points) Use a Riemann sum with 2 subintervals to approximate
4
0
(
x
+ 1)
dx
with the
rightendpoint Rule.
(b) (8 points) Evaluate
4
1
2
√
t
(1 + 2
√
t
)
dt
.
(c) (7 points) Evaluate
xe
x

x
+1
x
dx
.
6. (10 points) Show that of all the rectangles with given area
A
, the one with the smallest perimeter
is a square.
7. (10 points) A baseball diamond is a square with side 90 ft (see below). A batter hits the ball and
runs toward ﬁrst base with a speed of 24 ft/s as shown below. At what rate is his straightline
distance to the
second base
decreasing when he is 45 ft from home plate?
8. (20 points: 5 each) Consider the following graph of
df
dx
, where
f
(
x
) is a continuous function with
domain [0
,
6].
(a) On what intervals is
f
(
x
) increasing? Decreasing?
(b) On what intervals is
f
(
x
) concave up? Concave down?
(c) Find all local maximum and minimum points.
(d) Carefully sketch a possible graph of
f
(
x
) on its entire domain using the information from
above.
1
st
2
nd
3
rd
Home Base
(a) Problem 7
1
1
2
2
3
4
5
6

1
df
dx
x
(b) Problem 8
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