See the attached file for questioins
Thehonest
2.docDIRECTIONS: Show as much work as possible within each question as I grade on both the process and the final answer. TI-89’s are wonderful calculators, but they don’t show me if you know anything about calculus! You may have 2 hours maximum to complete the exam. Show all work on the exam itself, you should not use any outside paper, notes, etc.
1. (7 pts each) Determine the following antiderivatives (don’t worry about simplifying, just show the rules)
a.
x2 x3 − 3( )2.3 dx∫
x
2
x
3
-3
()
2.3
dx
ò
b.
x2 +1 x4
dx∫
x
2
+1
x
4
dx
ò
c.
x459 + 9+ e2πx( )dx∫
x
459
+9+e
2px
()
dx
ò
2. (7 pts each) Calculate the value of each definite integral (Show work!):
a.
3 x + 5 x4
− 8x⎛ ⎝⎜
⎞ ⎠⎟ dx
1
2
∫
3
x
+
5
x
4
-8x
æ
è
ç
ö
ø
÷
dx
1
2
ò
b.
2x2 +1( ) 2x3 + 3x( )dx 0
1
∫
2x
2
+1
()
2x
3
+3x
()
dx
0
1
ò
c.
9 x4 dx
3
∞
∫
9
x
4
dx
3
¥
ò
3. (6 pts each) a. Approximate the area under the curve
f x( ) = 4 − x2
fx
()
=4-x
2
and above the x-axis by splitting the region from x 0 to x 2 into 4 equal subintervals (rectangles) and using the left endpoints of thesubintervals as the heights.
b. Use the Trapezoidal Rule with n 4 to estimate area under the curve
f x( ) = 4 − x2
fx
()
=4-x
2
c. Use geometry to find the exact value of
4 − x2 dx 0
2
∫
4-x
2
dx
0
2
ò
, and compare with the answer obtained from part (a), and (b). Which method is more accurate, part A or part B?4. (8 pts) Determine the area between the curves
f x( ) = x
fx
()
=x
and
g x( ) = x3
gx
()
=x
3
.5. (8 pts) A stock analyst plots the price per share of a certain stock as a function of time and finds that it can
be modeled by the function S(t) 255e0.01t where t is the time (in years) since the stock was purchased.
Find the average price of the stock over the first six years of its purchase.
6. (8 pts) Use the consumer’s surplus formula
D q( )− p0⎡⎣ ⎤⎦ 0
q0
∫ dq
Dq
()
-p
0
é
ë
ù
û
0
q
0
ò
dq
to determine the consumer’s surplus if the demand function for extra virgin olive oil is given by
D q( ) = 32000 2q + 8( )3
Dq
()
=
32000
2q+8
()
3
if the supply and demand are in equilibrium at q 67. (8 pts) Sketch the region and then calculate the volume of the solid of revolution formed by rotating the region bounded by f (x) x 1 y 0, x 1, and x 5 around the x-axis.
8. The function f (x) 390 represents the rate of flow of money in dollars per year. Assume a 3-year period for t and a rate r of 10% compounded continuously and determine the following:
a. (6 pts) The present value
P = f x( )e−rx dx 0
t
∫ ⎛ ⎝⎜
⎞ ⎠⎟
P=fx
()
e
-rx
dx
0
t
ò
æ
è
ç
ö
ø
÷
b. (2 pts) The accumulated amount
A = ert f x( )e−rx dx 0
t
∫ ⎛ ⎝⎜
⎞ ⎠⎟
A=e
rt
fx
()
e
-rx
dx
0
t
ò
æ
è
ç
ö
ø
÷
