Exam # 3
Fall 2005
MATH 1100-01
Instructor: Oana Veliche
Time: 50 minutes
NAME:
ID#:
INSTRUCTIONS
(1) Fill in your name and your student ID number.
(2) There are 10 problems, each worth 10 points.
(3) Justify all you answers. Correct answers with no justification will not be given any credit.
(4) Use (non-graphing) calculators only on the problems marked with c .
(5) A list of definitions and formulas is attached at the end.
Problem #
1
2
3
4
5
# Points
1
6
7
8
9
10
Total
2
Problem 1. (a) Find the indefinite integral:
Z
√
1
3 x − √ dx.
x
(b) Find the equation of the function f whose graph passes through the point (1, 2), if
√
1
f′ = 3 x − √ .
x
3
c Problem 2. Find the change in profit if the marginal profit is given by
2000
dP
= 0.4 −
dx
x
and the number of units increases by 100 from x = 5000.
4
Problem 3. Using the general power rule, or the method of substitution find the indefinite integral:
Z
3x(2x2 + 1)3 dx.
5
1
Problem 4. Use the definite integral to find the area of the region bounded by the graph of y = 2 ,
x
the x-axis and the vertical lines x = 1 and x = 2.
6
Problem 5. Use the definite integral to find the area of the region bounded by the curves y = x2
and y = 3x. (Hint: Set the functions equal to each other in order to find the intersection points
and then graph the functions.)
7
Problem 6. Using the method of substitution evaluate the definite integral:
Z 1
x
dx.
3
0 (x + 1)
8
Problem 7. Using the method integration by parts find the indefinite integral:
Z
x3 ln x dx.
9
Problem
Z 8. Find the indefinite integrals:
3
(a) x2 ex +1 dx.
(b)
Z
x2 + 1
dx.
x3 + 3x
10
c Problem 9. A company expects its income c during the next 10 years to be modeled by
c(t) = 450. Assuming an annual inflation rate of 4%, what
Z is the present value of this income?
t1
c(t)e−rt dt.)
(Hint: You may use the formula of the Present value=
0
11
Problem 10. Using partial fractions, find the indefinite integral:
Z 3
x + x2 + 1
dx.
x2 + x
12
(5 points) Bonus Problem. Evaluate the definite integral
Z
3
−2
|4x − 2| dx.
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Useful formulas
(1)
Z
Z
xn dx =
xn+1
+ C, for n 6= −1.
n+1
1
dx = ln |x| + C.
Z x
(3)
ex dx = ex + C.
(2)
Z
du
un+1
(4) The general power rule:
un
dx =
+ C for n 6= −1.
dx
n+1
Z
Z ′
1 du
u
dx =
dx = ln |u| + C.
(5)
u dx
Z u
du
(6)
eu
dx = eu + C.
dx
Z
Z
(7) Integration by parts:
u dv = uv − v du.
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