Exam # 1
Spring 2005
MATH 1220-01
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) No books, notes or calculators may be used.
(5) A list of formulas is attached at the end.
Problem #
1
2
3
4
5
# Points
1
6
7
8
9
10
Total
2
Problem 1. Let y = ln
√
2x + 1
.
xex
(a) Simply the function y, using the properties of the natural logarithm.
(b) Calculate:
dy
.
dx
3
Problem 2. Calculate the following differentials:
(a) Dx [32x + log3 (2x)].
(b) Dx (x2x ) .
4
Problem 3. Calculate the following definite integral:
Z
1
2
cosh(ln x)
dx.
x
5
Problem 4. Calculate the following definite integral:
Z
2
1
√
t t − 1.
6
Problem 5. Calculate the following integrals:
Z 1
x
dx.
(a)
4
0 1+x
(b)
Z
x3
dx.
1 + x4
7
Problem 6. A tank initially contains 150 gallons of brine with 75 pounds of salt. Brine containing
2 pounds of salt per gallon is entering the tank at the rate of 3 gallons per minute and is flowing
out at the same rate; the mixture in the tank is kept uniform by constant stirring. Let y denote
the quantity of salt at time t.
(a) Write a first order linear equation in y(t) that describes the above situation.
(b) Find the function y(t).
8
Problem 7. Compute the following integrals:
Z
(a) tan x sec3 x dx.
(b)
Z
0
π
2
sin x sin(2x) dx.
9
Problem 8. Consider the following integral:
Z
2x + 1
√
dx.
5 − x2 − 4x
(a) Complete the square inside the square root and find the trigonometric substitution that helps
you compute the integral.
(b) Write the integral in terms of the new variable.
(c) Compute the integral.
10
Problem 9. Calculate the following definite integral:
Z
0
−1
(x + 1)e3x dx.
11
Problem 10. A radioactive substance has a half-life of 750 years. If there were 15 grams initially,
how much would be after 100 years ? Hint: use that the decay equation is y(t) = y(0)ekt.
12
Useful formulas
(1) Half-angle identities:
1 − cos 2x
2
1
+
cos
2x
cos2 x =
2
sin 2x = 2 sin x cos x
sin2 x =
(2)
1
sin mx cos nx = [sin(m + n)x + sin(m − n)x]
2
1
sin mx sin nx = [cos(m + n)x − cos(m − n)x]
2
1
cos mx cos nx = [cos(m + n)x + cos(m − n)x]
2
(3) Pythagorean Identity:
1 + tan2 x = sec2 x
Z
du
u
√
(4)
+C
= sin−1
a a2 − u2
Z
1
1
du
|u|
a
−1
−1
√
= sec
(5)
+ C = cos
+C
2 − a2
a
a
a
|u|
u
u
Z
(6)
sec x dx = ln | sec x + tan x| + C