Sample Midterm Problems Math 1220
Fall 2010
Instructor: Rémi Lodh
Sept. 28th 2010
r
x−1
dy
, find
.
1. If y = ln
x
+
1
dx
√
3
x2 + 1
dy
2. If y = √
, find
by logarithmic differentiation.
dx
R x+3
3. Find tan(x)dx.
R
x
4. Find
dx.
x2 + 1
R
7
dx.
5. Find
2x + 5
6. Show that the function f (x) = x5 + 4x3 + 2x has an inverse (you don’t need to
find the inverse, just show that it exists!).
2x + 1
7. Find the inverse of the function f (x) :=
. Check that your answer is
x−1
indeed the inverse function.
8. If f −1 (y) is the inverse of the function f (x) = x5 + 4x3 + 2x, find (f −1 )0 (7).
R
2
9. Find xex dx.
R x2
10. Find x2 dx.
d
((2x)x ).
11. Find
dx
R
1
12. Find (x1/2 + x )dx.
2
13. Bacteria in a lab grows exponentially (i.e. the rate of growth is proportional
to the population). Suppose that at time t = 0 there were 15000 bacteria, and at
time t = 3 there were 60000. Predict the number of bacteria at time t = 12.
14. Solve the differential equation
dy
+ xy = x.
dx
15. Solve the differential equation
x
dy
+ y = cos(x)
dx
given that y = π if x = π.
16. Consider the differential equation
dy
= xy.
dx
1
Let y be the solution to this equation satisfying y(0) = 1. By applying Euler’s
Algorithm with h = 1/4, approximate y(1). Give your answer as a fraction or a sum
thereof.
R
1
dx.
17. Find √
4 − 9x2
x
R
e
dx.
18. Find
1 + e2x
R
dx
p
19. Find
.
x 1 − (ln x)2
R
sin(x)
20. Find
dx.
2
R 1 + cos2 (x)
21. Find R x tan(x )dx.
22. Find R x sin(x)dx.
23. Find R ln(x)dx.
24. Find R sin−1 (x)dx.
25. Find R ex cos(x)dx.
26. Find R sin2 (x)dx.
3
27. Find R tan
(x)dx.
√
3
2x
+ 1dx.
28. Find x
R√
29. Find R 1 − x2 dx.
30. Find 10x + 5(x − 2)2 (x + 3)dx.
R 5x2 + 15x + 10
31. Find
dx.
(x + 7)(x2 + 1)
tan(x)
32. Find limx→0
.
1 + ln(x)
2
cos(x ) + 1
33. Find limx→0
.
x3
−1
sin (x) − x
34. Find limx→0
.
x2
ln(x)
35. Find limx→0+
.
cot(x)
36. Find limx→0+ (1 + x)cot(x) .
R ∞ ln(x)
37. Find 1
dx.
x2
R ∞ dx
.
38. Find 1
x
R ∞ dx
39. Find 1
.
R ∞ x2
40. Find 0 cos(x)dx.
R∞
dx
41. Find −∞
.
1 + x2
R 1 dx
42. Find 0 √ .
x
R 1 dx
43. Find −2 2 .
x
R 10
dx
.
44. Find 0
(x − 3)4/5
2