Math 1210-1 Midterm 2
Name:
Instructions: There are TEN problems in this exam. Answer the questions
in the spaces provided on the question sheets. If you need more space, use
the bottom of this page, but remember to indicate which problem you are
doing. Partial credit will be awarded. Calculators are NOT allowed. This
exam is closed book and closed notes. Show you work on each problem unless
otherwise declared.
The following values of trigonometric functions are provided for your convenience:
3π
π
= 1,
sin π = 0,
sin
= −1,
2
2
3π
π
cos 0 = 1, cos = 0,
cos π = −1, cos
= 0,
2
2
π
3π
tan 0 = 0, tan = DNE, tan π = 0,
tan
= DNE.
2
2
sin 0 = 0,
sin
The following special sums are provided for your convenience:
n
X
i=
i=1
n
X
n(n + 1)
2
n(n + 1)(2n + 1)
6
i=1
µ
¶2
n
X
n(n + 1)
3
i =
2
i=1
i2 =
1
1. Suppose f (x) = x3 − 3x + 1 defined on the closed interval [−2, 3].
(a) (6 points) Find all critical points.
(b) (4 points) Find the maximum value and the minimum value.
2. Suppose f (x) = 2x3 − 6x + 1 defined on all real numbers.
(a) (5 points) Find the interval where f (x) is decreasing.
(b) (5 points) Find the interval where f (x) is concave up.
2
3. (10 points) Suppose f (x) = x4 − 4x3 . Find the local maximum and local minimum. You may run
either test you like.
4. (10 points) Evaluate the following definite integral:
Z √π
2
x · sin(x2 ) dx
0
5. (10 points) Find the following derivative of definite integral:
d
dx
Z
0
tan x
Simplify your answer to get 2 bonus points.
sin x
Hint: tan x =
.
cos x
3
1
dt
1 + t2
Z
6. Suppose
Z
2
Z
3
f (x) dx = 1,
3
g(x) dx = −1 and
0
0
Z
g(x) dx = 1.
2
2
(a) (4 points) Evaluate
g(x) dx.
0
Z
(b) (6 points) Evaluate
2
(5f (x) − 3g(x) + 1) dx.
0
7. (10 points) The graph of g(x) is given in the following picture.
(a) g(x) is increasing on the intervals [
,
] and [
(b) g(x) is concave up on the interval [
,
].
(c) The maximal value is achieved at x =
.
The local maximal values are achieved at x =
(d) The inflection points are x =
(e) The singular point is x =
and x =
.
You don’t have to show your work in this problem.
4
,
and x =
.
].
.
8. (10 points) Find the function f (x) such that f 0 (x) =
x
and f (1) = 2.
f (x)
9. (10 points) Simplify the following sum:
n
X
(i + 1)(i + 2)
i=2
Notice that the sum is taken from i = 2 instead of i = 1.
5
10. (Bonus 10 points) Suppose


2
when x < 0





DNE
when x = 0


2x
when 0 < x < 1
f (x) =


 DNE when x = 1



 1

when x > 1
x2
Find the unique continuous function F (x) such that F 0 (x) = f (x) and F (−1) = 1.
Hint 1: Since f (x) is given piece by piece, your answer should also be written piece by piece.
Hint 2: Sketching a graph of F (x) will help a lot.
6