Math 1210-1 Midterm 1
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 the last 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 given for your convenience:
π
3π
= 1,
sin π = 0,
sin
= −1,
2
2
3π
π
cos π = −1, cos
= 0,
cos 0 = 1, cos = 0,
2
2
π
3π
tan 0 = 0, tan = DNE, tan π = 0,
tan
= DNE.
2
2
sin 0 = 0,
sin
1
1. (10 points) Find the following derivatives. You don’t have to show your work.
(a) Dx (x1025 ) =
√
(b) Dx ( 4 x) =
(c) Dx (x−3 ) =
(d) Dx (2x3 + 3x2 + 5x − 1) =
(e) Dx401 (sin x) =
2. (10 points) Suppose

2

if x < 0
 x
f (x) =
1
if x = 0


x − 1 if x > 0
Find the following value and limits. You don’t have to show your work.
(a) f (0) =
(b) lim+ f (x) =
x→0
(c) lim− f (x) =
x→0
(d) lim f (x) =
x→0
(e) lim f (x) =
x→1
3. (10 points) Find the following limit:
lim
x→0
2
cos x
x2
4. (10 points) Find the following limit:
x2 + 2x − 3
x→1
x2 − 1
lim
5. (10 points) Find the following derivative:
Dx
x2
x−1
6. (10 points) Find the following limit:
x4 + x2 + 1
x→∞ x3 − x + 100
lim
3
7. (a) (5 points) Find dy/dx if y is given by the following equation x2 + y 2 = 1, i.e., the unit circle.
(b) (3 points) Find the slope of tangent line of the unit circle at
(c) (2 points) Find the slope of the radius from the origin to
between this radius and the tangent line in (b)?
4
√
√
3 1
2 , 2
3 1
2 , 2
.
. What is the relation
8. (10 points) Find the equation of the tangent line of f (x) = sin(πx) at (1,0).
9. (a) (5 points) Estimate
(b) (5 points) Estimate
√
8.97
√
3
8.03
5
10. (Bonus) Suppose that f (x) is an odd function and the positive part of the graph is as follows:
b
a
-a
b
f(x)
(a) (2 points) Sketch the negative part of the graph.
(b) (2 points) Sketch the tangent lines at (a, 0) and (−a, 0), respectively.
(c) (3 points) Suppose that f 0 (a) = −1. Find f 0 (−a).
Hint: You are going to use the identity f (−x) = −f (x) for odd functions.
(d) (3 points) Is f 0 (x) an even/odd/neither function? Circle your answer and justify it.
Hint: Compare f 0 (−x) with f 0 (x).
6