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Math 165 F3. Test 1 (Homework) Name:
This is a homework test. You should do it on your own. Please attach addi-
tional sheets of paper with your solutions. Circle two problems that you want
me to grade. Show all your work. No credit is allowed for mere answers with
no work shown.
- (24 points) Evaluate the following limits:
(a) lim
z→ 3
z
2 − 5 z + 6
√
3 z − 3
(b) lim
z→ 0
z
2 − 4
3 z
3
2 − 2 z
(c) lim
z→ 2 −
z
2 − 4
3 z
3
2 − 2 z
(d) lim
x→ 0
5 x csc 3x
cos 5x
(e) lim
x→
π 2
−
3 x
sin 2x
(f) lim
x→∞
3 x
2
3 x
(g) lim
x→ 0
x
2
tan 3x
(h) lim
x→∞
5 x
5
2
x(3 − 2 x
2 )
- (16 points) Let f (x) =
x
if x < 0 ,
x
2
x + 1 if x > 1.
and g(x) =
−x
2
sin (3x)
x
if 0 < x ≤ 1 ,
2 x − 1 if x > 1.
Determine if the functions are continuous at each of the points 0 and 1. Justify your answer.
- (20 points) Use the definition of the derivative to compute
d
dx
2 x) and
d
dx
[
x + 3
]
- (24 points) Find f
′ (x) using the rules.
You DO NOT need to simplify your answer.
(a) f (x) =
3 x
5 − 1
3 − cos x
(b) f (x) =
3
x
3 +^
x
3
3
(sec x + 5 cot x)
(c) f (x) =
csc(x) − csc
x
(d) f (x) =
cos 2x +
x
10
(e) f (x) = tan
[
sin (3x
2
7
]
(f) f (x) =
x
2 − 1
x
2
5
x
- (a) (16 points) Write an equation of the tangent line to the curve
x
3 y
3 = 2x
2
2 − 1 at the point (2, 1).
(b) Write an equation of the tangent line to the curve
x sin y −
x √
2
= 0 at the point (2,
π
4
