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MTH 121 — Summer — 2005 Essex County College — Division of Mathematics
Test # 1^1 — Created May 31, 2005
Name:
Signature:
Show all work clearly and in order, and box your final answers. Justify your answers alge-
braically whenever possible. You have at most 110 minutes^2 to take this 100 point exam. No
cellular phones allowed.
- For what values of x does the graph of f (x) = x + 2 sin x have a horizontal tangent? 9 points
- Find f ′^ (a) using the definition. 9 points
f (z) =
2 z + 1
z + 3
(^1) This document was prepared by Ron Bannon using LATEX 2 ε. (^2) 8:30 a.m. until 10:20 a.m..
- If a ball is thrown into the air with a velocity of 50 ft/s, its height in feet after t seconds
is given by h (t) = 50t − 16 t^2.
(a) Find the average velocity for the time period beginning at t = 3 and lasting
i. 1 second 2 points
ii. 0.1 second 2 points
iii. 0.01 second 2 points
(b) Predict the instantaneous velocity when t = 3. 1 points
- Find the number(s) at which f is discontinuous. At which of these number(s) is f contin-
uous from the right, from the left, or neither? 8 points
f (x) =
x + 2 if x < 0
2 x^2 if 0 ≤ x ≤ 1 2 − x if x > 1
- Show by means of an example that 5 points
lim x→a
[f (x) · g (x)]
may exist even though neither
lim x→a
f (x) nor lim x→a
g (x)
exists.
- Differentiate. 5 points
f (x) =
x^2
3 x^2 − 2 x + 1
- If f (x) = x^3 − x^2 + x, show thet there is a number c such that f (c) = 10. 5 points
- Evaluate the limit, if it exists. 6 points
lim x→ 0
x
1 + x
x
- Use the definition of the derivative to show that f is not differentiable at x = 0. 8 points
f (x) = |x|
- Differentiate. 4 points
f (x) = (1 + 4x)
3 + x − x^2
