Math 125
Carter
Test 1 Fall 2010
General Instructions: Write your name on only the outside of your blue book. Put your
test paper inside your blue book as you leave. Solve each of the following problems point
values are indicated on the problems.
To make a delicious peanut butter and jam sandwich start with a hearty whole grain
bread, some all-natural (no hydrogenated oil) peanut butter, and a high quality or home
made jam or preserves. Spread the peanut butter and jam thickly each on a single piece of
bread. Close sandwich. Not recommended for people with peanut allergies.
1. Use the rules for computing the limit for each of the following problems (8 points each).
(a)
x2 − 25
x→−5 x + 5
lim
(b)
1
1
− 2
2
(x + h)
x
1
lim
h→0 h
!
(c)
tan (6x)
x→0
5x
lim
2. Use the rules for differentiation (power, product, and quotient rules) to compute the
derivatives for the following functions (8 points each).
(a)
f (x) = 2x4 − 4x3 + 10x2 − 12x + 2
(b)
f (x) = (x2 + x + 1)ex
(c)
f (x) =
x2 − 5
x4 − 4
(d)
f (x) = (x + 2) sin (x)
3. (10 points) Use the rules of differentiation to compute the equation of the line tangent
to the curve y = x2 − 3x + 5 at a = 1.
4. (10 points) Solve the inequality
8 < |2x − 6|
5. (10 points) Give a proof that
lim+
h→0
sin (h)
= 1.
h
1
6. (10 points) Is the function f (x) given below continuous at x = 2? Explain.
(
f (x) =
x2 − 3 if x ≤ 2
x − 1 if 2 < x
7. (10 points) Complete the square and sketch the graph of the parabola
y = 3x2 − 24x − 10.
2