Math 125
Carter
Test 1 Summer 2011
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 peel a hard boiled egg that is hot from the boiling water, immerse the egg in icecold water for 60 seconds in order to contract the white of the egg away from the shell.
Subsequently, immerse the egg in the hot water from the boiling pot for 10 seconds. This
allows the shell to expand away from the cooled egg white. Peel immediately.
1. Use the rules for computing the limit for each of the following problems (5 points each).
(a)
x3 − 4x
lim
x→2 x − 2
(b)
lim
1
(3+h)
−
1
3
h
h→0
(c)
lim
x→0
sin (2x)
sin (3x)
2. (5 points) What does the expression
lim f (x) = L
x→c
mean? State the definition of a limit.
3. (10 points) Give a proof that
lim+
h→0
sin (h)
= 1.
h
4. (5 points) Use the limit in Problem # 3 to prove that
cos (h) − 1
= 0.
h→0
h
lim
5. Use the rules for differentiation (power, product, chain, and quotient rules) to compute
the derivatives for the following functions (5 points each).
(a)
f (x) = 3x3 − 4x2 +
(b)
√
5x − 14
√
f (x) = sin (x) x2 + 1
(c)
f (x) =
1
ex
cos (x)
(d)
f (x) = e−x
(e)
2
√
f (x) = sin ( x2 + 1)
(f)
f (x) = x2 tan (x)
(g)
f (x) = x + x−1
6. (10 points) Use the rules of differentiation to compute the equation of the line tangent
to the curve f (x) = −5x2 + 40x + 30 at the point x = 9.
7. (10 points) Complete the square and sketch the graph of the parabola
y = −5x2 + 20x + 17.
8. (10 points) A particle is shot upwards from the ground at the rate of 2 meters per
second. When does it reach its apex? How high does it go? The height of the particle
is given by s(t) = −5t2 + v0 t + s0 where s(t) is the distance above the ground measured
in meters, v0 indicates the initial velocity, and s0 indicates the initial position which,
in this case, is s0 = 0.
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