Math 125
Carter
Final Exam
Fall 2013
General instructions. Write your name on only the outside of your blue book. Do not
write on this test sheet. Put your test sheet into the blue book as you leave. Don’t forget
the +C. I have enjoyed having each of you in my class, and I wish you real success for the
rest of your college career. Obey local traffic laws.
1. (20 points) Match the four statements with the corresponding term or theorem name.
A. lim f (x) = L.
x→c
B. The derivative of y = f (x) at x = c.
C. The Fundamental Theorem of Calculus.
D. A function is continuous at x = c.
I. Let y = f (t) denote a continuous function for any t ∈ [a, b]. Define
Z x
f (t)dt.
A(x) =
a
Then
A0 (x) = f (x).
II. For any > 0 there is a δ > 0 such that 0 < |x − c| < δ implies that
|f (x) − L| < .
III.
lim f (x) = f (c).
x→c
IV.
f (c + h) − f (c)
.
h→0
h
lim
1
2. Compute the following limits (you may use L’Hospital’s rule if it applies)(5 points
each):
(a)
lim
θ→0
1
3+h
− 31
lim
h→0
h
h
3 −1
lim h
h→0 5 − 1
(b)
(c)
3. Compute the derivatives (f 0 (x) or
(a)
(b)
(c)
(d)
sin (5x)
tan (2x)
dy
)
dx
of the following (5 points each):
f (x) = x4 − x3 + x2 − x + 4
y = e(x
2 +1)
f (x) = x arcsin x
f (x) =
h
sin (x)+1
sin (x)−1
i3
(e)
x3 y + xy 3 = 625
(f)
f (x) =
Rx
0
2
et dt
4. (10 points) A hot air balloon rising straight up from a level field is tracked by a range
finder 150 meters from the liftoff point. At the moment that the range finder’s elevation
angle is π/4, the angle is increasing at the rate of 0.14 rad/min. How fast is the balloon
rising at that moment?
150 meters
2
5. (10 points) Compute the equation of the line tangent to the curve y =
√
x − 2 at the
point x = 27.
6. (10 points) A 5 foot tall woman walks at night away from a 20 foot tall lamppost at
the rate of 7 feet per second. How fast is the length of her shadow increasing?
7. (10 points) A box with an open top is to be constructed from a rectangular piece of
cardboard that is 14 inches by 22 inches by cutting out equal squares of side length
x from each corner and folding up the sides as indicated. Find the box of Maximum
volume that can be constructed in this way.
x
x
x
x
x
x
14”
x
x
22”
8. (10 points each) Sketch the graphs of the following functions. Compute x-intercepts,
critical points (critical values when feasible), regions of increase and decrease, and
compute the second derivative and regions in which the function is concave up and
concave down.
(a)
f (x) = x3 (x − 2)
(b)
f (x) =
x−2
x+1
(c)
f (x) =
x+1
(x − 2)(x − 3)
(d)
f (x) = x sin (x) for x ∈ [0, 2π].
3
9. (5 points each) Compute the following definite and indefinite integrals.
(a)
7
Z
(3x − 2) dx
1
(b)
(t2 + 1)
√
dt
t
Z
(c)
π
Z
sin (x) dx
0
(d)
Z
0
4
3
e2x dx