Math 125
Carter
Final Exam
Fall 2012
General instructions. Write your name on only the outside of your blue book. Do not
write on this test sheet. However there is a separate sheet attached. Write your
solutions to those problems alone on that sheet. Write your name on the back
of that sheet. Each of these is worth 10 points. Scrambled eggs and toast with coffee
makes a nice breakfast.
1. Compute the following limits (you may use L’Hospital’s rule if it applies)(5 points
each):
sin (3x)
sin (2x)
√
√
3+h− 3
lim
h→0
h
h
e −1
lim
h→0
h
(a)
lim
θ→0
(b)
(c)
2. Compute the derivatives (f 0 (x) or
dy
)
dx
of the following (5 points each):
(a)
f (x) = x3 − 2x2 + 3x − 5
(b)
y = sin (x2 + 1)
(c)
y = ecos (x)
(d)
f (x) = e2x tan (x)
x
(x2 −1)
(e)
f (x) =
(f)
x2 y + xy 2 = 4
3. (5 points) Prove the product rule:
(f g)0 (x) = f (x)g 0 (x) + f 0 (x)g(x).
4. (5 points) Give the -δ definition of the phrase:
lim f (x) = L.
x→x0
1
5. (10 points) Prove by induction,
1 + 2 + ··· + n =
n(n + 1)
.
2
6. (10 points) Compute the equation of the line tangent to the curve y =
√
x + 1 at the
point x = 15.
7. (10 points) A 6 foot tall man walks at night away from a 15 foot tall lamppost at the
rate of 6 feet per second. How fast is the length of his shadow increasing?
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) =
x2
x
−1
(b)
f (x) = x3 − x
(c)
f (x) = 2 sin (3x) for x ∈ [−2π, 2π].
9. (5 points each) Compute the following definite and indefinite integrals.
(a)
5
Z
(4x + 7) dx
2
(b)
Z
(t + 1)
√
dt
t
(c)
Z
sec (x) tan (x) dx
2
(d)
π/3
Z
sin (x) dx
0
(e)
Z
3
e3x dx