Math 125
Carter
Final Exam
Summer 2011
General instructions. Write your name on only the outside of your blue book. Do not
write on this test sheet. Do all your work inside your blue book. Write neat complete answers
to each of the problems below. If you work the test out of order, make sure that you label
the problems carefully.
The fundamentals of Thai curry are simple. Saute garlic and onion in hot oil, add Thai
curry paste. Add meat, vegtables, and/or tofu. When the meat is fully cooked, add a can of
coconut milk. You might want to add some fish saute (nuoc mam) and sugar. In addition
to serving over hot rice, you might like wrapping the mixture in warm corn tortillas.
1. Compute the derivatives (f 0 (x) or
dy
)
dx
of the following (5 points each):
(a)
f (x) = 3x2 − 2x + 7
(b)
y = cos (x2 + 1)
y = e(x
(c)
2 +1)
f (x) = x2 sin (ex )
(d)
(e)
f (x) =
(f)
f (x) =
Rx
0
sin (x)
(x2 +1)
√
(t + 1)/ t dt
x3 + xy = 4
(g)
2. (5 points) Give the -δ definition of the phrase:
lim f (x) = L.
x→x0
3. (5 points) Prove the product rule:
(f g)0 (x) = f (x)g 0 (x) + f 0 (x)g(x).
4. (10 points) Prove by induction,
n(n + 1)
1 + 2 + ··· + n =
2
3
3
3
1
2
.
5. (10 points) Compute the equation of the line tangent to the curve y =
√
x at the point
x = 25.
6. (10 points) Compute the limit
1
x+h
−
h
lim
h→0
1
x
7. (10 points) Water drips from a cone that has a radius of 10 centimeters at its top and
a height of 15 centimeters of at a rate of 2cm3 per second. At what rate is the height
changing when the height is 6 centimeters. (The volume of a cone is V = π3 r2 h where
r indicates the radius and h indicates the height).
8. (10 points) The area of a circle is increasing at the rate of 10 square centimeters per
second. How fast is the radius increasing when the radius is 10 centimeters? (The area
of a circle is A = πr2 ).
9. (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) =
x−1
x+1
(b)
f (x) = 3x2 + 5x − 17
(c)
f (x) = 3 sin (2x) for x ∈ [−2π, 2π].
10. (5 points each) Compute the following definite and indefinite integrals.
(a)
5
Z
(3x + 7) dx
2
2
(b)
Z
(x + 1)
√
dx
x
(c)
Z
sec2 (x) dx
(d)
Z
π/2
cos (x) dx
−π/2
(e)
Z
e10x dx
11. (10 points) Find the maximum value of the quantity A = xy subject to the constraint
that 3x + 2y = 12.
3