Math 125
Carter
Take Home Quiz
Instructions. This quiz is singular. Take it home this weekend, and solve each problem. I
will grade each problem on an “all correct” or not scale: 1 or 0. These problems are edited
and culled from the last two finals that I gave. They are not indicative of the entire final,
but a substantial subset thereof. So you should work these to begin studying for the final.
Now is a good time to start.
1. 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 ).
2. Coffee drips at a rate of 5 cubic centimeters per second from a 18 centimeter tall Melita
cone whose radius is 9 centimeters. How fast is the height of the coffee decreasing when
the height is 5 centimeters? The volume of a cone is V = π3 r2 h.
3. A rectangle is inscribed with two vertices on the parabola y = 48 − x2 and with its
base along the x-axis. What are the dimensions of the rectangle of maximal area that
fits this description?
4. Sketch the graphs of the following functions.
(a)
f (x) = 3x2 + 5x − 17
(b)
f (x) = x2 (x − 4)
(c)
f (x) =
x−1
x2 + x
f (x) =
x2 + x
x−1
f (x) =
x−1
x+1
(d)
(e)
(f)
f (x) = 2 sin (3x − π)
for x ∈ [0, 2π].
1
(g)
f (x) = 3 sin (2x) for x ∈ [−2π, 2π].
5. (5 points each) Compute the following anti-derivatives and definite integrals
(a)
4
Z
x3 dx
1
(b)
Z
(x + 1)
√ dx
x
(c)
Z
dx
2x
(d)
Z
e2x dx
(e)
π/2
Z
cos (x)dx
−π/2
(f)
5
Z
(3x + 7) dx
2
(g)
Z
sec2 (x) dx
(h)
Z
π/2
cos (x) dx
−π/2
(i)
Z
2
e10x dx