3. [14] A brushfire is burning on the prairie. The
fireline, enclosing the area which is burning, is a
circle. The radius r of the fireline is increasing at
the rate of 3 feet per minute. At the instant when
Second Test Answers
the area A which is burning is 4900 square feet,
Math 120 Calculus I
what is the rate at which that area A is increasing?
November, 2014
Since A = πr2 , taking the derivative with respect
to t gives
dr
dA
Scale. 90–100 A, 79–89 B, 60–78 C, 40–59 D. Me= 2πr .
dian 90.
dt
dt
p
When A = 4900, then r = 4900/π. So the rate
1. [14] Consider the curve whose equation is x3 + of change of A at that time is
y 3 = x − y.
p
da
dy
= 2π 4900/π 3
. (You don’t have to
a. Find the derivative
dt
dx
simplify it.)
square feet per minute. It’s not necessary to simDifferentiate the given equation with respect to
plify that on this test.
x to get
dy
dy
3x2 + 3y 2
=1−
4. [40; 8 points each part] Differentiate the foldx
dx
lowing functions. (Do not simplify your answers.)
dy
Solve for
to get
√
dx
a. f (x) = 5x3/2 + 6 x.
1 − 3x2
dy
= 2
15
6
dx
3y + 1
f 0 (x) = x1/2 + √
2
2 x
b. The point (x, y) = (1, 0) lies on that curve. Find b. f (x) = ln(sec x).
the slope of the tangent line there.
1
Evaluate the derivative in part a to get −2.
f 0 (x) =
sec x tan x
sec x
2. [14] Prove that the derivative of the function c. f (x) = (x2 + 5) tan x.
f (x) = x3 is f 0 (x) = 3x2 using only the definition
f 0 (x) = 2x tan x + (x2 + 5) sec2 x
of derivative in terms of limits and properties of
limits. Recall that we defined the derivative of a
ex
f
(x
+
h)
−
f
(x)
d.
f
(x)
=
.
2 + 5x + 2
function f by f 0 (x) = lim
.
x
h→0
h
0
f (x) =
=
=
=
(x + h)3 − x3
lim
h→0
h
x3 + 3x2 h + 3xh2 + h3 − x3
lim
h→0
h
2
2
3x h + 3xh + h3
lim
h→0
h
2
lim (3x + 3xh + h2 ) = 3x2
f 0 (x) =
ex (x2 + 45x + 2) − ex (2x + 5)
(x2 + 5x + 2)2
e. f (x) = xsin x . (Suggestion: logarithmic differention)
Take natural logs of the given equation to get
ln f (x) = sin x ln x.
h→0
1
Differentiate to get
f 0 (x)
sin x
= cos x ln x +
f (x)
x
Solve for f 0 (x) to get
0
f (x) = x
sin x
sin x
cos x ln x +
x
5. [10] Suppose that the domain of f is (−∞, ∞)
and f 0 (x) = 8x3 − 3x2 + 4x + 5. Determine all the
possible functions that f could be.
By inspection, one such is
f (x) = 2x4 − x3 + 2x2 + 5x.
Since any two functions that have the same derivative differ by some constant C, therefore all of them
are of the form
f (x) = 2x4 − x3 + 2x2 + 5x + C.
6. [10] Suppose that the local maximum values of
a function f occur at x = 2 and x = 6 while a local
minimum value occurs at x = 5. Sketch the graph
of such a function f that has local maximum values
at x = 2 and x = 6 and a local minimum value at
x = 5.
[Answer omitted since it’s a sketch]
2