Math 2250 Written HW #6 Solutions
1. Roughly speaking, we can approximate the human-equivalent age of a dog which is y years
old by the function
y2
√
h(y) = 16 y + .
15
This isn’t perfect, but it’s certainly more accurate than claiming that a 1-year-old dog is
equivalent to a 7-year-old human.
Anyway, assuming this function is correct:
(a) What information does the function h0 (y) convey? To guide your thinking, consider the
following: if h0 (2) ≈ 5.92, what does that really tell us about how a 2-year-old dog is
aging?
Answer: By definition, the derivative h0 (y) gives the rate of change of the function
h(y). Recall that h(y) does the following: given a dog which is y years old, h(y) tells
us the human-equivalent age of the dog. So the rate of change of that function tells us
how fast the human-equivalent age of a dog is changing. For example, h0 (2) ≈ 5.92 tells
us that when it is 2 years old a dog is aging by about 6 human years per actual year.
In other words, a 2-year-old dog is aging 6 times faster than a human. Whereas, since
8
h0 (4) = 4 + 15
(as we’ll see below), a 4-year-old dog is aging about 4.5 times faster than
a human.
In general, h0 (y) tells us how many times faster a y-year-old dog is aging than a human.
(b) What is h0 (4)?
Answer: We can re-write h(y) as
h(y) = 16y 1/2 +
1 2
y ,
15
so by splitting up and using the Power Rule, we see that
1 −1/2
1
16
2y
8
2y
0
h (y) = 16
y
+ (2y) = √ +
=√ + .
2
15
2 y 15
y 15
Therefore,
8
2·4
8
h0 (4) = √ +
=4+ .
15
15
4
1
2. Consider the function f (x) =
x2 e x
.
1+x2
What is f 0 (x)?
Answer: By the quotient rule,
f 0 (x) =
=
=
=
=
=
d
d
(1 + x2 ) dx
(x2 ex ) − x2 ex dx
(1 + x2 )
(1 + x2 )2
d
d
(1 + x2 ) dx
(x2 )ex + x2 dx
(ex ) − x2 ex (2x)
(1 + x2 )2
(1 + x2 )(2xex + x2 ex ) − x2 ex
(1 + x2 )2
x
2
(2xe + x ex + 2x3 ex + x4 ex ) − 2x3 ex
(1 + x2 )2
2xex + x2 ex + x4 ex
(1 + x2 )2
xex (2 + x + x3 )
(1 + x2 )2
3. Find the derivative of the function g(x) = sin(x) cos(x).
Answer: By the product rule,
d
d
(sin(x)) cos(x) + sin(x) (cos(x))
dx
dx
= (cos(x)) cos(x) + sin(x)(− sin(x))
g 0 (x) =
= cos2 (x) − sin2 (x).
If we use the double-angle identity cos(2θ) = cos2 (θ) − sin2 (θ), we could also write g 0 (x) as
g 0 (x) = cos(2x).
2