Fall 2007
ARE211
Problem Set #09 Answer key
First Calculus Problem Set
Please note: There are parts of one question in this assignment to which you may want
to reply: “I beg your pardon?” or something possibly ruder than this. That kind of
response is entirely appropriate, but you should add a sentence explaining why you
are led to make such a remark.
(1) Consider the function
f
(
x
) = 20
x
3

120
x
2

5
x
+ 36
a) What is the derivative of this function?
Ans:
f
prime
(
x
) = 60
*
x
2

240
*
x

5
b) What is the derivative of this function evaluated at
x
0
= 4?
Ans:
f
prime
(
x
0
= 4) = 60
*
4
2

240
*
4

5 = 240
*
4

240
*
4

5 =

5
c) What is the differential of this function at
x
0
= 4?
Ans:
df
=

5
dx
d) Approximate the change in the function when moving from
x
0
= 4 to
x
1
= 5.
Ans:
df

x
0
=4
=

5
*
1 =

5
e) What is the actual change in the function when moving from
x
0
= 4 to
x
1
= 5?
Ans:
Δ
f
=
f
(
x
1
= 5)

f
(
x
0
= 4) =

489

(

624) = 135
.
Not a very good approximation, huh! Any ideas why? Try plotting the function.
(2)
Multivariate calculus drill:
Simon and Blume, question 14.1, parts (a), (d) and (f);
Ans:
a)
∂f
∂x
= 8
xy

3
y
3
+ 6
,
∂f
∂y
= 4
x
2

9
xy
2
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2
d)
∂f
∂x
= 2
e
2
x
+3
Y
,
∂f
∂y
= 3
e
2
x
+3
y
.
f)
∂f
∂x
= 6
xy

7
√
y,
∂f
∂y
= 3
x
2

7
x
2
√
y
(3)
Gradients:
Simon and Blume, question 14.19
Ans:
Δ
y
2
e
3
x
= (3
y
2
e
3
x
,
2
ye
3
x
)
.At
(0
,
3)
, this vector is proportional to
(9
,
2)
. Normalizing to
length
1
, the vector is
(9
/
√
85
,
2
/
√
85)
.
(4)
More differential approximations:
Simon and Blume, question 14.4.
Ans:
a)
Q
= 5400
b)
Q
(998
,
216) = 5398
.
798
.
The approximation gives
Q
≈
5392
.
8
, which is in error by

0
.
002
.
c)
Q
(1000
,
217
.
5) = 5412
.
471
.
The approximation gives
Q
≈
5412
.
5
, which is in error by

0
.
029
.
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 Fall '07
 Simon
 Calculus, Derivative, lim, Blume

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