406 Appendix C
13. Let x
ij
= barrels of gasoline of type i used to make fuel of type j
F
j
= barrels of fuel of type j
Prot = Revenue – Cost
The objective function and constraints can be written as
Minimize
( ) ( )
( )
90 100 60 65
70
1 2 11 12 21 22
31 32
F F x x x x
x x
+ − + − +
− + −− +80
41 42
( )x x
subject to
75x
11
+ 85x
21
+ 90x
31
+ 95x
41
− 80F
1
≥ 0
75x
12
+ 85x
22
+ 90x
32
+ 95x
42
− 90F
2
≥ 0
F
1
+ F
2
≥ 6000
0 ≤ (x
11
+ x
12
) ≤ 3000
0 ≤ (x
21
+ x
22
) ≤ 4000
0 ≤ (x
31
+ x
32
) ≤ 5000
0 ≤ (x
41
+ x
42
) ≤ 4000
14. Functions at plots (a) and (d) are convex (see Figure C.4).
15. The Taylor series of a function f(x) at x = a is given by
f a f a x a
f a
x a( ) ( )( )
( )
!
( )+ − + − +
′
′′
2
2
The Taylor series for the function ln(x − 1) at x = 3 is given by
ln2
1
2
3
1
8
3
2
+ − − −( ) ( )x x
407Appendix C
16. The linear approximation of a function is given by the rst two terms
of the Taylor series. The linear expansion for the function (1 + x)
50
+
(1 − 2x)
60
at x = 1 is given by
L(x) = (1 + 2
50
) + (120 + 50 × 2
49
)(x − 1)
17. The Taylor series for the function e
x
at x = 3 is given by
e e x
e
x
e
x
3 3
3
2
4
3
3
2
3
6
3+ − + − + −( ) ( ) ( )
18. The Taylor series for the function e
cos x
at x = π is given by
1 1
2
2
e e
x+ −( )π
–4 –3 –2 –1 0 1 2 3 4
0
5
10
15
20
25
30
35
40
45
50
(a) (b)
(c) (d)
–3 –2.5 –2 –1.5 –1 –0.5 0 0.5 1
–70
–60
–50
–40
–30
–20
–10
0
–1.6–1.5–1.4–1.3–1.2–1.1 –1 –0.9–0.8
–600
–400
–200
0
200
400
600
–5 –4 –3 –2 –1 0 1 2 3 4 5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
figure C.4
Plot of four different functions.
408 Appendix C
19. The quadratic approximation of a function is given by the rst three
terms of the Taylor series. The quadratic expansion for the function
ln(1 + sin x) at x = 0 is given by
Q x x
x
( )
!
= −
2
2
20. The gradient of the function is given by
∇ =
−
+ −
−
f
x x x x
x x x x x
x x x x
2
2
2
1 2 2 3
2
1
2
2 3 1 3
2
2
2
1 2 3
The gradient at (1, 1, –1) is given by
∇ = −
f
1
2
3
Now
∇ = −
=f
T
( ) [ ]x u 1 2 3
1 14
2 14
3 14
6 14
/
/
/
/
21. Both functions are convex (see Figure C.5).
22.
i. The maximum value is 19,575/17 and occurs at x = 99/17 and
x=48/17 (see Figure C.6).
ii. The maximum value is 120 and occurs at x = 0 and x = 30 (see
Figure C.7).
23. The Jacobian is given by
J=
− + −
1 4 9
2 2
3 2 3 4
2 3
1 2 3
2
1
2
3
2
3 2 1
2
2 3 1 3
x x
x x x x x x x x
x x x x 22 4
1 2
x x+
409Appendix C
−3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3
0
10
20
30
40
50
60
70
80
90
(a)
0
0.5
1
1.5
2
2.5
3
(b)
figure C.5
Plot of two different functions.
Maximize 125*x + 150*y
–1 1 2 3 4 5 6 7 8 9 10
–1
1
2
3
4
5
6
7
figure C.6
Linear programming problem.
Maximize 3*x + 4*y
10
15
20
25
30
figure C.7
Linear programming problem.

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