CALCULATING THE DERIVATIVE - Grosse Pointe Public Schools
[Pages:58]Chapter 4
CALCULATING THE DERIVATIVE
4.1 Techniques for Finding Derivatives
1. y = 12x3 ? 8x2 + 7x + 5
dy = 12(3x3?1) ? 8(2x2?1) + 7x1?1 + 0 dx
= 36x2 ? 16x + 7
2.
y
=
8x3
?
5x2
?
x 12
dy dx
= =
8(3x3?1) ? 5(2x2?1) ?
24x2
?
10x
?
1 12
1 12
x1?1
3.
y
=
3x4
?
6x3
+
x2 8
+
5
dy dx
=
3(4x4?1)
?
6(3x3?1) +
1 8
(2x2?1)
+
0
=
12x3
?
18x2
+
1 4x
4. y = 5x4 + 9x3 + 12x2 ? 7x
dy = 5(4x4?1) + 9(3x3?1) dx
+ 12(2x2?1) ? 7(x1?1) = 20x3 + 27x2 + 24x ? 7
5. y = 6x3:5 ? 10x0:5
dy = 6(3:5x3:5?1) ? 10(0:5x0:5?1)
dx
=
21x2:5
? 5x?0:5
or
21x2:5
?
5 x0:5
6. f (x) = ?2x1:5 + 12x0:5 f 0(x) = ?2(1:5x1:5?1) + 12(0:5x0:5?1)
= ?3x0:5 + 6x?0:5 or
?
3x0:5
+
6 x0:5
7.
y
=
p 8x
+
6x3=4
= 8x1=2 + 6x3=4
?
?
dy = 8 dx
1 2
x1=2?1
+6
3 4
x3=4?1
=
4x?1=2
+
9 2
x?1=4
4
9
or x1=2 + 2x1=4
8. y = ?100px ? 11x2=3
= ?100x1=2 ? 11x2=3
?
?
dy = ?100 dx
1 2
x1=2?1
? 11
2 3
x2=3?1
=
?50x?1=2
?
22x?1=3 3
or
?50 x1=2
?
22 3x1=3
9. g(x) = 6x?5 ? x?1 g0(x) = 6(?5)x?5?1 ? (?1)x?1?1 = ?30x?6 + x?2
or
?30 x6
+
1 x2
10. y = 10x?3 + 5x?4 ? 8x
dy = 10(?3x?3?1) + 5(?4x?4?1) ? 8x1?1 dx
=
?30x?4
?
20x?5
?
8
or
?30 x4
?
20 x5
?
8
11. y = 5x?5 ? 6x?2 + 13x?1
dy = 5(?5x?5?1) ? 6(?2x?2?1) + 13(?1x?1?1) dx
= ?25x?6 + 12x?3 ? 13x?2
or
?25 x6
+
12 x3
?
13 x2
75 12. f(t) = t ? t3
= 7t?1 ? 5t?3 f 0(t) = 7(?1t?1?1) ? 5(?3t?3?1)
=
?7t?2
+
15t?4
or
?7 t2
+
15 t4
14 12 p
13. f(t) = t + t4 + 2
=
14t?1
+
12t?4
+
p 2
f 0(t) = 14(?1t?1?1) + 12(?4t?4?1) + 0
=
?14t?2
?
48t?5
or
?14 t2
?
48 t5
239
240
Chapter 4 CALCULATING THE DERIVATIVE
6 7 3p
14. y = x4 ? x3 + x + 5
=
6x?4
?
7x?3
+
3x?1
+
p 5
dy = 6(?4x?4?1) ? 7(?3x?3?1) + 3(?1x?1?1) + 0 dx
= ?24x?5 + 21x?4 ? 3x?2
or
?24 x5
+
21 x4
?
3 x2
317 15. y = x6 + x5 ? x2
= 3x?6 + x?5 ? 7x?2 dy = 3(?6x?7) + (?5x?6) ? 7(?2x?3) dx
= ?18x?7 ? 5x?6 + 14x?3
or
?18 x7
?
5 x6
+
14 x3
16. p(x) = ?10x?1=2 + 8x?3=2
?
?
p0(x) = ?10
?
1 2
x?3=2
+8
?
3 2
x?5=2
= 5x?3=2 ? 12x?5=2
or
5 x3=2
?
12 x5=2
17. h(x) = x?1=2 ? 14x?3=2
?
h0(x)
=
?
1 2
x?3=2
?
14
?
3 2
x?5=2
=
?x?3=2 2
+ 21x?5=2
or
?1 2x3=2
+
21 x5=2
18. y = 4p6 x = 6x?1=4
?
dy = 6 dx
?
1 4
x?5=4
=
?
3 2
x?5=4
?3 or 2x5=4
19. y = p?2 3x
=
?2 x1=3
=
?2x?1=3 ?
dy = ?2 dx
?
1 3
x?4=3
=
2x?4=3 3
or
2 3x4=3
20. f(x) = x3 + 5 x
= x2 + 5x?1 f 0(x) = 2x2?1 + 5(?1x?1?1)
= 2x ? 5x?2
or
2x
?
5 x2
21. g(x) = x3p? 4x x
=
x3 ? 4x x1=2
= x5=2 ? 4x1=2
?
g0(x)
=
5 2
x5=2?1
?
4
1 2
x1=2?1
=
5 2
x3=2
? 2x?1=2
or
5 2
x3=2
?
p2 x
22. g(x) = (8x2 ? 4x)2 = 64x4 ? 64x3 + 16x2
g0(x) = 64(4x4?1) ? 64(3x3?1) + 16(2x2?1) = 256x3 ? 192x2 + 32x
23. h(x) = (x2 ? 1)3 = x6 ? 3x4 + 3x2 ? 1
h0(x) = 6x6?1 ? 3(4x4?1) + 3(2x2?1) ? 0 = 6x5 ? 12x3 + 6x
24. A quadratic function has degree 2. When the derivative is taken, the power will decrease by 1 and the derivative function will be linear, so the correct choice is (b).
26. d (4x3 ? 6x?2) dx
= 4(3x2) ? 6(?2x?3) = 12x2 + 12x?3
=
12x2
+
12 x3
choice (c)
=
12x2(x3) x3
+
12
=
12x5 + x3
12
choice (b)
Neither choice (a) nor choice (d) equals
d (4x3 ? 6x?2): dx
Section 4.1 Techniques for Finding Derivatives
?
?
27. Dx
9x?1=2
+
2 x3=2
= Dx[9x?1=2 + 2x?3=2]
?
?
=9
?
1 2
x?3=2
+2
?
3 2
x?5=2
=
?
9 2
x?3=2
?
3x?5=2
or
?9 2x3=2
?
3 x5=2
?
?
28. Dx
p8 ? p3
4x
x3
= Dx [8x?1=4 ? 3x?3=2]
?
?
=8
?
1 4
x?5=4
?3
?
3 2
x?5=2
=
?2x?5=4
+
9x?5=2 2
or
?2 x5=4
+
9 2x5=2
29.
f (x)
=
x4 6
?
3x
=
1 6
x4
?
3x
f 0(x)
=
1 6
(4x3)
?
3
=
2 3
x3
?
3
f 0 (?2)
=
2 3
(?2)3
?
3
=
?
16 3
?
3
=
?
25 3
30.
f (x)
=
x3 9
?
7x2
=
1 9
x3
?
7x2
f 0 (x)
=
1 9
(3x2)
?
7(2x)
=
1 3
x2
?
14x
241
f 0(3)
=
1 (3)2 3
?
14(3)
= 3 ? 42 = ?39
31. y = x4 ? 5x3 + 2; x = 2 y0 = 4x3 ? 15x2
y0(2) = 4(2)3 ? 15(2)2 = ?28
The slope of tangent line at x = 2 is ?28: Use m = ?28 and (x1; y_1) = (2; ?22) to obtain the equation.
y ? (?22) = ?28(x ? 2) y = ?28x + 34
32. y = ?3x5 ? 8x3 + 4x2 y0 = ?3(5x4) ? 8(3x2) + 4(2x) = ?15x4 ? 24x2 + 8x
y0(1) = ?15(1)4 ? 24(1)2 + 8(1) = ?15(1)4 ? 24(1)2 + 8(1) = ?15 ? 24 + 8 = ?31
y(1) = ?3(1)5 ? 8(1)3 + 4(1)2 = ?7
The slope of the tangent line at x = 1 is ?31: Use m = ?31 and (x1; y1) = (1; ?7) to obtain the equation.
y ? (?7) = ?31(x ? 1) y + 7 = ?31x + 31 y = ?31x + 24
33.
y = ?2x1=2 + x3=2
?
y0 = ?2
1 2
x?1=2
+
3 2
x1=2
=
?x?1=2
+
3 2
x1=2
=
?
1 x1=2
+
3x1=2 2
y0(9)
=
?
1 (9)1=2
+
3(9)1=2 2
=
?
1 3
+
9 2
=
25 6
The
slope
of
the
tangent
line
at
x
=
9
is
25 6
:
242
34. y = ?x?3 + x?2 y0 = ?(?3x?4) + (?2x?3) = 3x?4 ? 2x?3
=
3 x4
?
2 x3
y0(2)
=
3 (2)4
?
2 (2)3
=
3 16
?
2 8
=
?
1 16
The
slope
of
the
tangent
line
at
x
=
2
is
?
1 16
.
35. f (x) = 9x2 ? 8x + 4 f0(x) = 18x ? 8
Let f 0(x) = 0 to ...nd the point where the slope of the tangent line is zero.
18x ? 8 = 0 18x = 8
x
=
8 18
=
4 9
Find the y-coordinate.
f (x) = 9x2 ? 8x + 4
f
4 9
?
=
9
4 9
?2
?
8
4 9
?
+
4
?
=9
16 81
?
32 9
+
4
=
16 9
?
32 9
+
36 9
=
20 9
T? 49h;e29s0l?o.pe of the tangent line is zero at one point,
36. f (x) = x3 + 9x2 + 19x ? 10 f0(x) = 3x2 + 18x + 19
If the slope of the tangent line is ?5, then the f 0(x) = ?5:
3x2 + 18x + 19 = ?5 3x2 + 18x + 24 = 0 3(x2 + 6x + 8) = 0 3(x + 2)(x + 4) = 0
x = ?2 or x = ?4
f(?2) = (?2)3 + 9(?2)2 + 19(?2) ? 10 = ?8 + 36 ? 38 ? 10 = ?20
f(?4) = (?4)3 + 9(?4)2 + 19(?4) ? 10 = ?64 + 144 ? 76 ? 10 = ?6
Chapter 4 CALCULATING THE DERIVATIVE
Thus, the points where the slope of the tangent line is ?5 are (?2; ?20) and (?4; ?6):
37. f(x) = 2x3 + 9x2 ? 60x + 4 f 0(x) = 6x2 + 18x ? 60
If the tangent line is horizontal, then its slope is zero and f0(x) = 0:
6x2 + 18x ? 60 = 0 6(x2 + 3x ? 10) = 0 6(x + 5)(x ? 2) = 0 x = ?5 or x = 2
Thus, the tangent line is horizontal at x = ?5 and x = 2:
38. f(x) = x3 + 15x2 + 63x ? 10 f 0(x) = 3x2 + 30x + 63
If the tangent line is horizontal, then its slope is zero and f0(x) = 0:
3x2 + 30x + 63 = 0 3(x2 + 10x + 21) = 0
3(x + 3)(x + 7) = 0 x = ?3 or x = ?7
Therefore, the tangent line is horizontal at x = ?3 or x = ?7:
39. f(x) = x3 ? 4x2 ? 7x + 8 f 0(x) = 3x2 ? 8x ? 7
If the tangent line is horizontal, then its slope is zero and f0(x) = 0:
3x2 ? 8x ? 7 = 0
p
x= 8?
64 + 84 6
p x = 8 ? 6 148
p
x
=
8
?
2 6
37
p
x
=
2(4
? 6
37)
p
x=
4? 3
37
p
Thus,
the
tangent
line
is
horizontal
at
x
=
4? 3
37 :
Section 4.1 Techniques for Finding Derivatives
40. f (x) = x3 ? 5x2 + 6x + 3 f0(x) = 3x2 ? 10x + 6
If the tangent line is horizontal, then its slope is zero and f 0(x) = 0:
3x2 ? 10x + 6 = 0
p
x
=
10
?
100 p6
?
72
x
=
10 ? 6
28 p
x
=
10
?2 p6
7
x
=
5
? 3
7
p
The
tangent
line
is
horizontal
at
x
=
5? 3
7:
41. f (x) = 6x2 + 4x ? 9 f0(x) = 12x + 4
If the slope of the tangent line is ?2; f 0(x) = ?2:
12x + 4 = ?2 12x = ?6
x ?
=
?
1 2
f
?
1 2
=
?
19 2
The
slope
of
the
tangent
line
is
?2
at
? ?
1 2
;
?
19 2
?
:
42. f (x) = 2x3 ? 9x2 ? 12x + 5 f0(x) = 6x2 ? 18x ? 12
If the slope of the tangent line is 12, f0(x) = 12:
6x2 ? 18x ? 12 = 12 6x2 ? 18x ? 24 = 0 6(x2 ? 3x ? 4) = 0 6(x ? 4)(x + 1) = 0
x = 4 or x = ?1
f (4) = ?59 and f (?1) = 6 The slope of the tangent line is ?12 at (4; ?59) and (?1; 6):
43. f (x) = x3 + 6x2 + 21x + 2 f0(x) = 3x2 + 12x + 21
If the slope of the tangent line is 9, f 0(x) = 9:
3x2 + 12x + 21 = 9 3x2 + 12x + 12 = 0 3(x2 + 4x + 4) = 0
3(x + 2)2 = 0 x = ?2
f (?2) = ?24
243
The slope of the tangent line is 9 at (?2; ?24):
44. f(x) = 3g(x) ? 2h(x) + 3 f 0(x) = 3g0(x) ? 2h0(x) f 0(5) = 3g0(5) ? 2h0(5)
= 3(12) ? 2(?3) = 42
45.
f (x)
=
1 2
g(x)
+
1 4
h(x)
f 0(x)
=
1 2
g0
(x)
+
1 4
h0(x)
f 0(2)
=
1 2
g0
(2)
+
1 4
h0(2)
=
1 2
(7)
+
1 4
(14)
=
7
46. (a) From the graph, f(1) = 2; because the curve goes through (1; 2):
(b) f0(1) gives the slope of the tangent line to f at 1. The line goes through (?1; 1) and (1; 2):
m
=
1
2?1 ? (?1)
=
1 2;
so
f 0(1)
=
1 2:
(c) The domain of f is [?1; 1) because the xcoordinates of the points of f start at x = ?1 and continue in...nitely through the positive real numbers.
(d) The range of f is [0; 1) because the y-coordinates of the points on f start at y = 0 and continue in...nitely through the positive real numbers.
49. f(x) = 1 ? f(x) kk
Use the rule for the derivative of a constant times
a function.
??
?
?
d f(x) = d 1 ? f(x)
dx k
dx k
= 1 f 0(x) k
= f0(x) k
50. Graph the numerical derivative of f (x) = 1:25x3 + 0:01x2 ? 2:9x + 1
for x ranging from ?5 to 5. (a) When x = 4, the derivative equals 57.18.
(b) The derivative crosses the x-axis at approximately ?0:88 and 0:88.
244
51. The demand is given by q = 5000 ? 100p:
Solve for p:
p
=
5000 ? 100
q
?
R(q) = q
5000 ? q 100
= 5000q ? q2 100
R0(q)
=
5000 ? 100
2q
(a)
R0(1000) =
5000 ? 2(1000) 100
= 30
(b)
R0(2500) =
5000 ? 2(2500) 100
=0
(c)
R0(3000) =
5000 ? 2(3000) 100
= ?10
52. C(q) = 3000 ? 20q + 0:03q2
?
R(q) = qp = q
5000 ? q 100
=
q
? 50
?
q 100
?
(from Exercise 51)
The pro...t equation is found as follows.
P =R?C
=
q
? 50
?
q? 100
?
(3000
?
20q
+
0:03q2)
=
50q
?
q2 100
?
3000
+
20q
?
0:03q2
=
50q
+
20q
?
q2 100
?
0:03q2
?
3000
= 70q ? 0:01q2 ? 0:03q2 ? 3000
Therefore, P = 70q ? 0:04q2 ? 3000: Now, the marginal pro...t is
P 0(q) = 70 ? 0:04(2q) ? 0 = 70 ? 0:08q:
(a) P 0(500) = 70 ? 0:08(500) = 70 ? 40 = 30
(b) P 0(815) = 70 ? 0:08(815) = 70 ? 65:2 = 4:8
Chapter 4 CALCULATING THE DERIVATIVE
(c) P 0(1000) = 70 ? 0:08(1000) = 70 ? 80 = ?10
53. S(t) = 100 ? 100t?1 S0(t) = ?100(?1t?2) = 100t?2
100 = t2
(a)
S0(1)
=
100 (1)2
=
100 1
=
100
(b)
S0(10)
=
100 (10)2
=
100 100
=
1
54.
p(q) =
1000 q2
+ 1000
If R is the revenue function, R(q) = qp(q):
?
R(q) = q
1000 q2
+
1000
R(q) = 1000q?1 + 1000q R0(q) = ?1000q?2 + 1000
R0(q)
=
1000
?
1000 q2
?
R0(q) = 1000
1
?
1 q2
?
R0(10) = 1000
1
?
1 102
?
R0(10) = 1000
99 100
= 990
The marginal revenue is $990.
55. Pro...t = Revenue ? Cost
P (q) = qp(q) ? C(q)
?
P (q) = q
1000 q2 + 1000
? (0:2q2 + 6q + 50)
= 1000 + 1000q ? 0:2q2 ? 6q ? 50 q
= 1000q?1 + 994q ? 0:2x2 ? 50 P 0(q) = ?1000q?2 + 994 ? 0:4q
=
994
?
0:4q
?
1000 q2
P 0(10)
=
994
?
0:4(10)
?
1000 (10)2
= 994 ? 4 ? 10 = 980
The marginal pro...t is $980.
Section 4.1 Techniques for Finding Derivatives
56.
C(x) = 2x;
R(x) = 6x ?
x2 1000
(a) C0(x) = 2
(b)
R0(x)
=
6?
2x 1000
=
6?
x 500
(c) P (x) = R(x) ? C(x)
=
6x
?
x2 ? 1000
?
2x
=
4x
?
x2 1000
P 0(x)
=
4
?
2x 1000
=
4
?
x 500
(d)
P 0(x) = 4 ?
x 500
=0
4
=
x 500
x = 2000
(e) Using part (d), we see that marginal pro...t is 0 when x = 2000 units.
P
(2000)
=
4(2000)
?
(2000)2 1000
= 8000 ? 4000 = 4000
The pro...t for 2000 units produced is $4000.
57. (a) 1982 when t = 50:
C(50) = 0:00875(50)2 ? 0:108(50) + 1:42 = 17:895 ? 17:9 cents
2002 when t = 70:
C(70) = 0:00875(70)2 ? 0:108(70) + 1:42 = 36:735 ? 36:7 cents
(b) C0(t) = 0:00875(2t) ? 0:108(1) = 0:0175t ? 0:108
1982 when t = 50:
C0(50) = 0:0175(50) ? 0:108 = 0:767 cents/year
2002 when t = 70: C0(70) = 0:0175(70) ? 0:108
? 1:12 cents/year
(c) Using a graphing calculator, a cubic function that models the data is C(t) = (?1:790 ? 10?4)t3 + 0:02947t2
? 0:7105t + 3:291:
245
Using the values from the calculator, the rate of change in 1982 is C0(50) ? 0:894 cents/year. Using the values from the calculator, the rate of change in 2002 is C0(70) ? 0:784 cents/year.
58. M (t) = 3:044t3 ? 379:6t2 + 14,274.5t ? 139,433
M 0(t) = 3:044(3t3?1) ? 379:6(2t2?1) + 14,274.5t1?1 ? 0
= 9:132t2 ? 759:2t + 14,274.5
(a) 1920 ? 1900 = 20 M 0(20) ? 2743
(b) 1960 ? 1900 = 60 M0(60) ? 1598
(c) 1980 ? 1900 = 80 M 0(80) ? 11,983
(d) 2000 ? 1900 = 100 M0(100) ? 29,675
(e) The amount of money in circulation was increasing at the rate of $2743 million/year in 1920 but then began to decrease. The decrease of money continued through the early 1950's. Since then, the amount has again been increasing.
59. N(t) = 0:00437t3:2 N 0(t) = 0:013984t2:2 (a) N 0(5) ? 0:4824 (b) N 0(10) ? 2:216
60. G(x) = ?0:2x2 + 450
(a) G(0) = ?0:2(0)2 + 450 = 450
(b) G(25) = ?0:2(25)2 + 450 = ?125 + 450 = 325
G0(x) = ?2(0:2)x = ?0:4x (c) G0(10) = ?0:4(10) = ?4
After 10 units of insulin are injected, the blood sugar level is decreasing at a rate of 4 points per unit of insulin.
(d) G0(25) = ?0:4(25) = ?10
After 25 units of insulin are injected, the blood sugar level is decreasing at a rate of 10 points per unit of insulin.
246
Chapter 4 CALCULATING THE DERIVATIVE
61. V (t) = ?2159 + 1313t ? 60:82t2
(a) V (3) = ?2159 + 1313(3) ? 60:82(3)2 = 1232.62 cm3
(b) V 0(t) = 1313 ? 121:64t V 0(3) = 1313 ? 121:64(3) = 948:08 cm3/yr
62.
w(c) =
c3 100
?
1500 c
(a) When c = 30; w = 220 g or 0.22 kg.
(b)
dw dc
=
3c2 100
+
1500 c2
When
c
=
30;
dw dc
=
28
2 3
g/cm.
When the cir-
cumference of the brain is 30 cm, it is increasing
by
28
2 3
g
with
every
centimeter
the
circumference
increases.
63. v = 2:69l1:86 dv = (1:86)2:69l1:86?1 ? 5:00l0:86 dl
64. l(x) = ?2:318 + 0:2356x ? 0:002674x2
(a) The problem states that a fetus this formula concerns is at least 18 weeks old. So, the minimum x value should be 18. Considering the gestation time of a cow in general, a meaninful range for this funcion is 18 ? x ? 44:
(b) l0(x) = 0:2356 ? (2)0:002674x = 0:2356 ? 0:005348x
(c) l0(25) = 0:2356 ? 0:005348(25) = 0:1019 cm/week
65. t = 0:0588s1:125
(a) When s = 1609; t ? 238:1 seconds, or 3 minutes, 58.1 seconds.
(b) dt = 0:0588(1:125s1:125?1) ds = 0:06615s0:125
When
s
=
100;
dt ds
? 0:118
sec/m.
At
100
meters,
the fastest possible time increases by 0.118 sec-
onds for each additional meter.
(c) Yes, they have been surpassed. In 2000, the world record in the mile stood at 3:43.13. (Ref: )
66. V = C(R0 ? R)R2 = CR0R2 ? CR3
dV dR
=
2CR0R ? 3CR2 CR(2R0 ? 3R)
= =
0 0
CR = 0 or 2R0 ? 3R = 0
R=0
R
=
2 3 R0
Discard R = 0, since a closed windpipe produces
no
airow.
Velocity
is
maximized
when
R
=
2 3
R0
.
67.
BMI =
703w h2
(a) 60200 = 74 in.
703(220) BMI = 742 ? 28
(b)
BMI
=
703w 742
=
24:9
implies
w
=
24:9(74)2 703
?
194:
A 220-lb person needs to lose 26 pounds to get
down to 194 lbs.
(c)
If
f (h)
=
703(125) h2
=
87,875h?2,
then
f 0(h) = 87,875(?2h?2?1)
=
?175,750h?3
=
?
175,750 h3
(d)
f 0 (65)
=
?
175,750 653
?
?0:64
For a 125-lb female with a height of 65 in. (50500),
the BMI decreases by 0.64 for each additional inch
of height.
(e) Sample Chart
ht=wt 140 160 180 200 60 27 31 35 39 65 23 27 30 33 70 20 23 26 29 75 17 20 22 25
68. s(t) = 11t2 + 4t + 2
(a) v(t) = s0(t) = 22t + 4
(b) v(0) = 22(0) + 4 = 4 v(5) = 22(5) + 4 = 114 v(10) = 22(10) + 4 = 224
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