12-4 Derivatives
Find the derivative of each function.
8. z(n) = 2n 2 + 7n
SOLUTION: ANSWER: z′(n) = 4n + 7
10. SOLUTION: ANSWER: 12. n(t) = + + + 4
SOLUTION: ANSWER: 14. q(c) = c9 − 3c5 + 5c2 − 3c
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ANSWER: 12-4 Derivatives
14. q(c) = c9 − 3c5 + 5c2 − 3c
SOLUTION: ANSWER: 8
4
q′(c) = 9c −15c + 10c − 3
16. f (x) = −5x3 − 9x4 + 8x5
SOLUTION: ANSWER: 2
3
f ′(x) = −15x − 36x + 40x
4
17. TEMPERATURE The temperature in degrees Fahrenheit over a 24-hour period in a certain city can be defined
3
2
as f (h) = −0.0036h − 0.01h + 2.04h + 52, where h is the number of hours since midnight.
a. Find an equation for the instantaneous rate of change for the temperature.
b. Find the instantaneous rate of change for h = 2, 14, and 20.
c. Find the maximum temperature for 0 ≤ h ≤ 24.
SOLUTION: a. The instantaneous rate of change for the temperature is equivalent to f ’(h).
b.
c. Determine when f ’(h) = 0.
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12-4 Derivatives
c. Determine when f ’(h) = 0.
Since
, choose only the positive value and the endpoints. Evaluate f at h = 0, 12.85, and 24.
Thus, the maximum temperature is 68.92°.
ANSWER: 2
a. f ′(h) = −0.0108h − 0.02h + 2.04
b. f ′(2) = 1.96; f ′(14) = −0.36; f ′(20) = −2.68
c. 68.92°F
Find the derivative of each function.
28. SOLUTION: 2
Let g(x) = 4x + 3 and h(x) = x + 9. So, f (x) = g(x)h(x).
Use g(x), g′(x), h(x), and h′(x) to find the derivative of f (x).
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f ′(x) = 12x + 6x + 36
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ANSWER: 2
a. f ′(h) = −0.0108h − 0.02h + 2.04
f ′(2) = 1.96; f ′(14) = −0.36; f ′(20) = −2.68
12-4b.Derivatives
c. 68.92°F
Find the derivative of each function.
28. SOLUTION: 2
Let g(x) = 4x + 3 and h(x) = x + 9. So, f (x) = g(x)h(x).
Use g(x), g′(x), h(x), and h′(x) to find the derivative of f (x).
ANSWER: 2
f ′(x) = 12x + 6x + 36
32. SOLUTION: Let
and . So, g(x) = f (x)h(x).
Use g(x), g′(x), h(x), and h′(x) to find the derivative of f (x).
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ANSWER: 2
12-4f ′(x)
Derivatives
= 12x + 6x + 36
32. SOLUTION: and Let
. So, g(x) = f (x)h(x).
Use g(x), g′(x), h(x), and h′(x) to find the derivative of f (x).
ANSWER: g′(x) =
36. f (x) = (1.4x5 + 2.7x)(7.3x9 − 0.8x5)
SOLUTION: 5
9
5
f(x) = (1.4x + 2.7x)(7.3x − 0.8x )
5
9
5
Let g(x) = 1.4x + 2.7x and h(x) = 7.3x − 0.8x . So, f (x) = g(x)h(x).
Use g(x), g′(x), h(x), and h′(x) to find the derivative of f (x).
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ANSWER: 12-4g′(x)
Derivatives
=
36. f (x) = (1.4x5 + 2.7x)(7.3x9 − 0.8x5)
SOLUTION: 5
9
5
f(x) = (1.4x + 2.7x)(7.3x − 0.8x )
5
9
5
Let g(x) = 1.4x + 2.7x and h(x) = 7.3x − 0.8x . So, f (x) = g(x)h(x).
Use g(x), g′(x), h(x), and h′(x) to find the derivative of f (x).
ANSWER: Use the Quotient Rule to find the derivative of each function.
40. SOLUTION: Let d(n) = 3n + 2 and f (n) = 2n + 3. So,
.
Use d(n), d′(n), f (n), and f ′(n) to find the derivative of g(n).
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ANSWER: 12-4 Derivatives
Use the Quotient Rule to find the derivative of each function.
40. SOLUTION: Let d(n) = 3n + 2 and f (n) = 2n + 3. So,
.
Use d(n), d′(n), f (n), and f ′(n) to find the derivative of g(n).
ANSWER: g′(n) =
42. SOLUTION: 4
2
3
Let a(q) = q + 2q + 3 and b(q) = q – 2. So,
.
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Use a(q), a′(q), b(q), and b′(q) to find the derivative of m(q).
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ANSWER: 12-4g′(n)
Derivatives
=
42. SOLUTION: 4
2
3
Let a(q) = q + 2q + 3 and b(q) = q – 2. So,
.
Use a(q), a′(q), b(q), and b′(q) to find the derivative of m(q).
ANSWER: m′(q) =
46. SOLUTION: 3
2
3
Let f (r) = 1.5r + 5 – r and g(r) = r . So,
.
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ANSWER: =
12-4m′(q)
Derivatives
46. SOLUTION: 3
2
3
Let f (r) = 1.5r + 5 – r and g(r) = r . So,
.
Use f (r), f ’(r), g(r), and g’(r) to find the derivative of q(r).
ANSWER: q′(r) =
48. SOLUTION: 5
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4
3
Let f (x) = x + 3x and g(x) = −x – 2x – 2x – 3. So, m(x) =
.
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ANSWER: 12-4q′(r)
Derivatives
=
48. SOLUTION: 5
4
3
Let f (x) = x + 3x and g(x) = −x – 2x – 2x – 3. So, m(x) =
.
Use f (x), f ’(x), g(x), and g’(x) to find the derivative of m(x).
ANSWER: m′(x) =
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