Unformatted text preview: −f (2)g (2)
(g (2))2 = 5(4)−3(−2)
42 = 26
16 = 13
8. 7. If you invest P dollars in a bank account at an annual interest rate of r%, then after t years you will have B
dollars, where
rt
B =P 1+
.
100
(a) Find dB/dt, assuming P and r are constant. In terms of money, what does dB/dt represent?
(b) Find dB/dr, assuming P and t are constant. In terms of money, what does dB/dr represent?
Solution:
r
rt
dB
dB
ln 1 +
= P 1+
. The expression
tells us how fast the amount of money in the bank
dt 39 Page: 153100
100 10) [ex6]
dt
Job: chap3temp Sheet:
(March 8, 2012 10 :
is changing with respect to time for ﬁxed initial investment P and interest rate r.
dB
r t−1 1
dB
(b)
= Pt 1 +
. The expression
indicates how fast the amount of money changes with
dr
100
100
dr
respect to the interest rate r , assuming ﬁxed initial investment P and time t.
(a) 3.6 THE CHAIN RULE AND INVERSE FUNCTIONS 6w5153 153 8. Use the following graph to calculate the derivative. 36w59
In Problems 60–62, use Figure 3.32 to calculate the derivative. 68. Figure 3.33 shows the number of motor vehicles,7 f (t),
in millions, registered in the world t years after 1965.
With units, estimate and interpret
(2.1, 5.3) (a) f (20)
(c) f −1 (500) f (x ) (b)
(d) f (20)
(f −1 ) (500) 2 4 6 (millions) (2, 5) 800 ins36w5153fig 8 600 ! 10 400 Figure 3.32 12 3 36w51
36w52
36w53
36w54 (a) h (2) if h(x) = (f (x))
200
Solution: Since the point (2, 5) is on the curve, we know f (2) = 5. The point (2.1, 5.3) is on the tangent
14
3
60. h (2) if h(x) = (f (x))
line, so−1
(year)
36w59fig
61. k (2) if k(x) = (f (x))
’65 ’70 ’753 − 5 ’850.3’90 ’95 2000
5. ’80
16
Slope tangent =
=
= 3.
−1
62. g (5) if g (x) = f (x)
2.1 − 2 3.33
Figure 0.1
63. (a)
(b)
(c)
(d) Given that Thus, f 3 ,2) = f (2).
f (x) = x ( ﬁnd 3.
Find f −1 (x).
By the chain rule
36w60 69. Using Figure 3.34, where f (2) = 2.1, f (4) = 3.0,
Use your answer from part (b) to ﬁnd (f −1 ) (8).
2
h (2) =f3(6)(2))27, f (2) = 4.2,5ﬁnd3 f −1 ) (8).
· ( = 225.
( f = 3. · f (8) = 3 ·
How could you have used your answer from part (a)
−1
to ﬁnd (f k)((8)? k (x) = (f (x))−1
(b)
2) if 18 20 22 36w55 64. (a) For f (x) = 2x5 + 3x3Sinceﬁnd f point (2, 5) is on the curve, 24 know f (2) = 5. The point (2.1, 5.3) is on the tangent
Solution: + x, the (x).
we
(b) How can you use your answer to part (a) to deterf (x )
24
line, so
16
mine if f (x) is invertible?
0.3
5.3 − 5
Slope tangent 8 =
=
= 3.
(c) Find f (1).
2.1 − 2
0.1
26
(d) Find f (1).
x
36w60fig
(e) Find (f −1 ) (6).
2
4
6
8 36w57 65. Use the table and the fact that f (x) is invertible and differentiable everywhere to ﬁnd (f −1 ) (3). 28 Figure 3.34
30 x f (x ) f (x ) 3 1 7 3miscw118 70. If f is increasing and f (20) = 10, which of the two options, (a) or (b), must be wrong?
−1 32 Thus, f (2) = 3.
By the chain rule
k (2) = −(f (2))−2 · f (2) = −5−2 · 3 = −0.12. (c) g (5) if g (x) = f −1 (x)
Solution: Since the point (2, 5) is on the curve...
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 Fall '11
 fisherselin
 Derivative, Differential Calculus, Slope

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