5128_Ch03_pp098-184

178 Chapter 3 Derivatives
[–5, 10] by [–10, 30]
Figure 3.59 The graph of dPdt, the rate
of spread of the flu in Example 8. The
graph of P is shown in Figure 3.58.
(b) To find the rate at which the flu spreads, we find dPdt. To find dPdt, we need to
invoke the Chain Rule twice:
d P d
1001 e
dt
d t
3t 1 100 • 11 e 3t 2 d
• 1 e d t
3t
1001 e 3t 2 • 0 e 3t d
• 3 t d t
1001 e 3t 2 e 3t • 1
100e
1 e
3t
3t
2
At t 3, then, dPdt 1004 25. The flu is spreading to 25 students per day.
(c) We could estimate when the flu is spreading the fastest by seeing where the graph of
y Pt has the steepest upward slope, but we can answer both the “when” and the
“what” parts of this question most easily by finding the maximum point on the graph of
the derivative (Figure 3.59).
We see by tracing on the curve that the maximum rate occurs at about 3 days, when
(as we have just calculated) the flu is spreading at a rate of 25 students per day.
Now try Exercise 51.
Quick Review 3.9 (For help, go to Sections 1.3 and 1.5.)
1. Write log 5 8 in terms of natural logarithms. l n
8
ln
5
2. Write 7 x as a power of e. e x ln 7
In Exercises 3–7, simplify the expression using properties of exponents
and logarithms.
3. ln e tan x tan x 4. ln x 2 4 ln x 2 ln (x 2)
5. log 2 8 x5 3x 15 6. log 4 x 15 log 4 x 12 54
7. 3lnx ln 3x ln 12x 2 ln (4x 4 )
13. 3 csc x (ln 3)(csc x cot x)
Section 3.9 Exercises
1
1
23. y log 2 1x, x 0 24. y 1log 2 x
x l n2
x(ln 2)( log 2 x) 2
25. y ln 2 • log 2 x 26. y log 3 1 x ln 3
1 x , x 0 1
1
26., x
1 x ln 3 ln 3
In Exercises 1–28, find dydx. Remember that you can use NDER
to support your computations.
1. y 2e x 2e x 2. y e 2x 2e 2x
3. y e x e x 4. y e 5x 5e 5x
5. y e 2x3 2 3 e2x3 6. y e x4 1 4 ex/4
7. y xe 2 e x e 2 e x 8. y x 2 e x xe x x 2 e x xe x e x
9. y e x ex/2x 10. y e x2 2xe (x2 )
11. y 8 x 8 x ln 8 12. y 9 x 9 x ln 9
13. y 3 csc x 14. y 3 cot x 3 cot x (ln 3)(csc 2 x)
15. y ln x 2 2 x 16. y ln x2 2l n x
x
17. y ln 1x See page 180. 18. y ln 10x See page 180.
1
19. y ln ln x x l
20. y x ln x x ln x
n x
21. y log 4 x 2 See page 180. 22. y log 5 x See page 180.
In Exercises 8–10, solve the equation algebraically using logarithms.
Give an exact answer, such as ln 23, and also an approximate answer
to the nearest hundredth.
8. 3 x 19 x l n 19
ln
3
2.68
9. 5 t ln 5 18 x ln 18 ln (ln 5)
1.50
ln 5
10. 3 x1 2 x ln 3
x 2.71
ln 2 ln 3
27. y log 10 e x 1
28. y ln 10 x ln 10
ln 10
29. At what point on the graph of y 3 x 1 is the tangent line
parallel to the line y 5x 1? (1.379, 5.551)
30. At what point on the graph of y 2e x 1 is the tangent line
perpendicular to the line y 3x 2? (1.792, 0.667)
31. A line with slope m passes through the origin and is tangent to
y ln (2x). What is the value of m? 2e 1
32. A line with slope m passes through the origin and is tangent to
y ln (x3). What is the value of m? 1 3 e1
In Exercises 33–36, find dydx.
33. y x p x 1 34. y x 12 (1 2)x 2
35. y x 2 2x 21 36. y x 1e (1 e)x e
In Exercises 37–42, find f(x) and state the domain of f .
1
37. f (x) ln (x 2) , x 2
x 2
38. f (x) ln (2x 2)
1
, x 1
x 1