Chapter 4
A. Sketch the graphs of the following functions, find the extrema and inflection points:
(i) f(x) = x ex
(ii) f(x) = ex / x
(iii) f(x) = x exp(-x2)
B. Given f(x) = ex + x2 - 2 , prove that
(i) f(x) has at least one root in the interval (0, 1), [ c is a root of f(x) if f(c)=0].
(ii) f(x) has only one root in the interval (0, 1).
§ 4.1 Exponential Function
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6, 18, 20, 34
6. Evaluate the given expressions.
a. (128)3 / 7
 27 
b.  
 64 
2/3
 64 
 
 25 
3/ 2
18. Use the properties of exponents to simplify the given expressions.
a. ( x 2 y 3 z )3
1/ 6
 x 3 y 2 
b.  4 
 z 
20. Find all real numbers x that satisfy the equation 3 x 2 2 x  144 .
34. POPULATION GROWTH
It is estimated that the population of a certain
country grows exponentially. If the population was 60 million in 1997 and 90 million
in 2002, what will the population be in 2012?
§ 4.2 Logarithmic Function
Page 314
6, 8, 20, 31, 36, 43
In Problems 6 and 8, evaluate the given expression using properties of the logarithm.
6. e 2 ln 3
8. ln
e3 e
e1/ 3
4


x
20. Use logarithmic rules to simplify ln 

3
2
 x 1 x 
In Problems 31 and 36, solve the given equation for x.
1
31. ln x  (ln 16  2 ln 2)
3
5
3
36.
1  2e x
43. COMPOUND INTEREST
How quickly will money double if it is invested at
an annual interest rate of 6% compounded continuously?
§ 4.3
Differentiation of Logarithmic and Exponential Function
Page 330
7, 36, 55
7. Differentiate the functions: (i) f ( x)  ( x 2  3x  5)e 6 x , (ii) ln(1+ x2) / [1 + exp(-x2)]
(iii) (ex –e-x) / (ex + e-x)
36. Find the largest and smallest values of the function F ( x)  e x 2 x over interval
[0, 2].
2
55. Given f ( x)  5 x , use the logarithmic differentiation to find the derivative
f ' ( x) .
2
§ 4.4 Additional Exponential Models
Page342 25, 27, 38, 39
25. THE SPREAD OF AN EPIDEMIC
Public health records indicate that t
weeks after the outbreak of a certain form of influenza, approximately
2
f (t ) 
thousand people had caught the disease.
1  3e 0.8t
a. Sketch the graph of f (t )
b. How many people had the disease initially?
c. How many had caught the disease by the end of 3 weeks?
d. If the trend continues, approximately how many people in all will contract the
disease?
27. EFFICIENCY
The daily output of a worker who has been on the job for t
weeks is given by a function of the form Q(t )  40  Ae  kt . Initially the worker could
produce 20 units a day, and after 1 week the worker can produce 30 units a day. How
many units will the worker produce per day after 3 weeks?
39. Newton's law of cooling states that the temperature of a hot object will cool down to the
temperature of its surrounding following an exponential decay. In symbols, if an object is
at a temperature T at time t and the surrounding is at a constant temperature
(e for
external), then T can be expressed as a function of time t:
T(t) = Te + (T0 – Te) exp(-kt).
If a piece of steak leaves an oven at a temperature of 200oC in a room at 20oC. It cools to
100oC in 15 minutes. How long does it take to cool further from 100 oC to 40oC?
§ 4.5 Differentiation of Trigonometric and Exponential Function
C. Differentiate the functions, where a & b are constants: (1) sin 2(ax) + cos 2(bx);
(2)ex sin(ax); (3) (sin(ax) + cos(bx)) exp(-x2) ; (4)cos(ax) / (ex + e-x).
§ 4.6 Taylor Expansion
D. Expand the following functions around x=0:
(ii) sin(x)
(i) ln(1+x),
(iii)
f ( x)  x 2  2 x 3
(iv)
f ( x)  3xe x
2
/2
E. Use the Taylor Expansion to evaluate the following: (i) ln(1.5); (ii) sin(30o). How many
terms you need in order to achieve accuracy up to 3 decimal places.