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Chapter 4 Exponential and Logarithmic Functions (指数函数和对数函数 )

In this Chapter, we will In this Chapter, we will encounter encounter some important conceptssome important concepts

Exponential Functions(Exponential Functions( 指数函数指数函数 ))

Logarithmic Functions(Logarithmic Functions( 对数函数对数函数 ))

Differentiation of Logarithmic andDifferentiation of Logarithmic and Exponential FunctionsExponential Functions

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Section 4.1 Exponential FunctionsExponential function(Exponential function( 指数函数指数函数 ):): If If bb is a is a positive number other than positive number other than 11 ( (bb>0, >0, bb≠1≠1), there is ), there is a unique function called the exponential function a unique function called the exponential function with with bbase ase bb that is defined by that is defined by ff((xx))=b=bxx for every real number for every real number xx

NOTENOTE: Such function can be used to describe exponential : Such function can be used to describe exponential and logistic growth and a variety of other important and logistic growth and a variety of other important quantities. quantities.

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Definition of for Rational Values of Definition of for Rational Values of nn (and (and b>0b>0))

Integer powers: If Integer powers: If nn is a positive integer, is a positive integer,

Fractional powers: If Fractional powers: If nn and and mm are positive integers, are positive integers,

where denotes the positive where denotes the positive mmth root. th root.

Negative powers: Negative powers:

Zero power: Zero power:

nb

facters n

n bbbb

m nnmmn bbb /

m b

nn

bb

1

10 b

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Example 1

Solution:Solution:

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Figure below shows graphs of various members of the Figure below shows graphs of various members of the family of exponential functions family of exponential functions

xby

NOTENOTE: Students often confuse the : Students often confuse the powerpower function function

with the with the exponentialexponential function function

bxxp )(xbxf )(

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7

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Definition:Definition: The natural exponential function is The natural exponential function is

WhereWhere

n 10 100 1000 10,000 100,000

2.59374 2.70481 2.71692 2.711815 2.71827

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Continuous Compounding of Interest(Continuous Compounding of Interest( 连续利息连续利息 ))

If If PP is the initial investment (the principal) and is the initial investment (the principal) and rr is the interest is the interest rate (expressed as a decimal), the balance rate (expressed as a decimal), the balance BB after the interest is after the interest is added will beadded will be

B=P+PB=P+Prr=P=P((1+r1+r) dollars) dollars

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Example 2

Suppose Suppose $1,000$1,000 is invested at an annual interest rate of is invested at an annual interest rate of 6%.6%. Compute the balance after Compute the balance after 1010 years if the interest is compounded years if the interest is compounded

a. Quarterly b. Monthly c. Daily d. Continuously a. Quarterly b. Monthly c. Daily d. Continuously

Solution:Solution:

a.a. To compute the balance after 10 years if the interest is To compute the balance after 10 years if the interest is compounded quarterly, using the formula compounded quarterly, using the formula

with with t=10t=10, , p=1,000p=1,000, , r=0.06r=0.06, and , and k=4k=4: :

kt

k

rptB

1)(

02.814,1$4

06.01000,1)10(

40

B

to be continued

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b. This time, take b. This time, take t=10t=10, , p=1,000p=1,000, , r=0.06r=0.06, and , and k=12k=12 to get to get

40.819,1$12

06.01000,1)10(

120

B

c. Take c. Take t=10t=10, , p=1,000p=1,000, , r=0.06r=0.06, and , and k=365k=365 to obtain to obtain

03.822,1$365

06.01000,1)10(

650,3

B

d. For continuously compounded interest use the formula d. For continuously compounded interest use the formula rtpetB )(

with with t=10t=10, , p=1,000p=1,000, and , and r=0.06r=0.06::

12.822,1$000,1)10( 6.0 eB

This value, This value, $1,822.12$1,822.12, is an upper bound for the possible balance. , is an upper bound for the possible balance. No matter how often interest is compounded, No matter how often interest is compounded, $1,000$1,000 invested at an invested at an annual interest rate of annual interest rate of 6%6% can not grow to more than can not grow to more than $1,822.12$1,822.12 in in 1010 years. years.

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Present Value is the interest accumulation process in reverse. Rather than adding interest to a principal to determine a sum, it is in effect subtracted from a sum to determine a principal.The present value P of future value F with interest rate r per conversion period for n period is given by the formula

Present Value (Present Value ( 现值现值 ) & Future Value) & Future Value

is the present value interest factor (PVIFn) (or discount factor) of n periods.

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If the interest rate is i*100% per year and the interest is compounded continuously for t years, then the present value P for a given amount of future value F can be calculated from

where exp(−it) is the continuous PVIF.

i = annualized interest rate, t = number of yearsm = compound periods per year

Present Value (Present Value ( 现值现值 ) & Future Value) & Future Value

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Present value of $1.00 when interest (i=5%) is compounded quarterly for a year

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Example 3

Sue is about to enter college. When she graduates Sue is about to enter college. When she graduates 44 years from years from now, she wants to take a trip to Europe that she estimates will now, she wants to take a trip to Europe that she estimates will cost cost $5,000$5,000. How much should she invest now at . How much should she invest now at 7%7% to have to have enough for the trip if interest is compounded: enough for the trip if interest is compounded:

a. Quarterly b. Continuouslya. Quarterly b. Continuously

Solution:Solution:

The required future value is The required future value is F=$5,000 F=$5,000 in in t=4t=4 years with years with r=0.07r=0.07. .

a.a. If the compounding is quarterly, then If the compounding is quarterly, then k=4k=4 and the present value and the present value is is

to be continued

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08.788,3$4

07.01000,5

)4(4

P

b. For continuous compounding, the present value isb. For continuous compounding, the present value is

92.778,3$000,5 )4(07.0 eP

Thus, Sue would have to invest about $9 more if interest is Thus, Sue would have to invest about $9 more if interest is compounded quarterly than if the compounding is continuous. compounded quarterly than if the compounding is continuous.

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Example4 The present value of a future expense. Suppose that you plan to buy a luxury car four years from now and that car will cost $55,000 at that time. If your bank’s savings deposit interest rate is 5% per year, what is the amount that you have to deposit now (present value) in your savings account in order to grow enough interest and with the principal to buy your dream car in the future? Assume the interest is compounded: (a) annually; (b) quarterly; (c) weekly; or (d) continuously, for a year. (e) Show your results in a table of PV calculation with separate FV, PVIF, and PV columns.

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Answer:(a) annually P = $55,000(1 + 0.05)−4 = $45,248.64;(b) quarterly P = $55,000(1 + 0.05/4)−4∗4 = $45,086.05;(c) weekly P = $55,000(1 + 0.05/52)−52∗4 = $45,034.52;(d) continuously P = $55, 000e−0.05∗4 = $45,030.19;(e)

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Pricing the Cash Flows of a BondApplication of present value

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For a typical bond that promises to pay a fixed coupon payment of C/2 every six months and repay the par amount FV (face value) at maturity, current price P is:

Price of a bond is equal to the sum of present values of future coupons plus that of the redemption value.

where r /2 is the appropriate semiannual yield rate and n is the remaininglife of the bond measured in the unit of the coupon payment period (six months).

Coupon Bond valuation equation

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Present value of annuityv=1/(1+i)

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Example: How to value a coupon bond?1

Suppose a bond has a face value of $1000, will pay semi-

annual interest at the (coupon) rate of 5%, and will be

due 10 years after issue.

This means that the owner of one such bond will receive

$25 (=1000*5%/2) every six months for a total of 20

payments, together with a payment of $1000 10 years

after the date of issue.

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If the purchaser wishes to receive a 6% return,

compounded semiannually, then the present value at

issue is

A purchaser content with a return of 4% compounded

semi-annually would value the bond at

Example: How to value a coupon bond?2

Price of a bond is equal to the present value of future coupons plus that of

the redemption value.

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Sensitivity of Bond Prices to Interest Rate Movements

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Bond price elasticity =

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Vdi

idV

Discounted Cash-flow Valuation:

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A share of preferred stock is like a perpetual bond, and has no maturity. It receives a pre-specified regular dividend.

Present value of preferred stock =

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指数增长和指数下降

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Section 4.2 Logarithmic Functions(对数函数 )

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Example 4

Use logarithm rules to rewrite each of the following expressions Use logarithm rules to rewrite each of the following expressions in terms of .in terms of . 3log and 2log 55

a. b. c. a. b. c.

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5log5 8log5 36log5

Solution:Solution:

a. a.

15log since 3log1

rulequotient 3log5log3

5log

55

555

b.b. rulepower 2log32log8log 53

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c. c.

rulepower 3log222log

ruleproduct 3log2log)32(log36log

55

25

25

2255

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Example 5

Solution:Solution:

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Exponential & Logarithmic functions: important limits Lim (x0) (exp(x) – 1 ) / x = 1

Or exp(x) 1 + x as x0 , and Lim (x0) ln(1+x) / x = 1

From these, we can prove d/dx ( exp(x) ) = exp(x) , d/dx ( ln(x) ) = 1/x

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Section 4.3 Differentiation of Logarithmic Section 4.3 Differentiation of Logarithmic and Exponential Function and Exponential Function

Example 6

Solution:Solution:

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Example 7

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Solution:Solution:

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Differentiate both sides of the equationDifferentiate both sides of the equation

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Example 8

Solution:Solution:

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Solution:Solution:

Example 9

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Taking the derivatives of some complicated functions Taking the derivatives of some complicated functions can be simplified by using logarithms. This is called can be simplified by using logarithms. This is called logarithmic differentiationlogarithmic differentiation..

Example 10

Solution:Solution:

to be continued

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Example 11

Differentiate each of these functionDifferentiate each of these function

a. b. a. b. xy 2 xy 3log

Solution:Solution:

a. To differentiate , we use logarithmic differentiation a. To differentiate , we use logarithmic differentiation as follows: as follows:

xy 2

to be continued

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b. b.

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The relative rate of change of a quantityThe relative rate of change of a quantity Q(x) Q(x) can be can be computed by finding the derivative ofcomputed by finding the derivative of lnlnQ.Q.

'( )(ln )

( )

d Q xQ

dx Q x

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Example 12

Solution:Solution:

to be continued

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Summary Exponential Functions, Basic Properties of Exponential Exponential Functions, Basic Properties of Exponential Functions, The Natural Exponential Base Functions, The Natural Exponential Base e.e.

Compound Interest,Compound Interest, Continuously Compounded Continuously Compounded Interest, Present Value.Interest, Present Value.

Exponential Growth and Decay.Exponential Growth and Decay.

Logarithmic Functions, The Natural Logarithm.Logarithmic Functions, The Natural Logarithm.

Differentiation of logarithmic and Exponential Differentiation of logarithmic and Exponential Functions. Functions.

Optimal holding time.Optimal holding time.

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Trigonometric functions: a very important limit

From this we can prove:

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Derivatives of trigonometric functions Using the previous results we can derive

d/dx sin(x) = cos(x) d/dx cos(x) = - sin(x) d/dx tan(x) = ? d/dx sec(x) = ?

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Hyperbolic functions

sinh (x) = ( exp(x) – exp(-x) )/ 2 cosh (x) = ( exp(x) + exp(-x) )/ 2 d/dx sinh(x) = cosh(x) d/dx cosh(x) = sinh(x)

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Indeterminate forms

Expressions of the form 0/0, ∞/∞, 0 ×∞, ∞−∞, 0^∞ and ∞^0 are called indeterminate forms.

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L’Hospital’s Rule – for indeterminate forms

If f’(x) and g’(x) exist, and if g’(x) is nonzero, then we have the following result, known as L’Hospital’s rule:

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Examples: