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- 1 Chapter 4 Exponential and Logarithmic Functions ( ) In this Chapter, we will encounter some important concepts Exponential Functions( ) Logarithmic Functions( ) Differentiation of Logarithmic and Exponential Functions Exponential Functions
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- 2 Section 4.1 Exponential Functions Exponential function( ): If b is a positive number other than 1 (b>0, b1), there is a unique function called the exponential function with base b that is defined by f(x)=b x for every real number x NOTE: Such function can be used to describe exponential and logistic growth and a variety of other important quantities.
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- 3 Definition of for Rational Values of n (and b>0) Integer powers: If n is a positive integer, Fractional powers: If n and m are positive integers, where denotes the positive mth root. Negative powers: Zero power:
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- 4 Example 1 Solution: Solution:
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- 5 Figure below shows graphs of various members of the family of exponential functions NOTE: Students often confuse the power function with the exponential function
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- 8 Definition: The natural exponential function is Where n 10100100010,000100,000 2.593742.704812.716922.7118152.71827
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- 9 Continuous Compounding of Interest( ) If P is the initial investment (the principal) and r is the interest rate (expressed as a decimal), the balance B after the interest is added will be B=P+P r =P(1+r) dollars
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- 13 Example 2 Suppose $1,000 is invested at an annual interest rate of 6%. Compute the balance after 10 years if the interest is compounded a. Quarterly b. Monthly c. Daily d. Continuously Solution: a.To compute the balance after 10 years if the interest is compounded quarterly, using the formula with t=10, p=1,000, r=0.06, and k=4: with t=10, p=1,000, r=0.06, and k=4: to be continued
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- 14 b. This time, take t=10, p=1,000, r=0.06, and k=12 to get c. Take t=10, p=1,000, r=0.06, and k=365 to obtain d. For continuously compounded interest use the formula with t=10, p=1,000, and r=0.06: This value, $1,822.12, is an upper bound for the possible balance. No matter how often interest is compounded, $1,000 invested at an annual interest rate of 6% can not grow to more than $1,822.12 in 10 years.
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- 15 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 ( ) & Future Value is the present value interest factor (PVIFn) (or discount factor) of n periods.
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- 16 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 years m = compound periods per year Present Value ( ) & Future Value
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- 17 Present value of $1.00 when interest (i=5%) is compounded quarterly for a year
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- 18 Example 3 Sue is about to enter college. When she graduates 4 years from now, she wants to take a trip to Europe that she estimates will cost $5,000. How much should she invest now at 7% to have enough for the trip if interest is compounded: a. Quarterly b. Continuously Solution: The required future value is F=$5,000 in t=4 years with r=0.07. a.If the compounding is quarterly, then k=4 and the present value is to be continued
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- 19 b. For continuous compounding, the present value is Thus, Sue would have to invest about $9 more if interest is compounded quarterly than if the compounding is continuous.
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- 20 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 banks 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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- 21 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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- 22 Pricing the Cash Flows of a Bond Application of present value
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- 25 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 remaining life of the bond measured in the unit of the coupon payment period (six months). Coupon Bond valuation equation
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- 26 Present value of annuity v=1/(1+i)
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- 27 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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- 28 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 30
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- Bond price elasticity = 31
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- Discounted Cash-flow Valuation: 32 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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- 34 Section 4.2 Logarithmic Functions( )
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- 36 Example 4 Use logarithm rules to rewrite each of the following expressions in terms of. a. b. c. Solution: a. b. c.
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- 39 Example 5 Solution:
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- 40 Exponential & Logarithmic functions: important limits Lim (x 0) (exp(x) 1 ) / x = 1 Or exp(x) 1 + x as x 0, and Lim (x 0) 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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- 41 Section 4.3 Differentiation of Logarithmic and Exponential Function Example 6 Solution:
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- 42 Example 7
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- 43 Solution:
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- 44 Differentiate both sides of the equation
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- 45 Example 8 Solution:
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- 47 Solution: Example 9
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- 48 Taking the derivatives of some complicated functions can be simplified by using logarithms. This is called logarithmic differentiation. Example 10 Solution: to be continued
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- 50 Example 11 Differentiate each of these function a. b. Solution: a. To differentiate, we use logarithmic differentiation as follows: to be continued
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- 51 b.
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- 54 The relative rate of change of a quantity Q(x) can be computed by finding the derivative of lnQ.
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- 55 Example 12 Solution: to be continued
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- 57 Summary Exponential Functions, Basic Properties of Exponential Functions, The Natural Exponential Base e. Compound Interest, Continuously Compounded Interest, Present Value. Exponential Growth and Decay. Logarithmic Functions, The Natural Logarithm. Differentiation of logarithmic and Exponential Functions. Optimal holding time.
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- 58 Trigonometric functions: a very important limit From this we can prove:
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- 59 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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- 60 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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- 61 Indeterminate forms Expressions of the form 0/0, /, 0 , , 0^ and ^0 are called indeterminate forms.
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- 62 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 LHospitals rule:
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- 63 Examples: