4.1 Exponential Functions. For Formula 1, the value of n is based on the frequency of compounding. Common frequencies include:

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1 4.1 Exponential Functions Hartfield MATH 2040 Unit 4 Page 1 Recall from algebra the formulas for Compound Interest: Formula 1 For Discretely Compounded Interest A t P 1 r n nt Formula 2 Continuously Compounded Interest rt A t Pe t = units of time in years P = principle (initial value) r = rate of interest as a decimal A(t) = amount at/after time t n = number of compounds per year For Formula 1, the value of n is based on the frequency of compounding. Common frequencies include: annually (n = 1), semiannually (n = 2), quarterly (n = 4), bimonthly (n = 6), monthly (n = 12), biweekly (n = 26), weekly (n = 52), daily (n = 360 or 365). In economics, financial institutions have to report an annual percentage change even though their nominal rate of interest is for some other interval of time. A formula may ask for a future amount when given a present amount (given P, find A(t)) or it may ask for a present amount when given a future amount (given A(t), find P).

2 Hartfield MATH 2040 Unit 4 Page 2 Ex. 1: Kelsey has $2000 to save for three years. She goes to three different banks and finds she can lock in the following rates. Rounding down to the penny, how much would she have with each bank and thus which bank is the best option? Bank A: Bank B: Bank C: 4.20% interest per year compounded quarterly 4.18% interest per year compounded daily 4.15% interest per year compounded continuously

3 Hartfield MATH 2040 Unit 4 Page 3 Ex. 2: To the nearest thousandth of a percent, what is the Annual Percentage Yield on each bank? Bank A: Bank B: Bank C: 4.20% interest per year compounded quarterly 4.18% interest per year compounded daily 4.15% interest per year compounded continuously

4 Ex. 3: A $24,000 automobile depreciates by 40% per year. Find its value in 6 months and then in 3 years. Hartfield MATH 2040 Unit 4 Page 4

5 4.3a Differentiation of Exponential Functions Hartfield MATH 2040 Unit 4 Page 5 Rule 8: Derivative of a Natural Exponential Ex. 1: Find the derivative. d e dx x e x f x 2 x 3 e If the exponential function has a function within the exponent, we can extend rule 8 using Chain Rule: Rule 8a: Generalized Derivative of a Natural Exponential d e f x e f x f x dx

6 Hartfield MATH 2040 Unit 4 Page 6 Ex. 2: Find the derivative. Then evaluate the derivative at the given x-value and approximate it to three decimal places. Ex. 3: Find the second derivative. f x e x 3 2, x f x xe e f (1) x

7 Applications Hartfield MATH 2040 Unit 4 Page 7 Ex. 1: The percentage P(t) of people surviving to age t years in ancient Rome can be approximated by P(t) = 92e t. Calculate P (22) and explain what the result indicates. (Source: Finite Mathematics & Calculus Applied to the Real World (1996) p. 972, #73) Ex. 2: A cup of coffee brewed at 200 degrees, if left in a 70-degree room, will cool to T(t) = e 2.5t degrees in t hours. Determine the temperature of the coffee in 1 hour and the rate of change in the temperature at that time. (Source: 5 th edition p. 302, #76)

8 4.2 Logarithms & Exponential Equations Hartfield MATH 2040 Unit 4 Page 8 Review from algebra about solving exponential equations. Ex.: Solve 2x 1 2e Isolate an exponential expression on one side. 2. Take the natural logarithm (or common logarithm) of both sides. 3. Use the laws of logarithms to rewrite the exponential expression so that no variable remains in the exponent. 4. Apply basic algebraic and arithmetic manipulation to solve for x. 5. Use the laws of logarithms to simplify the solution and approximate the solution. 6. Check your solution.

9 Solving exercises with exponential equations Hartfield MATH 2040 Unit 4 Page 9 Ex. 1: To the tenth of a year, how long would it take an account to double if the interest rate is 9.9% A: compounded monthly. B: compounded continuously.

10 Hartfield MATH 2040 Unit 4 Page 10 Ex. 2: To the tenth of a year, how long would it take an account with an interest rate of 16.79% compounded daily (n=365) to A: triple in value. B: increase by 40%.

11 Hartfield MATH 2040 Unit 4 Page 11 Ex. 3: Using information from a recent Edmunds.com report, the resale value of a 2008 SUV is expected to depreciate up to 20% per year. To the tenth of a year, how long will it take value of an SUV to decrease by half?

12 4.3b Differentiation of Logarithmic Functions Hartfield MATH 2040 Unit 4 Page 12 Rule 9: Derivative of a Natural Logarithm d lnx dx 1 x Derivatives of Exponential & Logarithmic Functions of other bases Rule 10: Derivative of Exponential (base a) If the logarithmic function has a function within the logarithm, we can extend rule 9 using Chain Rule: d a dx x ln a a x Rule 9a: Generalized Derivative of a Natural Logarithm d ln f x dx f x f x Rule 11: Derivative of a Logarithm (base a) d log dx a x 1 lna x

13 Hartfield MATH 2040 Unit 4 Page 13 Ex. 1: Find the derivative. Ex. 2: Find the derivative. 2 f( x) ln x 2x 3 x 5 f( x) ln 1

14 Hartfield MATH 2040 Unit 4 Page 14 Ex. 3: Find the derivative. x f( x) e ln x 1 Ex. 4: Find the derivative. Then evaluate the derivative at the given x-value. lnx fx ( ) f (1) 2 x

15 4.3-c & 4.4 Economic Applications Hartfield MATH 2040 Unit 4 Page 15 Definition: Let D(p) be the consumer demand at price p. Then the consumer expenditure E is E(p) = p D(p) Ex. 1: If consumer demand for a commodity is given by the function D(p) = 8000e 0.05p, where p is the selling price in dollars, find the price that maximizes consumer expenditure. (Source: 4 th edition p. 290, #66)

16 Hartfield MATH 2040 Unit 4 Page 16 A similar exercise for maximizing revenue can be approached where the independent variable is the quantity x sold and price is a function of x (as opposed to the inverse relationship in consumer expenditure): R(x) = p(x) x Ex. 2: Find the quantity x (in thousands) and the price p (in dollars) where revenue is maximized if p(x) = 200e 0.25x.

17 Relative Rates of Change Hartfield MATH 2040 Unit 4 Page 17 While all derivatives are rates of change, not all derivatives consider the relative values of a function at a given point of time. In absolute terms a $10,000 product increasing at a rate of $100 a year has a higher rate of change than a $100 product increasing at $20 a year. Taking into respect the current value of each product however tells us that the $100 object has a higher relative rate of change. Definition: The relative rate of change in a function is the derivative of the natural log of the function; that is, Ex. 1: Find the absolute rate of change and the relative rate of change in f. Then evaluate the relative rate of change at the given value of t. f(t) = t 3, t = 10 d dt f() t ln ft ( ). f () t

18 Hartfield MATH 2040 Unit 4 Page 18 Applications of Relative Rates of Change Ex. 2: Find the absolute rate of change and the relative rate of change in f. Then evaluate the relative rate of change at the given value of t. f(t) = e t³, t = 5 Ex 1: The gross domestic product of a developing country is forecast to be G(t) = 5 + 2e 0.01t million dollars t years from now. Find the relative rate of change in the GDP 20 years from now. (Source: 4 th edition p. 303, #14)

19 Hartfield MATH 2040 Unit 4 Page 19 Ex 2: The population (in millions) of a city t years from now is given by the function P(t) = e 0.05t. (Source: 4 th edition p. 303, #14) a. Find the relative rate of change of the population 8 years from now. b. Will the relative rate of change ever reach 1.5%?

20 Demand Functions & the Elasticity of Demand Hartfield MATH 2040 Unit 4 Page 20 Definition: The demand function x = D(p) gives the quantity x of an item that will be demanded by consumers at price p. Statement: The Law of Downward-Sloping Demand states that since demand generally falls as prices rise, the slope of the demand function is negative. Since revenues are a product of prices and demand, a balance point should exist between increasing prices and increasing demand so that revenues are maximized. Definition: For a demand function D(p), the elasticity of demand is pd( p) Ep ( ). Dp ( ) If E(p) > 1, then demand is elastic and prices should be lowered to increase revenue. If E(p) < 1, then demand is inelastic and prices should be raised to increase revenue. At maximum revenue, E(p) = 1 and demand is unitary.

21 Hartfield MATH 2040 Unit 4 Page 21 Ex. 1: For the demand function D(p) = 100 p 2, determine where demand is elastic, inelastic, or unitary. Ex. 2: A liquor distributor wants to increase its revenues by discounting its bestselling liquor. If the demand function for this liquor is D(p) = 60 3p, where p is the price per bottle, and the current price is $15, will the discount succeed? (Source: 4 th edition p. 303, #28)

22 Hartfield MATH 2040 Unit 4 Page 22 Ex. 3: At a market level, the demand function for distilled spirits is defined by D(p) = 3.509p Determine where demand is elastic, inelastic, or unitary and analyze the relationship implied between taxes, liquor consumption, and government revenues. (Source: Journal of Consumer Research 12)