1 College Algebra K/DC Monday, 28 March 2016 OBJECTIVE TSW use properties of exponents to solve exponential equations. ASSIGNMENTS DUE TOMORROW –Sec. 4.1: p. 396 (59-81 odd, all) –Sec. 4.2: p. 410 (61-86 all) TODAY’S ASSIGNMENT (due on Friday, 04/01/16) –Sec. 4.2: pp (87-96 all, 98 omit D, 99, 100) QUIZ: Sec. 4.1 – 4.2 is tomorrow, Tuesday, 03/29/16. TEST: Sec. 4.1 – 4.3 is on Friday, 04/01/16. ROOM 2266Relocation (6 th period only) for Thursday, 03/31/16: ROOM 2266 COLD Water for sale (25¢) ! ! !

Due tomorrow, Tuesday, 29 March ASSIGNMENT: Sec. 4.2: p. 410 (61-86 all) Show all work. Due tomorrow, Tuesday, 29 March Write the problem and solve. Use solution sets.

4-3 Exponential Functions 4.2 Exponents and Properties ▪ Exponential Functions ▪ Exponential Equations ▪ The Number e ▪ Compound Interest ▪ The Number e and Continuous Compounding ▪ Exponential Models and Curve Fitting

4-4 Properties of Exponents (previously given) n times

4-5 If, find each of the following. Evaluating an Exponential Expression (a) (b) (c) (d)

Using a Property of Exponents to Solve an Equation Definition of negative exponent Set exponents equal. Solution set: {–3} Write 125 as a power of 5. An equation is being solved, so use solution sets. The variable is the exponent, so get a common base first. Once the bases are the same, set the exponents equal to each other.

4-7 Using a Property of Exponents to Solve an Equation Solution set: {7} Write 9 as a power of 3. Set exponents equal. Solve for x.

4-8 Using a Property of Exponents to Solve an Equation Raise both sides to the 2/5 power. Write 243 as a power of 5. Simplify.

4-9 Using a Property of Exponents to Solve an Equation It is necessary to check all proposed solutions in the original equation when both sides have been raised to a power. Check b = 9. Solution set: {9}

4-10 The Number e e is a real number, not a variable. To nine decimal places, e ≈ e is irrational.

4-11 Simple Interest and Compound Interest The simple interest formula is I = Prt, where P is the principal (amount deposited) r is the annual rate of interest (expressed as a decimal) t is the time in years. Compound interest is interest paid on both principal and interest.

4-12 Compound Interest (Don’t copy this yet.) Suppose t = 1 yr. Some amount P is invested. At the end of one year, the total amount is the original principal plus interest. If this new balance earns interest at the same rate for another year, the balance at the end of that 2 nd year will be Previous balanceInterest on previous balance

4-13 Compound Interest (Don’t copy this yet.) After the third year, this will grow to This leads to the formula for interest compounded annually: And this last formula gives us the compound interest formula.

4-14 Compound Interest Formula (Copy this.) If P dollars are deposited in an account paying an annual rate of interest r compounded (paid) n times per year, then after t years the account will contain A dollars, where Memorize this formula!!!

4-15 Using the Compound Interest Formula Suppose $2500 is deposited in an account paying 6% per year compounded semiannually (twice per year). Find the amount in the account after 10 years with no withdrawals. P = 2500, r =.06, n = 2, t = 10 Compound interest formula Round to the nearest hundredth. There is $ in the account after 10 years. Application problem; answer in context.

4-16 Using the Compound Interest Formula How much interest is earned over the 10-year period? The interest earned over the 10 years is $ – $2500 = $

4-17 Finding Present Value Leah must pay a lump sum of $15,000 in 8 years. What amount deposited today at 4.8% compounded annually will give $15,000 in 8 years? Compound interest formula If Leah deposits $10, now, she will have $15,000 when she needs it. A = 15,000, r =.048, n = 1, t = 8 Simplify, then solve for P. Round to the nearest hundredth.

4-18 Finding Present Value If only $10,000 is available to deposit now, what annual interest rate is necessary for the money to increase to $15,000 in 8 years? Compound interest formula An interest rate of about 5.20% will produce enough interest to increase the $10,000 to $15,000 by the end of 8 years. A = 15,000, P = 10,000, n = 1, t = 8 Simplify, then solve for r. Use a calculator.

4-19 How e Is Determined Suppose $1 is invested at 100% interest per year, compounded n times per year. What happens if the compounding (n) increases? Look what happens: What is A getting close to as n gets bigger and bigger? e!!!!!! n A

4-20 Continuous Compounding If P dollars are deposited at a rate of interest r compounded continuously for t years, the compound amount in dollars on deposit is Memorize this formula!!!

4-21 Solving a Continuous Compounding Problem Suppose $8000 is deposited in an account paying 5% interest compounded continuously for 6 years. Find the total amount on deposit at the end of 6 years. There will be about $10, in the account at the end of 6 years. Continuous compounding formula P = 8000, r =.05, t = 6 Round to the nearest hundredth. Answer in context.

4-22 Comparing Interest Earned as Compounding is More Frequent Suppose $2500 is invested at 6% in an account for 10 years. Find the amounts in the account at the end of 10 years if the interest is compounded quarterly, monthly, daily, and continuously.

4-23 Using Data to Model Exponential Growth If current trends of burning fossil fuels and deforestation continue, then future amounts of atmospheric carbon dioxide in parts per million (ppm) will increase as shown in the table. What will be the atmospheric carbon dioxide level in 2015? The data can be modeled by the function In 2015, the atmospheric carbon dioxide level will be 441 ppm.