Exponential Functions Copyright Scott Storla 2014

Power functions vs. Exponential functions Power functionExponential function Copyright Scott Storla 2014

The formula for continuous compounding, is a specific example of the more general exponential growth and decay formula. Future amount at time t. Rate of growth or decay. Initial amount Copyright Scott Storla 2014

Vocabulary for exponential functions Copyright Scott Storla 2014

If as the value of x increases the value of y increases you have exponential growth. If as the value of x increases the value of y decreases you have exponential decay. Copyright Scott Storla 2014

Horizontal asymptote Copyright Scott Storla 2014

Discuss whether the graph shows exponential growth or decay, the intercepts, the domain, the horizontal asymptote and the range. Copyright Scott Storla 2014

Discuss whether the graph shows exponential growth or decay, the intercepts, the domain, the horizontal asymptote and the range. Copyright Scott Storla 2014

Discuss whether the graph shows exponential growth or decay, the intercepts, the domain, the horizontal asymptote and the range. Copyright Scott Storla 2014

Discuss whether the graph shows exponential growth or decay, the intercepts, the domain, the horizontal asymptote and the range. Copyright Scott Storla 2014

A procedure to help graph exponential functions Procedure – Graphing an Exponential Function 1)The domain for the exponential function is all real numbers. 2)Find any x and y intercepts. 3)Find a few points on the graph. Make sure to include both positive and negative exponents. 4)Estimate the horizontal asymptote. 5)Draw the graph. Copyright Scott Storla 2014

Graph the exponential function. Copyright Scott Storla 2014

The formula for continuous compounding, is a specific example of the more general exponential growth and decay formula. Future amount at time t. Rate of growth or decay. Initial amount Copyright Scott Storla 2014

Assume the growth constant, k, for the population of the United States is If the population was million in 2000 estimate the population in A sample of the paint used in a cave painting in France is found to have lost 82% of its original carbon-14. Estimate the age of the painting. The value of k for Carbon-14 is – A cup of coffee contains approximately 96 mg of caffeine. When you drink the coffee, the caffeine is absorbed into the bloodstream and is eventually metabolized by the body. If the rate of decay is – 0.14, how many hours does it take for the amount of caffeine to be reduced to 12 mg?

Copyright Scott Storla 2014 Carbon-14 is often used to date objects that were alive in the past. Between 1947 and 1956 the Dead Sea scrolls were discovered in 11 caves along the northwest shore of the Dead Sea. If the Dead Sea scrolls are authentic then they should date to around 2000 years old. If 78.5% of the original carbon-14 was left, was their age appropriate to being authentic? The value of k for Carbon-14 is Over the last 60 years a number of North American snail species have become extinct. If there are currently 650 species, and the extinction follows exponential decay, how many species were there originally? The value of k is

Finding the rate of growth or decay Radioactive half-life is the time necessary for a radioactive substance to decay to one-half the original amount. Carbon-14 has a half life of 5715 years. Find the decay rate for cabon-14. From 1970 to 1980 the population of the United States in millions went from to Find the growth rate. $17,000 grows to $23,000 in 8.5 years. Find the growth rate.

Copyright Scott Storla 2014 The half-life of silver-110 is 24.6 seconds. How long will it take for only 3% of the original sample of silver-110 to remain? From 1980 to 1990 the population of the United States in millions went from to Estimate the population in bacteria grow to 600 bacteria in 2 hours. Find the number of bacteria in 2 days. Dinosaur bones were dated using potassium-40 which has a half-life of approximately 1.31 billion years. Analysis of certain rocks surrounding the bones found that 94% of the original potassium-40 was still present. Estimate the age of the bones. $350,000 is currently in an account which increased its value by 40% over the last 5 years. Estimate the number of years before there is $1,000,000 in the account.

The Power Property of Logarithms Copyright Scott Storla 2014

Solve Copyright Scott Storla 2014

Compound Interest Copyright Scott Storla 2014

Compound Interest Compound interest occurs when interest is reinvested as principal and itself begins earning interest. Find the amount at the end of five years if $8,000 is invested at 2.15%. Copyright Scott Storla 2014

Future Value Copyright Scott Storla 2014

$30,000 is invested at 2% compounded quarterly. How much is the investment worth after 30 years? A 20 year old student inherits $10,000 and decides to invest the money for retirement at 5% annual interest compounded monthly. The student retires at 70. How much is the original $10,000 worth? An investment of $7,500 loses 4% per year for 3 years. If the investment was compounded semi-annually how much is the investment worth at the end of three years? Copyright Scott Storla 2014

Present Value Copyright Scott Storla 2014

In five years a couple wants $50,000 for a down payment. How much should they save today at 3% compounded monthly for that to happen? A great grandparent wanted to leave you a million dollars. How much would they need to have saved 150 years ago at 5% compounded quarterly for the account to hold a million dollars today? Copyright Scott Storla 2014

How long will it take for $75,000 to become $92,000 at 4.3% compounded monthly? Find the time it takes at 1.7% compounded daily for $15,000 to become 18,000. Copyright Scott Storla 2014

Continuous Compounding Copyright Scott Storla 2014

In 1800 Great Great Grandma put $50 in an account paying 5.5% interest compounded continuously? How much was in the account in 1900? How much was in the account in 2000? How much will be in the account this year? I would like to have $25,000 in an account in 17 years. What rate would I need to make this happen if I have $12,800 to invest? Assume continuous compounding. Copyright Scott Storla 2014

To retire in 14 years I want to have $1,500,000 in savings. If I’m able to get 4.2% compounded continuously how much do I need to have saved by today to make this happen? Upon retiring I don’t want to use any of my principal. How much in interest will I be able to live on each subsequent year? Assume 4.2% compounded continuously. Unfortunately. Copyright Scott Storla 2014