AP Calculus Assignment #3; Exponential Functions Name: The equation identifies a family of functions called exponential functions. Notice that the ratio of consecutive amounts of outputs always stay the same. As a sequence, exponential functions are known as geometric sequences. This fact is an important feature of exponential curves that has widespread application. EXPLORATION Exponential Functions 1. Graph the function for a = 2, 3, 5, in a [-5,5] by [-2,5] viewing window. 2. For what values of x is it true that? 3. For what values of x is it true that? 4. For what values of x is it true that? 5. Graph the function for a = 2, 3, 5. 6. Repeat parts 2-4 for the functions in part 5. DEFINITION Exponential Function Let a be a positive real number other than 1. The function is the exponential function with base a. The domain of is and the range is. If a > 1, the graph of f looks like: If 0 < a < 1, the graph of f looks like: EXAMPLE Graphing an Exponential Function Graph the function. State its domain and range. Solution The graph to the right shows the graph of the function y. It appears that the domain is. The range is because for all x.
EXAMPLE Finding Zeros Find the zeros of graphically. Solution The figure to the right suggests that f has a zero between x = 1 and x = 2, closer to 2. We can use our Nspire to find that the zero is approximately 1.756. Open a graph page and graph the equation from above. You will see the graph at the right. Select Menu, 6:Analyze Graph, 1:Zero, enter. Move the lower bound line to the left of the zero, press enter, then move the upper bound line to the right of the zero, press enter. A point will be placed at the zero and a coordinate point will be displayed. To show more decimal places, you must go to settings and select a float of at least 5. You try Graph the function. Use your Nspire to find the zeros of the State its domain and range. function. Graph the function below, labeling the value of the zeros. Exponential functions obey the rules for exponents. Rules for Exponents If a > 0 and b > 0, the following hold for all real numbers x and y. 1. 2. 3. 4. 5.
You try Use the table below and an exponential regression model to predict the population of the United States in the year 2010. If you do not remember how to find a regression equation on you Nspire, ask in the fall, or ask one of your peers. United States Population Year Population (millions) Ratio 1998 276.1 279.3/276.1 1.0116 1999 279.3 282.4/279.3 1.0111 2000 282.4 285.3/282.4 1.0102 2001 285.3 288.2/285.3 1.0102 2002 288.2 291.0/288.2 1.0097 2003 291.0 Source: Statistical Abstract of the United States 2004-2005 Regression equation: Predicted amount in 2010: Exponential Decay Exponential functions can also model phenomena that produce a decrease over time, such as happens with radioactive decay. The half-life of a radioactive substance is the amount of time it takes for half of the substance to change from its original radioactive state to a nonradioactive state by emitting energy in the form of radiation. EXAMPLE Modeling Radioactive Decay Suppose the half-life of a certain radioactive substance is 20 days and that there are 5 grams present initially. When will there be only 1 gram of the substance remaining? Solution Model The number of grams remaining after 20 days is The number of grams remaining after 40 days is.. The function models the mass in grams of the radioactive substance after t days. Solve graphically The graph to the right shows that the graphs of and approximately 46.438 or 46.439. (for 1 gram) intersect when t is Interpret There will be 1 gram of the radioactive substance left after approximately 46.438 days, or about 46 days 10.5 hours.
You try The half-life of phosphorus-32 is about 14 days. There are 6.6 grams present initially. (a) Express the amount of phosphorus-32 as a function of time t. (b) When will there be 1 gram remaining? Interpret your result in one complete sentence. DEFNITIONS Exponential Growth, Exponential Decay The function is a model for exponential growth if a > 1, and a model for exponential decay if 0 < a < 1. Compound interest investments, population growth, and radioactive decay are all examples of exponential growth and decay. Growth rate vs growth factor For exponential growth and decay, the base, a, is the growth factor. The annual rate, r, is: a = 1 + r. For example: If 1880, then the annual rate of growth can be interpreted as: represents the US population growth every 10 years beginning in The annual rate of growth is about 1.4%. The Number e Many natural, physical, and economic phenomena are best modeled by an exponential function whose base is the famous number e, which is 2.718281828 to nine decimal places. We can define e to be the number that the function approaches as (x approaches infinity). The graph and table below strongly suggest that such a number exists. As, The exponential functions and are frequently used as models of exponential growth or decay. For example, interest compounded continuously uses the model, where P is the initial investment, r, is the interest rate as a decimal, and t is time in years.
Practice; Use your own paper 1. Match the function with its graph. Try to do so without using your grapher. a. b. c. d. e. f. i ii iii iv v vi 2. Determine how much time is required for an investment to double in value if interest is earned at the rate of 6.25% compounded continuously. 3. Investors use the rule of 72 to estimate how much time it will take to double their money. The rule of 72 says that the time it takes for your money to double can be estimated by:, where r is the interest rate and t is the time to double your money. Use the rule of 72 to see how close you are to your answer in #2 above. 4. True or False The number is negative. Justify your answer. 5. Assume that the graph of the exponential function passes through the two points. Find the values of a and k. a) (1,4.5), (-1,0.5) b) (1,1.5), (-1,6) 6. True or False If, then a = 6. Justify your answer. 7. Let. a. Find the domain of f. b. Find the range of f. c. Find the zeros of f. 8. For the functions below, identify the initial quantity and the growth rate. a. b.
9. Identify the x-intervals on which the function graphed in Figure 1.21 is: (a) Increasing and concave up (c) Decreasing and concave up (b) Increasing and concave down (d) Decreasing and concave down 10. Give a possible formula for the function graphed below. 11. Let. (i) Find the base a. (ii) Find the percentage growth rate, r. a) and b) and 12. Write the functions in the form. Which represents exponential growth? decay? a) b) 13. The table below shows some values of a linear function f and an exponential function g. Find exact values (not decimal approximations) for each of the missing entries. x 0 1 2 3 4 10? 20?? 10? 20?? 14. When the Olympic Games were held outside Mexico City in 1968, there was much discussion about the effect the high altitude (7340 feet) would have on the athletes. Assuming air pressure decays exponentially by 0.4% every 100 feet, by what percentage is air pressure reduced by moving from sea level to Mexico City?
15. (a) The half-life of radium-226 is 1620 years. Write a formula for the quantity, Q, of radium left after t years, if the initial quantity is. (b) What percentage of the original amount of radium is left after 500 years? 16. Match the functions,, and, whose values are in the table below with the formulas,,, assuming a, b, and c are constants. Note that the function values have been rounded to two decimal places. 2 1.06 1 2.20 3 3.47 3 1.09 2 2.42 4 3.65 4 1.13 3 2.66 5 3.83 5 1.16 4 2.93 6 4.02 6 1.19 5 3.22 7 4.22 17. The median price, P, of a home rose from $50,000 in 1970 to $100,000 in 1990. Let t be the number of years since 1970. (a) Assume the increase in housing prices has been linear. Give an equation for the line representing price, P, in terms of t. Use this equation to complete column (a) of the table below. Use units of $1,000. (b) If instead the housing prices have been rising exponentially, determine an equation of the form which would represent the change in housing prices from 1970-1990, and complete column (b) of the table. (c) On the same set of axes, sketch the function represented in column (a) and column (b) of the table. (d) Which model for the price growth do you think is more realistic? t (a) Linear growth price in $1,000 units (b) Exponential growth price in $1,000 units 0 50 50 10 20 100 100 30 40