Precalculus Notes: Exponential Functions



PreCalculus Notes ELF – 1: Exponential Functions

Properties of exponents (review). If a is a positive number, then

[pic] = [pic]=

[pic] = [pic] =

[pic] = [pic] =

Properties of exponential functions of the type y = ax, a > 0 (should be review of A2T).

a.

b. Domain:

Range:

c. Intercepts:

d. Asymptotes:

e. Increasing function if

As a increases

Decreasing function if

As a increases

f. For a > 1,

1) As x ( (, ax ( 2) As x ( -(, ax (

For 0 < a < 1,

1) As x ( (, ax ( 2) As x ( -(, ax (

Review transformations of graphs.

Ex: Let f(x) = ax, a > 1.

a. y = -f(x) b. y = f(-x)

c. y = f(x) + k d. y = f(x – h)

e. y = af(x) f. [pic]

PreCalculus Notes ELF – 2: Log Functions

Inverses (Review)

Ex: Solve the following for x; then write the inverse function:

[pic] ( If[pic], then [pic]

y = 3x ( If [pic], then [pic]

y = x3 ( If [pic], then [pic]

y = 3x ( If[pic], then [pic]

Ex: Write an equivalent equation:

a. [pic] b. [pic]

c. [pic] d. [pic]

e. [pic] f. [pic]

g. [pic] h. [pic]

i. [pic] j. [pic]

Ex: Write the inverse of each function:

a. [pic] b. [pic]

c. [pic] d. [pic]

e. [pic] f. [pic]

Properties of inverses: inverse functions

x + 3 – 3 = [pic]

[pic] [pic]

[pic]

[pic] [pic]

Ex: Simplify

a. [pic] b. [pic] =

c. [pic] d. [pic]

e. eln(2x – 3) = f. [pic] =

g. [pic] h. [pic]

PreCalculus Notes ELF – 3: Properties of Logs

Exponent Property Log Property

1. [pic] 1.

2. [pic] 2.

3. [pic] 3.

4. [pic] 4.

5. [pic] 5.

6. [pic] 6.

7. [pic] 7.

Ex: Expand [pic]

Ex: Expand [pic]

Ex: Expand log(x + y)

Ex: Write as a single log: [pic]

Ex: Write as a single log: [pic]

Evaluating logs

Ex: a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

Change of Base Formula

Solve for x using log base b: [pic]

PreCalculus Notes ELF – 4: Exponential and Log Equations

(Review) Rules for Solving Simple Equations:

1.

2.

Ex: x + 3 = 7 Ex: 3x = 7 Ex: x3 = 7 Ex: log3x = 7 Ex: 3x = 7

Def: Exponential and log equations are equations where the variable appears in either an exponent or in a log.

Ex: [pic], [pic], [pic], and [pic] but not [pic] or [pic].

Ex: Solve for x:

a. [pic]

b. [pic]

c. 2log4x – 1 = log4(24 – x)

d. [pic]

e. [pic]

PreCalculus Notes ELF – 6: Log Graphs

Recall: The graph of [pic] is

Function: [pic] (a > 1)

Domain:

Range:

y-intercept:

Roots:

Asymptote:

[pic] [pic]

Inverse:

Domain:

Range:

y-intercept:

Roots:

Asymptote:

[pic] [pic]

Ex: Sketch a graph of [pic] where a > 1.

Domain:

Vertical asymptote:

y-intercept:

Root:

[pic]

[pic]

PreCalculus Notes ELF – 7: Exponential Growth Models, Part I

Ex: On January 5, two alien blobs, Blobby and Bloberta, took up residence in room 309. On that date, they each weighed 50 grams. Since then, they have been feeding on the intense brainwaves put out by students in Mr. Clemens’s math classes and have been growing steadily.

a. Blobby is growing at a rate of 2 grams per day. Write an equation for Blobby’s mass, m as a function of t, the number of days since Jan 5.

b. Bloberta has been growing at a rate of 2% per day.

Write an equation for Bloberta’s mass, m, as a

function of time t.

Exponential Growth Models

An exponential growth (or decay) model has the form [pic] (or sometimes [pic]).

A =

t =

A0 =

b =

r =

k =

Ex: In 2006, about 8.4 gigatonnes of carbon was released into the earth’s atmosphere through the use of fossil fuels. This was about a 6.5% increase over the previous year. Write an exponential model for the amount of carbon released from fossil fuels each year since 2006.

Ex: On the island of Evilmathland, the golden sand cats used to prey on the three-toed voles. After the sand cats were eradicated by hunting, the vole population exploded. The number of voles on the island can be modeled by [pic] where P is the vole population in thousands and t is the number of years since the last cat was killed.

a. How many voles were on the island when the last cat was killed?

b. How fast is the vole population growing?

c. Could this population trend continue?

Ex: Outside on a cold day, Quinn sets a cup of hot McDonalds coffee on the hood of his truck. The temperature of coffee is modeled by [pic] where T is the temperature in (C and t is time in minutes since he set it down.

a. What was the temperature of the coffee when Quinn set it down?

b. How fast is the coffee cooling?

c. Under these conditions, what will be the final temperature of the coffee?

PreCalculus Notes ELF – 8: Exponential Growth Models, Part II

Ex: Jason is studying the population growth of a colony of small organisms which he calls Voorheeses in honor of himself. After much observation, Jason concludes that the Voorhees population grows at a rate of 14.87% an hour. Suppose he starts with an initial population of 100 Voorheeses.

a. Write an exponential model for P(t), the Voorhees population after t hours.

b. According to your model, what will the Voorhees population be after 50 hours?

Ex:. Freddy is studying the population growth of a colony of small organisms which he calls Kruegers in honor of himself. After much observation, Fred concludes that the Krueger population doubles every five hours. Suppose he starts with an initial population of 100 Kruegers.

a. What will the population be after

(1) 5 hours?

(2) 10 hours?

(3) 200 hours?

b. Write a function to give P(t), the population after t hours.

c. According to your equation, what will the Krueger population be after 50 hours?

Ex:. Thomas is studying the population growth of a colony of small organisms which he whimsically calls Leatherfaces because calling them Hewitts sounds stupid. After much observation, Tom concludes that the Leatherface population can be modeled by the equation[pic].

a. How many Leatherfaces did Tom have initially?

b. According to Tom’s equation, what will the Leatherface population be after 50 hours?

Review: The simplest exponential growth (or decay) models have the form [pic] where the base

b = 1 + r and r is the percent growth rate (expressed as a decimal) per time period. However, there may be times when it is easier or preferable to use some different base. A more flexible model is[pic] where b is some convenient base and k is a horizontal dilation factor.

Ex: In the year 2000, the population of Metropolis was 1.8 million and growing at a rate of 3.5% per year. Write an exponential model for the population as a function of time in years since 2000 assuming the same growth rate continued.

Ex: The half-life of Radium-224 is about 3.66 days.

a. Write an exponential model for the percent that remains from a sample of Ra-224 after t days.

b. What is the daily percent loss rate of Ra-224?

Ex: A new social networking service called Hebetude is launched in 2015. By the end of the year, it has 200,000 members. Six years later it has 600,000,000 members worldwide.

a. Write an exponential growth model using base e for the number of Hebetude members as a function of time.

b. What is the annual percent growth rate of Hebetude membership?

PreCalculus Notes ELF – 9: Exponential Growth Models, Part III

Ex: By diligently soliciting door-to-door during the evening of October 31st, Lara was able to accumulate 90 oz. of assorted sweets for later consumption. After much thought, she decided to eat exactly one third of her stash each week for as long as it lasted.

a. Write an equation for the amount of candy (in oz.) Lara has left as a function of time in days since Halloween.

b. When will Lara’s candy be gone?

Ex: Chuck Rocker sold his collection of moon rocks for $800; this he put into a Certificate of Deposit (CD) which earned 5% annual interest.

a. Find the value of Chuck’s CD as a function of time.

b. how long will it takes Chuck’s money to double?

Ex: Professor Tzap has discovered a totally new element which he has named the “element of surprise” (proposed chemical symbol Wtf). As it happens, Wtf is radioactive. One day after Dr. Tzap discovered it, only 85% of the original amount remained.

a. Write an exponential equation base e for the amount of Wtf remaining as a function of time.

b. Dr. Tzap originally found 20 grams of Wtf. Unfortunately for him, the radioactivity proved deadly. If it is safe to go into his lab when the amount of Wtf drops below 1 gram, how long will it be before Dr Tzap’s body can be safely recovered?

Ex: Major news events are communicated in a medium sized city according to the model

[pic], k > 0

where N is the number of people that have heard about the event and t is time in hours since the event took place.

a. As t ( (, how many people have heard about the news event? Logically, what does this number represent in the context of the problem?

b. Eight hours after news event occurred, 100,000 people have hear about it. Find the value of k.

c. After about how many hours have 80% the people in the city heard about the event?

-----------------------

x

y

x

y

y

x

y

x

y

x

y

x

y

x

y

x

0 10 20 30 40 50 60 70 80 90

240

220

200

180

160

140

120

100

80

60

40

20

0

| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |YZxy?‘’“œ?´µ¶·½¾ÕÖ×Øáâùúûü üõñüìüäüÑÇäüäü´ªäüäü—?äüäüzpäüäü]$j¤t

@[pic]hù

ÂCJ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

t

m

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download