Name
8
Name _________________________________________________
5.1 Exponential Functions
Essential Question(s):
• How do you graph exponential functions?
• How do you solve problems involving compound interest?
Exponential Function [pic] where b > 0 and b[pic]1
| | | |
| |Domain |Real Numbers |
| | |[pic] |
| | | |
|[pic] | | |
| | | |
| |Range |Positive Real Numbers |
| | |[pic] |
| | |[pic] |
| |y-intercept | |
| | |The x-axis |
| |Asymptote | |
| | |increasing |
| |If b > 1 | |
| | |decreasing |
| |If 0 < b < 1 | |
Examples:
| | | | | | | |
|Graph [pic] |x |y | |Graph [pic] |x |y |
| |
|[pic] |
| |
|[pic] is a reflection of [pic]across the y-axis |
The natural base e e = 2.718281828459
Note: e is the limit of the expression of [pic] as x gets larger and larger.
The natural exponential function ( [pic]
| | | |
|Sketch the graph of [pic]and [pic]. | | |
| | | |
| |x |y |
| | | |
|[pic] |-2 |0.135 |
| |-1 |0.368 |
| |0 |1 |
| |1 |2.718 |
| |2 |7.389 |
| | |
|Transformation |Equation ( effect on graph of [pic] |
|Vertical translation | |
| |[pic] ( upward shift |
| | |
| |[pic] ( downward shift |
|Horizontal translation | |
| |[pic] ( shift to the left |
| | |
| |[pic] ( shift to the right |
|Reflection | |
| |[pic]( reflection about x-axis |
| | |
| |[pic] ( reflection about y-axis |
|Vertical stretching or shrinking | |
| |[pic] ( stretch if c > 1, shrink if 0 < c < 1 |
|Horizontal stretching or shrinking | |
| |[pic]( shrink if c > 1, stretch if 0 < c < 1 |
For a and b positive, [pic], and m and n real:
1. Exponent laws:
[pic]
[pic]
[pic]
[pic]
[pic]
2. [pic]
3. [pic]
Examples:
Simplify.
1. [pic] 2. [pic] 3. [pic]
[pic] [pic] [pic]
Solve for x.
4. [pic] 5. [pic] 6. [pic]
[pic] [pic] [pic]
Compound Interest – interest computed on your original investment (principal) as well as on any accumulated interest
Compounded semiannually ( compound interest is paid twice a year
Compounded quarterly ( compound interest is paid four times a year
Compounded continuously ( always compounding
Compound interest formula: [pic], where
| | |
|[pic] |Value of account after t years |
| | |
|[pic] |Principal or amount invested |
| | |
|n = |Number of times compounded in one year |
| | |
|r = |Percent in decimal form |
| | |
|t = |Time in years |
Continuous compounding formula: [pic]
Example.
How much interest is earned on a $4000 investment a)compounded semiannually @ 4.5% for 24 months? b) compounded continuously @ 4.5% for 24 months?
| | | | |
|[pic] |? |Compounded semiannually |Compounded continuously |
| | | | |
|[pic] |4000 |[pic] | |
| | | |[pic] |
| | | | |
|n = |2 | | |
| | | | |
|r = |0.045 | | |
| | | | |
|t = |2 | | |
5.2 Exponential Models
Essential Question(s):
• How do you solve problems involving mathematical modeling?
• How do you find the regression equation for data?
| | |
|Doubling time |The time it takes for a population to double |
|Doubling time growth model | |
| |[pic] |
| | |
| | |
| |Alternative formula: |
| | |
| |[pic] |
| | |
| | |
|Half-life | |
| |The time it takes for half of a particular material to decay |
|Half-life decay model | |
| |[pic] |
| | |
| |Alternative formula: |
| |[pic] |
| | |
| | |
|Limited Growth Model |[pic] |
|Logistic Growth Model |[pic] |
|Exponential Regression | |
| |Used to fit a function of the form [pic] to a set of data points. |
|Logistic Regression |Used to fit a function of the form [pic] to a set of data points |
1. The bacteria in a certain culture double every 7.8 hours. The culture has 7,000 bacteria at the start.
a. Write an equation that gives the number of bacteria N in the culture after t hours.
[pic]
b. How many bacteria will the culture contain after 4 hours?
[pic]
2. The radioactive element americium-241 has a half-life of 432 years. Suppose we start with a 20-g mass of americium-241.
a. How much will be left after 200 years? Compute the answer to three significant digits.
[pic]
b. How much will be left after 490 years? Compute the answer to three significant digits.
[pic]
3. An employee is hired to assemble toys. The learning curve[pic] gives the number of toys the average employee is able to assemble per day after t days on the job.
a. How many toys can the average employee assemble per day after 6 days of training? Round to the nearest integer.
[pic]
b. How many toys can the average employee assemble per day after 18 days of training? Round to the nearest integer.
[pic]
c. Does N approach a limiting value as t increases without bound? Explain.
[pic]
Yes, 60 is the upper limit for the number of toys an employee can assemble per day.
Use the data for Radioactive Studies to answer questions 4 - 5:
4. Find an exponential regression model of the form y = abx for the data set.
[pic]
5. Estimate the amount of material remaining after 7 hours. Round to four decimal places.
[pic]
Use the data in the table to the right to answer questions 6 - 8:
|x |y |
|0 |8 |
|10 |20 |
|20 |54 |
|30 |104 |
|40 |191 |
|50 |359 |
|60 |412 |
|70 |429 |
6. Find a logistic regression model [pic] for the data.
[pic]
7. Use the model to find the approximate value of y when x = 53.
[pic]
8. Using the model, what is the projected value of y when x = 70? Why does this differ from the value in the table?
[pic]
The model is used to approximate the data.
5.3 Logarithmic Functions
Essential Question(s):
• How do you graph logarithmic functions?
• How do you change from logarithmic form to exponential form and vice versa?
• What are the properties of logarithmic functions?
Logarithmic functions are the inverse of exponential functions
A. Logarithmic function: [pic] (for x > 0, b > 0 and b [pic]1)
[pic] means…b raised to what power y equals x?
In other words, [pic] (logarithmic form) is equivalent to [pic] (exponential form)
| | | | |
|[pic] |means … |2 raised to what power equals 32? |[pic] |
| | | | |
|[pic] |means… |4 raised to what power equals 2? |[pic] |
| | | | |
|[pic] |means … |b raised to what power equals 1? |[pic] |
| | | | |
|[pic] |means … |b raised to what power equals x? |[pic] |
Examples
| | |Exponential |Logarithmic |
| | | | |
| | | | |
|[pic] | | | |
| |Equation |[pic] |[pic] |
| | |Reals |Positive Reals |
| |Domain | | |
| | |Positive Reals |Reals |
| |Range | | |
| | |[pic] |[pic] |
| |intercept | | |
| | |x-axis |y-axis |
| |Asymptote | | |
| | |increasing | |
| |If b > 1 | |increasing |
| | |decreasing | |
| |If 0 < b < 1 | |decreasing |
B. Natural Logarithm [pic]
Examples
| | |
|[pic] |[pic] |
| | |
|[pic] |[pic] |
C. Properties
| | |Common Logarithm |Natural Logarithm |
| |General |means … |means … |
| | |[pic] or log on calculator |[pic] or ln on calculator |
| | | | |
|1. |[pic] |[pic] |[pic] |
| | | | |
|2. |[pic] |[pic] |[pic] |
| | | | |
|3. |[pic] |[pic] |[pic] |
| | | | |
|4. |[pic] |[pic] |[pic] |
Examples: Evaluate
| |( |[pic] | |[pic] |( | |
|[pic] | | | | | |[pic] |
| |( |[pic] | |[pic] |( |[pic] |
|[pic] | | | | | | |
| |( |[pic] | |[pic] |( |[pic] |
|[pic] | | | | | | |
| |( |[pic] | | | | |
|[pic] | | | | | | |
The domain of [pic] consists of all x for which [pic]. Find the domains of the following logarithmic functions:
| |[pic] | |[pic] |[pic] |
|[pic] |Domain: [pic] | | |Domain: [pic] |
Logarithms are another way of expressing exponents…therefore, connect the laws of exponents with the laws of logarithms.
| | | |
|Product |[pic][pic] |When we multiply two powers with the same base |
|Rule of Logarithms | |we… |
| | | |
| | |ADD EXPONENTS |
| | | |
|Quotient Rule of |[pic][pic] |When we divide powers with the same base we… |
|Logarithms | | |
| | |SUBTRACT EXPONENTS |
| | | |
|Log of a power |[pic][pic] |When we have a power raised to a power we… |
| | | |
| | |MULTIPLY EXPONENTS |
|Expanding Logarithmic Expressions – “break apart” as much as |Condensing Logarithmic Expressions – write as a single logarithmic|
|possible |expression |
|Example |Example |
|Expand [pic] |Condense [pic] |
|[pic] |[pic] |
Change of base formula
[pic]
Example
Simplify [pic]
[pic] OR [pic]
5.4 Logarithmic Models
Essential Question(s):
• How do you find the regression equation for data?
|Decibel | |
| |[pic] |
| | |
| |where |
| |D is the decibel level of the sound |
| |I is the intensity of the sound measured in watts per square meter |
| |[pic] is the intensity of the least audible sound that an average healthy |
| |young person can hear (standardized to be [pic] watts per square |
| |meter). |
|Magnitude M on the Richter scale | |
| |[pic] |
| | |
| |where |
| |E is the energy released by the earthquake (measured in joules) |
| |[pic] is the energy released by a very small reference earthquake |
| |(standardized to be [pic] |
|Velocity of a rocket at burnout | |
| |[pic] |
| | |
| |where |
| |c is the exhaust velocity of the rocket engine |
| |[pic] is the takeoff weight |
| |[pic] is the burnout weight |
|pH scale | |
| |[pic] |
| | |
| |where |
| |[pic] is the hydrogen ion concentration in moles per liter |
| |pH < 7 ( acidic |
| |pH > 7 ( basic |
|Logarithmic regression | |
| |Used to fit a function of the form [pic] to a set of data points. |
Examples
|1. |A rock concert has a volume with an intensity of I = 1.0 × 10–1 W/m2. Find its rating in decibels. |
[pic]
|2. |An earthquake has an energy release of 6.83 × 108 joules. What was its magnitude on the Richter scale? |
[pic]
|3. |A rocket has a weight ratio Wt/Wb = 18.1 and an exhaust velocity c = 4.71 kilometers per second. What is its velocity at |
| |burnout? Compute the answer to two decimal places. |
[pic]
Use the following to answer questions 4-5:
A solution has a hydrogen ion concentration of [H+] = 2.5 × 10–5.
|4. |Find the pH of the solution. Round your answer to one decimal place. |
[pic]
|5. |Is the solution acidic or basic? |
acidic
Use the following to answer questions 6-8:
The following table shows the number of pounds a person lost since beginning a diet.
|Time |Pounds Lost |
|(days) | |
|7 |2 |
|14 |12 |
|21 |17 |
|28 |20 |
|35 |22.5 |
|42 |24 |
|49 |25 |
|56 |25.5 |
|6. |Find a logarithmic regression model for the data. |
[pic]
|7. |Use the regression model to estimate the person's total weight loss after 17 days. |
[pic]
|8. |According to the regression model, what is the projected weight loss for 67 days? |
[pic]
5.5 Exponential and Logarithmic Equations
Essential Question(s):
• How do you solve exponential and logarithmic equations?
Examples
1. Solve [pic] for x to four decimal places.
[pic]
2. Solve [pic] for x to three decimal places.
[pic]
3. Solve [pic], and check.
[pic]
Check:
[pic] [pic]
x = 20
4. Solve [pic], and check.
[pic]
[pic] [pic]
Check:
[pic] [pic]
x = 1, 100
5. Solve [pic] for [pic].
[pic]
-----------------------
Exponential Functions
[pic]
[?]89:ËÌÍÎåæ÷øùúý
! " # $ % ' ( * + üøôøíÞ×ÏËÃË´§ÃËŸ•Ÿ•ˆŸxhˆcŸËUËMjhñ`ýU[pic]jhñ`ýU[pic]mHnHu[pic] hñ`ý5?j
ÅÙ9[pic]h!Kìhñ`ý5?EHúÿU[pic]-j
ÅÙ9[pic]h!Kìhñ`ý5?U[pic]V[pic]jh!Kìhñ`ý5?U[pic]h!Kìhñ`ý5?6?h!Kìhñ`ý5?jh
G•hñ`ýEHöÿU[pic]jü>M[pic]hñ`ýCJU[pic]V[pic]aJjhñ`ýU[pic]hñ`ý
hñ`ýCJaJ
h…I®hñ`ýjThe Natural Base e
Transformations of Exponential Functions
Exponential Function Properties
Compound Interest
[pic]
where
P = Population at time t
[pic] = Population at time t = 0
d = doubling time
where
y = Population at time t
c = Population at time 0
k= Relative growth rate
where
A = Amount at time t
[pic]= Amount at time t = 0
h = half-life
where
y = Amount at time t
c = Amount at time 0
k= Relative growth rate
Radioactive Studies
|Time in |Grams of |
|Hours |Material |
|x |y |
|0.1 |2 |
|1 |1.2 |
|2 |0.6 |
|3 |0.3 |
|4 |0.2 |
|5 |0.1 |
|6 |0.06 |
[pic]
Note: Inverse functions interchange x and y (domain & range) and are reflections across the line y = x
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related searches
- company name and stock symbol
- why your name is important
- native american name generator
- why is my name important
- why is god s name important
- last name that means hope
- name for significant other
- name synonym list
- me and name or name and i
- name and i vs name and me
- name and i or name and myself
- name and i or name and me