UNIT #1: Transformation ofFunctions;ExponentialandLog Goals
Transcript of UNIT #1: Transformation ofFunctions;ExponentialandLog Goals
UNIT #1: Transformation of Functions; Exponential and Log
Goals:
• Review core function families and mathematical transformations.
Textbook reading for Unit #1: Read Sections 1.1–1.4
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Example: The graphs of ex, ln(x), x2 and x12 are shown below. Identify
each function’s graph.
x
y1
Unit 1 – Transformation of Functions; Exponential and Log 3
Comment on the properties of the graphs of
• inverse functions -
• exponentials -
• logarithms -
• powers of x -
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Knowing the graphs and properties of essential families of functions is crucial foreffective mathematical modeling.Name other families of functions.
Unit 1 – Transformation of Functions; Exponential and Log 5
Give examples of members of each family, and state some of their common
properties.
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The core families of functions can be made even more versatile by being trans-
formed.Example: Sketch the graph of y = x2, over the interval x ∈ [−4, 4].
On the same axes, sketch the graph of y = 4− 12(x + 1)2.
Unit 1 – Transformation of Functions; Exponential and Log 7
Review the four common types of function transformations.Type Form Example
Unit 1 – Transformation of Functions; Exponential and Log 9
Example: Consider the data shown below, showing the concentration of a
chemical produced in a reaction vessel, over time.
0 20 40 60 80 100
05
1015
Time (hours)
Con
cent
ratio
n (p
pm)
What family of functions would best describe this graph? Point out specific
features of the graph that make the choice a reasonable one.
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Give a general mathematical form for the func-
tion, based on the shape of the graph.
e.g. C(t) = ...
0 20 40 60 80 100
05
1015
Time (hours)
Con
cent
ratio
n (p
pm)
Unit 1 – Transformation of Functions; Exponential and Log 11
Determine as many of the numerical values in the
formula C(t) = ... as you can, given the graph.
Sketching related graphs along the way might be
helpful.
0 20 40 60 80 100
05
1015
Time (hours)
Con
cent
ratio
n (p
pm)
12
Looking closely at the graph, you see that after
30 hours, the concentration has reached almost
exactly 12 ppm. Determine the value for the fi-
nal missing parameter in your concentration func-
tion.0 20 40 60 80 100
05
1015
Time (hours)
Con
cent
ratio
n (p
pm)
Unit 1 – Transformation of Functions; Exponential and Log 13
Logarithm Review
Most students are quite comfortable with exponential functions, but many findlogarithms less familiar. To address this we will do a more comprehensive reviewof the logarithmic function and its use in transforming equations.
Log/Exponential Equivalency
ac = x means loga x = c
Simplify loga(a7).
Simplify aloga(25).
Unit 1 – Transformation of Functions; Exponential and Log 15
These problems suggest the following equations, which also follow from the factthat ax and loga(x) are inverse functions.
loga(ax) = x and aloga x = x
Rules for Computing with Logarithms
1. loga(AB) = logaA + logaB
2. loga(A/B) = logaA− logaB
3. loga(AP ) = P logaA
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Changing logarithmic bases
The functions ax and loga are not provided on calculators unless a = 10 or a = e(see next section of these notes). For other values of a, ax and loga can be expressedin terms of 10x and log10. To calculate loga x, we use the following formula:
Conversion of Log Bases
loga x =log x
log a
Prove the above formula, using the Rules for Computing Logarithms and the
fact that loga x = c means x = ac.
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Graphs of Logarithmic FunctionsSince the logarithm in base 10 is commonly used in science, we define log x (nosubscript) to mean log10 x, for brevity.
The graph of log x may be obtained from the graph of 10x by reversing the axes(that is, by reflecting the graph in the line y = x). (If drawing the graph of inversefunctions is unfamiliar, please read Section 1.3 in the text.)
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10
10
10x
log10(x)
What is the domain of log x? What is the range of log x?
Sketch the logarithm function for the bases e and 2.
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Classic Applications of Exponentials and Logarithms
Example: Based on H-H, Section 1.4 #48: A cup of coffee contains 100
mg of caffeine, which leaves the body at a continuous rate of 17% per hour.
Sketch the graph of caffeine level over time, after drinking one cup of coffee.
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There are two natural interpretations of the question statement which lead to
two different formulae for A(t). Write down both formulae.
Compare the predicted caffeine level after 10 hours, using each model. Based
on those values, how similar are these two models in practice?
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The key phrase continuous rate has a special meaning in mathematics and science,and it associated with the natural exponential form ert. It is typically associatedwith processes like chemical reactions, population growth, and continuously com-pounded interest.Common alternative statements about percentage growth or decay, where the rateis assumed to be measured at the end of one time period (hour, day year), areusually of the form (1± r)t.
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Write out an appropriate mathematical model for the following scenarios:
• Infant mortality is being reduced at a rate of 10% per year.
• My $10,000 investment is growing at 5% per year.
• A savings account offers daily compound interest, at a 4% annual rate.
• Bacteria are reproducing at a continuous rate of 125% per hour.
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We now return to our earlier modeling problem.Example: A cup of coffee contains 100 mg of caffeine, which leaves the
body at a continuous rate of 17% per hour. Write the formula for A(t).
What is the caffeine level at t = 4 hours?
At what time does the caffeine level reach A = 10 mg?
Find the half-life of caffeine in the body.