Transcript UNIT A - Mr. Santowski

UNIT A
PreCalculus Review
Unit Objectives
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1. Review characteristics of fundamental functions
(R)
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2. Review/Extend application of function models
(R/E)
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3. Introduce new function concepts pertinent to
Calculus (N)
A8 - Logarithmic Functions
Calculus - Santowski
Lesson Objectives
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1. Simplify and solve logarithmic expressions
2. Sketch and graph logarithmic fcns to find graphic
features
3. Explore logarithmic functions in the context of
calculus related ideas (limits, continuity, in/decreases
and its concavity)
4. Logarithmic models in biology (populations),
business (profit, cost, revenue)
Fast Five
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1. Solve ln(x + 2) = 3
2. Sketch f(x) = -ln(x + 2)
3. State the domain of f(x) = (ln(x - 1))0.5
4. Evaluate log232 + log31/81+ log0.2516
5. Sketch the inverse of f(x) = 3 - log2x
6. Find the domain of log3(9 - x2)
7. Evaluate log3324 - log34
4
a
log 2
bc
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8. Expand using LoL
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9. Evaluate log79 + log35 using GDC
10. Solve 3x = 11
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Explore
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Using your calculator for confirmation, and
remembering that logarithms are exponents,
explain why it is predictable that:
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(a) log 64 is three times log 4;
(b) log 12 is the sum of log 3 and log 4;
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(c) log 0.02 and log 50 differ only in sign.
(A) Logarithmic Fcns & Algebra
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(1) Find the inverse of f(t) = 67.38(1.026)t
(2) Solve e3-2x = 4
(3) Solve log6x + log6(x - 5) = 2
(3) Express 1 ln x  4ln y  lnx 2 1 as a single log
2
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(4) Solve log2x + log4x + log8x = 11
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(5) Simplify
 x 
ln

 x  1 
(B) Logarithmic Fcns &Graphs
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Be able to identify asymptotes, intercepts, end
behaviour, domain, range for y = logax
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Ex. Given the function y = log2(x - 1) - 2, determine the
following:
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- domain and range
- asymptotes
- intercepts
- end behaviour
- sketch and then state intervals of increase/decrease as well
as concavities
(B) Logarithmic Fcns & Graphs
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Ex 3. Given the functions f(x) = logx and
g(x) = x1/3,
(a) when is f(x) > g(x)
 (b) when is 100f(x) < g(x)
 (c) Which function increases faster,
100f(x) or g(x)?
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(C) Logarithmic Fcns &
Calculus Concepts
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Now we will apply the concepts of limits,
continuities, rates of change, intervals of
increase/decreasing & concavity to exponential
function
f (x)  lnx 1
2
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Ex 1. Graph
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From the graph, determine: domain, range, max
and/or min, where f(x) is increasing,
decreasing, concave up/down, asymptotes
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(C) Logarithmic Fcns &
Calculus Concepts
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Ex 2. Evaluate the following limits numerically or
algebraically. Interpret the meaning of the limit value.
Then verify your limits and interpretations graphically.
lim lnx  4 
x 
lim lnx  4 
x 4
lim lnsin x 
x 
lim log10 x 2  x 
x 1
(C) Logarithmic Fcns &
Calculus Concepts
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ln x
Ex 3. Given the function : f (x) 
x
(i) find the intervals of increase/decrease of f(x)
(ii) is the rate of change at x = 2 equal to/more/less than
the rate of change
 equal to/greater/less than the rate at
x = 1?
(iii) find intervals of x in which the rate of change of the
function is increasing. Explain why you are sure of your
answer.
(iv) where is the rate of change of f(x) equal to 0?
Explain how you know that?
(C) Logarithmic Fcns &
Calculus Concepts
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Ex 4. Given the function ,
f(x) between:
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ln x
f (x) 
x
find the average rate of change of
(a) 1 and 1.5
(b) 1.4 and 1.5
(c) 1.499 and
 1.5
(d) predict the rate of change of the fcn at x = 1.5
(e) evaluate limx1.5 f(x).
(f) Explain what is happening in the function at x = 1.5
(g) evaluate f(1.5)
(h) is the function continuous at x = 1.5?
(i) is the function continuous at x = 0?
(D) Applications of Logarithmic
Functions
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The population of Kenya was 19.5 million
in 1984 and was 32.0 million in 2004.
Assuming the population increases
exponentially, find a formula for the
population of Kenya as a function of time.
Then using logs, find the doubling time of
Kenya’s population
(E) Internet Links
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Logarithm Rules Lesson from Purple Math
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College Algebra Tutorial on Logarithmic
Properties from West Texas AM
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You can try some on-line word problems from U
of Sask EMR problems and worked solutions
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More work sheets from EdHelper's Applications
of Logarithms: Worksheets and Word Problems
(F) Homework
Text pages 113-117
 (1) Evaluate logs, Q19,20,23
 (1) properties of logs, Q31,33,35
 (2) solving eqns, Q39,41,51,53,60
 (3) apps, Q71,77,81
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