Transcript UNIT A - Mr. Santowski

UNIT A
PreCalculus Review
7/7/2015
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Unit Objectives
• 1. Review characteristics of fundamental functions
(R)
• 2. Review/Extend application of function models
(R/E)
• 3. Introduce new function concepts pertinent to
Calculus (N)
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A7 - Exponential Functions
Calculus - Santowski
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Lesson Objectives
• 1. Simplify and solve exponential expressions
• 2. Sketch and graph exponential fcns to find
graphic features
• 3. Explore exponential functions in the context of
calculus related ideas (limits, continuity,
in/decreases and its concavity)
• 4. Exponential models in biology (populations),
business (profit, cost, revenue)
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Fast Five
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1 Solve 2-x+2 = 0.125
2. Sketch a graph of y = (0.5)x + 3
3. Solve 4x2 - 4x - 15 = 0
4. Evaluate limx∞ (3-x)
5. Solve 3x+2 - 3x = 216
6. Solve log4(1/256) = x
7. Evaluate limx3 ln(x - 3)
8. Solve 22x + 2x - 6 = 0
9. State the exact solution for 2x-1 = 5 (2 possible
answers)
• 10. Is f(x) = -e-x an increasing or decreasing function?
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Explore
• Given 100.301 = 2 and 100.477 = 3, solve
without a calculator:
• (a) 10x = 6;
• (b) 10x = 8;
• (c) 10x = 2/3;
• (d) 10x = 1
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Explore
• A function is defined as follows:
 e x  a
x  2

f (x)   x  2 2  x  3
b  e x 3
x3

• (i) Evaluate limx-2 if a = 1
• (ii) Evaluate limx3 if b = 1
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• (iii) find values for a and b such f(x) is continuous at
both x = -2 and x = 3
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(A) Exponentials & Algebra
• (1) Factor e2x - ex
• (2) Factor and solve xex - 2x = 0 algebraically.
Give exact and approximate solutions (CF)
• (3) Factor 22x - x2 (DOS)
• (4) Express 32x - 5 in the form of a3bx (EL)
• (5) Solve 3e2x - 7ex + 4 = 0 algebraically. Give
exact and approximate solutions (F)
• (6) Solve 4x + 5(2x) - 12 = 0 algebraically.
Give exact and approximate solutions (F)
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(B) Exponentials & Their
Graphs
• Be able to identify asymptotes, intercepts,
end behaviour, domain, range for y = ax
• Ex. Given the function y = 2 + 3-x, determine
the following:
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- domain and range
- asymptotes
- intercepts
- end behaviour
- sketch and then state intervals of
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(B) Exponentials & Their
Graphs
• Be able to identify asymptotes, intercepts, end
behaviour, domain, range for y = ax
• Ex 1. Given the function y = 2 + 5(1 - ex+1), 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
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(B) Exponentials & Their
Graphs
• Ex 2. Given the graphs of f(x) = x5 and g(x) =
5x, plot the graphs and determine when f(x) >
g(x). Which function rises faster?
• Ex 3. Given the points (1,6) and (3,24):
• (i) determine the exponential fcn y = Cax that
passes through these points
• (ii) determine the linear fcn y = mx + b that passes
through these points
• (iii) determine the quadratic fcn y = ax2 + bx + c
that passes through these points
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(C) Exponentials & Calculus
Concepts
• Now we will apply the concepts of limits,
continuities, rates of change, intervals of
increase/decreasing & concavity to
exponential function
x 2
• Ex 1. Graph f (x)  e
• 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) Exponentials & Calculus
Concepts
• Ex 2. Evaluate the following limits numerically or
algebraically. Interpret the meaning of the limit value.
Then verify your limits and interpretations graphically.
lim2  ex 
x 
lim e
x
tan x

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2
3h 1
lim
h 0
h
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(C) Exponentials & Calculus
Concepts
• Ex 3. Given the function f(x) = x2e-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?
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(C) Exponentials & Calculus
Concepts
• Ex 4. Given the function f(x) = x2e-x, find the average rate of
change of f(x) between:
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(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 x2e-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?
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(D) Applications of
Exponential Functions
• The population of a small town appears to be increasing
exponentially. In 1980, the population was 35,000 and in 1990,
the population was 57,000.
• (a) Determine an algebraic model for the town’s population
• (b) Predict the population in 1995. Given the fact that the town
population was actually 74,024, is our model accurate?
• (c) When will the population be 100,000?
• (d) Find the average growth rate between 1985 and 1992
• (e) Find the growth rate on New Years day, 1992
• (f) Find on what day the growth rate was 6%
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(E) Internet Links
• Exponential functions from WTAMU
• Exponential functions from AnalyzeMath
• Solving Exponential Equations from
PurpleMath
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(F) Homework
• From our textbook, p99-103
• (1) for work with graphs, Q3-11
• (2) for work with solving eqns,
Q15,1619,10,21,22
• (3) for applications, Q35,40 (see pg95-6)
• (4) for calculus related work, see HO
(scanned copy on website)
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