Transcript UNIT A - Mr. Santowski

UNIT A
PreCalculus Review
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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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A9 - Trigonometric Functions
Calculus - Santowski
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Lesson Objectives
• 1. Simplify and solve trigonometric expressions
• 2. Sketch and graph trigonometric fcns to find
graphic features
• 3. Explore trigonometric functions in the context
of calculus related ideas (limits, continuity,
in/decreases and its concavity)
• 4. Trigonometric models in periodic applications
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Fast Five
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1. Calculate the degree measure of an angle of 7 radians
2. Sketch f(x) = 2cos(2x)
3. State the domain of f(x) = tan(x- /3)
4. Evaluate sin(60°) + cos(225°) - tan(-150°)
5. P(5,-12) determines an angle in standard position. Find
the values of the primary trig ratios
6. Find the exact value of sin(15°)
7. If sin(x) = 12/13, evaluate cos(2x)
8. Find the value of sin(/4 - /3)
9. Find the exact value of sin(50°)cos(20°) cos(50°)sin(20°)
10. Solve cos(x - /3) = 0.5 for x€R
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Explore
• Answer T or F on the following. Use algebraic,
graphic or numeric arguments to justify your choice
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(1) T or F: sin(/2-x) = cosx
(2) T or F: sin( + x) = -sinx
(3) T or F: 2sin(x) = sin(2x)
(4) T or F: sin(x + y) = sin(x) + sin(y)
(5) T or F: cos(x + 45) = cos(x) + cos(45)
• (6) if limxa(sinxcosx) = M, then
limx-a(sinxcosx) = M as well
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(A) Trigonometric Fcns &
Algebra
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(1) Solve 2cos2x + cosx - 1 = 0 on [0,2]
(2) Solve sec2(x) + 5tan(x) = -2
(3) Simplify 0.5(sin(x+y) + sin(x-y))
(4) Solve sin(2x) + 1 = 0 on [0,2]
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(B) Trigonometric Fcns
&Graphs
• Be able to identify amplitude, period, phase shift,
asymptotes, intercepts, end behaviour, domain,
range for trig functions.
• Ex. Given the function y = cos(2x - /2), determine
the following:
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- domain and range
- amplitude and period
- intercepts
- end behaviour
- sketch and then state intervals of increase/decrease as
well as concavities
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(B) Trigonometric Fcns &
Graphs
• Ex. Graph the function f(x) = sin-1(x)
• (a) Estimate/determine the instantaneous rate at
which f(x) is changing at x = n/8 using the TI-89
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• (b) Now graph g(x) 
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1
x
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• (c) Determine the values of g(x) at x = n/8
• (d) How are your answers in Qc related to your
answerin Qa?
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(C) Trigonometric Fcns &
Calculus Concepts
• Now we will apply the concepts of limits,
continuities, rates of change, intervals of
increase/decreasing & concavity to
exponential function
• Ex 1. Graph f(x) = x + sin(x)
• From the graph, determine: domain, range,
max and/or min, where f(x) is increasing,
decreasing, concave up/down, periodicity
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(C) Trigonometric Fcns &
Calculus Concepts
• Ex 2. Evaluate the following limits numerically.
Interpret the meaning of the limit value. Then verify
your limits and interpretations graphically.
lim tan(x)
x 
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sin(x)
lim
x 0
2x
sin(  h)  sin( )
lim
h 0
h
sin x
lim
x 0 x  sin x
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(C) Trigonometric Fcns &
Calculus Concepts
• Ex 3. Given the function RoC(t) = -2sin(2x). Let the
function RoC(t) represent the RATE OF CHANGE of an
original, unknown (for now) function.
• (1) Explain what RoC(/12) = -1 means about the original
function.
• (2) When is the original function increasing? How do you
know?
• (3) When is the original function neither increasing nor
decreasing?
• (4) When is the original function concave up?
• (5) Sketch the unknown function.
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(C) Trigonometric Fcns &
Calculus Concepts
ex
• Ex 4. Given the function ,f (x) 
find the average rate of
sin(x)
change of f(x) between:
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(a) 0 and 0.5
(b) 0 and 0.01
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(c) 0 and -0.01
(d) Is f(x) continuous at x = 0?
(e) evaluate limx0 f(x).
(f) Explain what is happening in the function at x = 0.
(g) Where else is the function discontinuous?
(h) Graph the function and confirm your previous results.
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(D) Applications of
Trigonometric Functions
• Delta Cephei is one of the most visible stars in the
night sky. Its brightness has periods of 5.4 days and
the average brightness is 4.0 with variations of +
0.35.
• (a) Find a formula that models the brightness of Delta
Cephei as a function of time, with t = 0 at its peak
brightness.
• (b) You view the star on the 3rd day. What is rate of
change of the brightness of the star at that time?
Explain how you determined your answer.
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(E) Internet Links
• On line worksheet from U of
Saskatchewan
• Trigonometry from The Mathpage
• Trig Review from WebTrig
• Trig Functions Review from Analyze
Math
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(F) Homework
• Work Sheet (From Stewart text)
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