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Section 3.6 A Summary of Curve Sketching
AP Calculus BC
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Warm-up:
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Warm-up:
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Warm-up -- Solution:
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Learning Objective: 1.) Analyze and sketch the graph of a function.
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Tools/concepts for curve sketching
All the different components of a graph that would help us determine its behavior are listed below. Some of these will be names of specific points on a graph, while some will be graph qualities or characteristics. x- and y-intercepts vertical & horizontal asymptotes relative and absolute max/min domain & range increasing & decreasing intervals continuity points of inflection differentiability concavity symmetry (even/odd functions)
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Steps for Curve Sketching:
Use Algebra β Find Asymptotes (use this to determine the domain/range) Find x-intercepts Find y-intercepts Use Calculus Find π β² (π₯) and critical value(s) Find where π π₯ is increasing/decreasing Find π β²β² (π₯) and point(s) of inflection(s) Find where π π₯ is concave up/concave down
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Example 1: Analyze and sketch the graph of: π π₯ = 2( π₯ 2 β9) π₯ 2 β4
Example on p. 207
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Example 2: Analyze and sketch the graph of: π π₯ = π₯ π₯ 2 +2
Example on p. 207
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Example 3: Analyze and sketch the graph of: π π₯ =2 π₯ 5 3 β5 π₯ 4 3
Example on p. 207
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Example 4: Analyze and sketch the graph of: π π₯ = π₯ 4 β12 π₯ π₯ 2 β64π₯ **Note: A polynomial function of even degree must have at least one relative extrema. Example on p. 207
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Example 5: Analyze and sketch the graph of: π π₯ = cosβ‘(π₯) 1+sinβ‘(π₯) .
Example on p. 207
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Homework: pg : #1-4 all, 5-23 odd, 37, 38, 49, 51
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