Sec 4 3 Curve Sketching 1 Curve Sketching
- Slides: 16
Sec 4. 3 Curve Sketching 1
Curve Sketching Problems Given: A function y = f(x). Objective: To sketch its graph. 2
Steps (1) Find a “Frame” for the graph Ø Domain Ø Asymptotes – Horizontal, Vertical, Slant (2) Find out how the graph “wiggles” Ø Derivative – intervals of increase/decrease; max/min Ø Second derivative – intervals for concave up/down; point(s) of inflection (3) Sketch 3
Example (1) Sketch Next Question: How does the graph wiggle between the two ends ? Frame: Domain: Asymptotes: Starts here Ends here 4
Wiggle: Derivative: – + – 2 nd derivative: – + Final Step: Put the wiggly graph onto the Frame. – + 5
Decreasing; Concave down Starts here Decreasing; Increasing; Concave up Concave down Decreasing; Concave up Concave down Increasing; Concave up Local max A “twist” : Concavity changes – a point of inflection Graph rebounds after a dip – a local min A “twist” : Concavity changes – a point of inflection Ends here A “twist” : Concavity changes – a point of inflection 6
Example (2) Next Question: How does the graph wiggle within each of the three sections ? Sketch Frame: ? Domain: Asymptotes: ? ? Starts here ? ? ? Ends here ? 7
Wiggle: Derivative: 2 nd derivative: 8
Example (3) Next Question: How does the graph wiggle within each of the three sections ? Sketch Frame: ? ? ? Domain: ? ? Asymptotes: ? Starts here ? ? ? Ends here 9
Wiggle: Derivative: 2 nd derivative: 10
Example (4) Sketch Frame: Domain: Asymptotes: Next Question: How does the graph wiggle between the two ends ? ? ? Ends here ? Starts here 11
Wiggle: Derivative: 2 nd derivative: 12
Example (5) Sketch ? Frame: Domain: Asymptotes: Next Question: How does the graph wiggle within the two regions ? ? ? Ends here ? Starts here ? ? 13
Wiggle: Derivative: 2 nd derivative: 14
Next Question: How does the graph wiggle in one of the regions ? Repeat here Example (6) Sketch Frame: ? Domain: ? Asymptotes: Periodicity: ? Repeat here ? 15
Wiggle: Derivative: 2 nd derivative: 16
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