Sec 4 3 Curve Sketching 1 Curve Sketching

  • Slides: 16
Download presentation
Sec 4. 3 Curve Sketching 1

Sec 4. 3 Curve Sketching 1

Curve Sketching Problems Given: A function y = f(x). Objective: To sketch its graph.

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,

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

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

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

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

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

Wiggle: Derivative: 2 nd derivative: 8

Example (3) Next Question: How does the graph wiggle within each of the three

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

Wiggle: Derivative: 2 nd derivative: 10

Example (4) Sketch Frame: Domain: Asymptotes: Next Question: How does the graph wiggle between

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

Wiggle: Derivative: 2 nd derivative: 12

Example (5) Sketch ? Frame: Domain: Asymptotes: Next Question: How does the graph wiggle

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

Wiggle: Derivative: 2 nd derivative: 14

Next Question: How does the graph wiggle in one of the regions ? Repeat

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

Wiggle: Derivative: 2 nd derivative: 16