# Curve Sketching - PowerPoint PPT Presentation

1 / 6

Curve Sketching. The Second Derivative and Points of Inflection. Graphs that are Concave Up on the interval a < x < b where f”(x) < 0. This means that the graph lies above the tangent line at each point on the interval. Graphs that are Concave Down on the interval a < x < b where f”(x) < 0.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Curve Sketching

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

#### Presentation Transcript

Curve Sketching

The Second Derivative and Points of Inflection.

Graphs that are Concave Up on the interval a < x < b where f”(x) < 0.

This means that the graph lies above the tangent line at each point on the interval.

Graphs that are Concave Down on the interval a < x < b where f”(x) < 0.

This means that the graph lies below the tangent line at each point on the interval.

Points of Inflection

A point (c, f(c)) on the graph f(x) is said to be a point of inflection if the graph has

a tangent line at x = c and if f’(x) does not change sign at x = c and f”(x) does

change sign at x = c.

It is essential to note that f”(x) = 0 does not necessarily correspond to a point of inflection. You must also verify the signs!

c

f”>0

f”< 0

• Two examples to illustrate this:

• Consider f(x) = x4.

• then f’(x) = 4x3 and f”(x) = 12x2.

• It follows that f”(0) = 0, but it can be

• seen that (0,0) is not an inflection

• point.

• Consider f(x) = . Then f”(x) = .

• It follows that f”(x) does change sign,

• nevertheless, x = 0 does not correspond

• to an inflection point since f(0) is

• undefined.

• Each of the graphs below represent the second derivative f”(x) of a function f(x).

• On what interval is the graph of f(x) concave up/down?

• List the x-coordinates of all points of inflection.

• Make a rough sketch of a possible graph of f(x) assuming that f(0) = 1.

1

1

2

CU on x< 1

CD on x > 1

P.O.I. at x = 1

CU on x<0, 0<x<1

CD on x > 1

P.O.I. on x = 1

CU on x<0,x>2

CD on 0<x<2

P.O.I. on x=0,x=2

• Create a sketch based on the following information:

• From A to B, y’ = 0.

• From B to D, y’ >0.

• At C, y” = 0.

• Between C and D, there is the only y-intercept.

• At D y’ = 0 and y” <0

• From D to F, y’ <0.

• At E, y” = 0.

• At F, y ‘ = 0 and y” >0.

• At G, y’ = 0 and y” < 0

• From F to G, y’ > 0.

• From G to H, y’ < 0.

• H is the only x-intercept

• Start your graph at A and finish at H.

A

H