Curve Sketching
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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.

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Curve Sketching

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Curve sketching

Curve Sketching


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.


Curve sketching

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


Curve sketching

  • 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.


Curve sketching

  • 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


Curve sketching

  • 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


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