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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.

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