5. Curve Sketching

by M. Bourne

NOTES:

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The kinds of things we will be searching for in this section are:

x-intercepts Use `y = 0`
NOTE: In many cases, finding x-intercepts is not so easy. If so, delete this step.
y-intercepts Use `x = 0`
local maxima Use `(dy)/(dx)=0`, sign of first derivative changes `+ → −`
local minima Use `(dy)/(dx)=0`, sign of first derivative changes ` − → +`
points of inflection Use `(d^2 y)/(dx^2)=0`, and sign of `(d^2 y)/(dx^2)` changes

Finding Maxima and Minima

Local maximum

A local maximum occurs when `y^' = 0` and `y^'` changes sign from positive to negative (as we go left to right).

Changing slope of a curve

Local minimum

A local minimum occurs when `y^' = 0 and y^'` changes sign from negative to positive.

Changing slope of a curve

The Second Derivative

The second derivative can tell us the shape of a curve at any point.

Concave up

If `(d^2y)/(dx^2) > 0`, the curve will have a minimum-type shape (called concave up)

d2y/dx2 - second derivative

Example 1

The curve y = x2 + 3x − 2 has `(dy)/(dx)=2x+3`.

Now `(d^2y)/(dx^2)=2` and of course, this is `> 0` for all values of x.

So it has a concave up shape for all x.

Concave down

If `(d^2y)/(dx^2) < 0`, the curve will have a maximum-type shape (called concave down)

 

Second derivative


Example 2

The curve y = x3 − 2x + 5 has `(dy)/(dx)=3x^2-2`. The second derivative is `(d^2y)/(dx^2)=6x` and this is `< 0` for all values of `x < 0`.

So the curve has a concave down shape for all `x < 0` (and it is concave up if `x > 0`).

Finding Points of Inflection

A point of inflection is a point where the shape of the curve changes from a maximum-type shape `(d^2y)/(dx^2) < 0` to a minimum-type shape `(d^2y)/(dx^2) > 0`.

Clearly, the point of inflection will occur when

`(d^2y)/(dx^2) = 0` and when there is a change in sign

(from plus ` →` minus or minus ` →` plus) of `(d^2y)/(dx^2)`.

Caoncave up and down of a curve

Example 3

Sketch the following curve by finding intercepts, maxima and minima and points of inflection:

`y=x^3-9x`

General Shapes

If we learn the general shapes of these curves, sketching becomes much easier. Of course, the following are "ideal" shapes, and there are many other possibilities. But at least this helps get us started.

Quadratic Cubic
Highest power of x: 2 Highest power of x: 3
Typical quadratic shape (parabola) Typical cubic shape
1 minimum, no maximum
[if it has a positive x2 term]
1 minimum, 1 maximum
no points of inflection 1 point of inflection
Quartic Pentic
Highest power of x: 4 Highest power of x: 5
Typical quartic shape Typical quintic shape
2 minimums, 1 maximum
[if it has a positive x4 term]
2 minimums, 2 maximums
2 points of inflection 3 points of inflection

Example 4

Sketch the curve and show intercepts, maxima and minima and points of inflection:

`y=x^4-6x^2`

Example 5

Sketch the curve and show intercepts, maxima and minima and points of inflection:

`y=x^5-5x^4`

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