5. Curve Sketching using Differentiation

by M. Bourne

NOTES:

  • There are now many tools for sketching functions (Mathcad, Scientific Notebook, graphics calculators, etc). It is important in this section to learn the basic shapes of each curve that you meet. An understanding of the nature of each function is important for your future learning. Most mathematical modelling starts with a sketch.

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  • You need to be able to sketch the curve, showing important features. Avoid drawing x-y boxes and just joining the dots.
  • We will be using calculus to help find important points on the curve.

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

Continues below

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).

A local maximum curve changing from positive to negative slope

Local minimum

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

A local minimum curve changing from negative to positive slope

The Second Derivative

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

Positive Second Derivative: Concave up

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

d2y/dx2 - second derivative positive curve

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.

Negative Second Derivative: Concave down

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

Second derivative negative curve (local maximum)

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)`.

A curve changes from concave up to concave down across a point of inflection

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.

Maximums and minimums are shown with a dot, while points of inflection have a "plus" symbol.

Quadratic

Highest power of x: 2

Typical quadratic shape (parabola). Curve is concave up for all `x`.

1 minimum, no maximum
[if it has a positive x2 term]

No points of inflection

Cubic

Highest power of x: 3

Typical cubic shape showing concave down to concave up.

1 minimum, 1 maximum

1 point of inflection

Quartic

Highest power of x: 4

Typical quartic shape showing local maxima and minima.

2 minimums, 1 maximum
[if it has a positive x4 term]

2 points of inflection

Pentic

Highest power of x: 5

Typical pentic shape showing local maxima and minima.

2 minimums, 2 maximums

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`