Chapter Contents

• Applications of Differentiation
• 1. Tangents and Normals
• 2. Newton's Method
• 3. Curvilinear Motion
• 4. Related Rates
• 5. Curve Sketching Using Differentiation
• 6. More Curve Sketching Using Differentiation
• 7. Applied Maximum and Minimum Problems
• 8. Radius of Curvature
• Applications of Differentiation Problem Solver

• Comments, Questions?

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5. Curve Sketching

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.

• 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

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).

Local minimum

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

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)

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)

 

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)`.

Example 3

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

`y=x^3-9x`

Answer

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
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
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`

Answer

Example 5

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

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

Answer