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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.
  • 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).

Positive slope Negative slope
`y' > 0`
`y' = 0`
`y' < 0`
Local maximum

Curve showing portion with positive slope in pink, and negative slope in green.

Local minimum

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

Positive
slope
Negative
slope
`y' > 0`
`y' = 0`
`y' < 0`
Local minimum

Curve showing positive slope in pink, and negative slope in green.

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)

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)

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)`.

+ Point of inflexion Concave down Concave up
`y'' < 0`
`y'' = 0`
`y'' > 0`

Curve showing concave up in green and concave down in pink.

Example 3

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

`y=x^3-9x`

Answer

The basic shape of a cubic when the coefficient (number before) `x^3` is:

Keeping this in mind helps with the sketching process.

1. x-intercepts:

`y=x^3-9x`

`=x(x^2-9)`

`=x(x+3)(x-3)`

Now `y=0` when

`x = 0`, `x = -3` and `x = 3`

2. y-intercepts:

When `x = 0`, `y = 0`.

3. maxima and minima?

`dy/dx=3x^2-9`

`=3(x^2-3)`

`=3(x+sqrt(3))(x-sqrt(3))`

`=0`

when

`x=-sqrt(3)` or `x=sqrt(3)`

So we have max or min at approximately `(-1.7,10.4)` and `(1.7,-10.4)`.

[We could check which is which by trying some points near `-1.7` and `+1.7` to determine what the sign changes are. But we need to find the second derivative anyway for points of inflection, so we use that to determine max or min.]

4. Second derivative:

`(d^2y)/(dx^2)=6x`

If `x = -1.7`, `y” < 0`, so MAX at `(-1.7,10.4)`

If `x = +1.7`, `y” > 0`, so MIN at `(1.7,-10.4)`

5. Point of inflection:

`(d^2y)/(dx^2)=6x`

Now

`(d^2y)/(dx^2)=0` when `x = 0`

and

`(d^2y)/(dx^2)`

changes sign from negative (concave down) to positive (concave up) as x passes through `0`.

So we are ready to sketch the curve:

1 2 3 4 -1 -2 -3 -4 20 40 -20 -40 x y
(-1.7,10.4)
(0,0)
(1.7,-10.4)

Graph of `y=x^3-9x`.

The following points are indicated with dots:

Local maximum (-1.7,10.4)

Point of inflection (0,0)

Local minimum (1.7,-10.4)

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`

Answer

1. x-intercepts

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

`=x^2(x^2-6)`

`=x^2(x+sqrt6)(x-sqrt6)`

`=0`

when

`x = 0`, `x=-sqrt(6)` and `x=sqrt(6)`

2. y-intercepts:

When `x = 0`, `y = 0`.

3. maxima and minima?

`(dy)/(dx)=4x^3-12x `

`=4x(x^2-3) `

`=4x(x+sqrt3)(x-sqrt3)`

`=0`

Now `(dy)/(dx)=0` when `x = 0` or `x=-sqrt(3)` and `x=sqrt(3)`

So we have max or min at `(0,0)` and `(-sqrt(3),-9)` and `(sqrt(3),-9)`.

4. Second derivative:

`(d^2y)/(dx^2)=12x^2-12`

Now `y” > 0` for `x = -sqrt3` so `(-sqrt3, -9)` is a local MIN

Now `y” < 0` for `x = 0` so `(0, 0)` is a local MAX

Now `y” > 0` for `x = sqrt3` so `(sqrt3, -9)` is a local MIN

5. Points of inflection:

We now use the second derivative to find points of inflection:

`(d^2y)/(dx^2)=12x^2-12`

`=12(x+1)(x-1)`

`=0`

when `x = -1` or `x = 1`

If `x < -1`, `y” > 0`, and for `-1 < x < 1`, we have `y” < 0`.

The sign of `y”` has changed, so `(-1, -5)` is a point of inflection.

If `x > 1`, `y” > 0`,

The sign of `y”` has changed, so `(1, -5)` is a point of inflection.

So we are ready to sketch the curve:

The following points are indicated with dots:

`x`-intercepts `(-sqrt(6),0)` and `(sqrt(6),0)` (green dots)

Local maximum, `x`-intercept and `y`-intercept `(0,0)` (green dot)

Points of inflection `(-1,-5)` and `(1,-5)` ("plus" signs)

Local minima `(-sqrt(3),-9)` and `(sqrt(3),-9)` (magenta dots)

Example 5

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

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

Answer

NOTE: This question is fairly sophisticated and is here to show you some of the complications that can occur. If you don't fully understand it now, don't worry!

1. x-intercepts

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

`=x^4(x-5)`

`=0`

when `x = 0`, `x = 5`

 

2. y-intercepts:

When `x = 0`, `y = 0`.

 

3. maxima and minima?

`(dy)/(dx)=5x^4-20x^3`

`=5x^3(x-4)`

`=0`

when `x = 0` or `x = 4`

So we have max or min at `(0,0)` and `(4,-256)`.

 

4. Second derivative:

`(d^2y)/(dx^2)=20x^3-60x^2`

Now `y” = 0` for `x = 0` so `(0,0)` is ???

Now `y” > 0` for `x = 4` so `(4,-256)` is a local MIN

 

5. Points of inflection

We now use the second derivative to find points of inflection:

`(d^2y)/(dx^2)=20x^3-60x^2`

`=20x^2(x-3)`

`=0`

when `x = 0` or `x = 3`

If `x < 0`, `y” < 0`, so we have a concave down shape:

If `0 < x < 3`, `y” < 0`, which also gives us a concave down shape:

There is no sign change, so at `x = 0`, there is NO point of inflection.

If `x > 3`, `y” > 0`, sp we have a concave up shape:

So the sign of `y”` has changed, so `(3,-162)` is a point of inflection.

Actually, at `x = 0`, we have a FLAT SPOT. It is not a local maximum, even though it may appear so.

So we are ready to sketch the curve:

The following points are indicated with dots:

`x`-intercepts `(0,0)` (the "flat spot"), and `(5,0)` (green dots)

Point of inflection `(3,-162)` ("plus" sign)

Local minimum `(4,-256)` (magenta dot)

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