# 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.

### Need Graph Paper?

- 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 ad ⇩

**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).

**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.

slope

slope

**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
*=* x*^{2}* +* 3*x *− 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 *=* x*^{3}
− 2*x *+ 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

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:

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 *x*^{2} 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 *x*^{4} 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:

Graph of `y=x^4-6x^2`.

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 orx= 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 when *x* = 0, `y'' = 0`, which is neither positive nor negative, so we can't conclude whether (0, 0) is is a local maximum or a local minimum using the second derivative test. However, since the first derivative changes from negative to positive as we take values on the left of 0 and then on the right of 0, we can conclude (0, 0) is a local MAX.

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 an interesting case where the second derivative is momentarily `0`, but is positive either side of `0`. It is a local maximum, even thoough the second derivative is not positive, which is usually the case.

So we are ready to sketch the curve:

Graph of `y=x^5-5x^4`.

The following points are indicated with dots:

`x`-intercepts (0, 0), and `(5,0)` (green dots)

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

Local maximumm (0, 0) (green dot)

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

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