# 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 |

**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
*=* 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*.

### 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`

## 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 x^{2}
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 x^{4}
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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