MAT01B1: Curve Sketching

Dr Craig

28 August 2017

My details:

I acraig@uj.ac.za

I Consulting hours:

Thursday 09h40 – 11h15

No Friday consulting this week

I Office C-Ring 508

https://andrewcraigmaths.wordpress.com/

(Or, just google ‘Andrew Craig maths’.)

Conference in Slovakia

I will be in Slovakia at a conference and then

a research visit from Friday 1 September to

Thursday 14 September.

My lectures and tuts will be taken by Mr

Mafunda and Dr Robinson. You can contact

me via email during this time.

Sick Test

I Tuesday 5 September

I 15h30 – 17h00

I Venue will be announced on Blackoard

and in lectures

I Same topics and format as Semester Test

I Forms and supporting documents to be

handed in by 12pm on Monday 4 Sep.

Increasing/Decreasing Test

(a) If f ′(x) > 0 on an interval, then f is

increasing on that interval.

(b) If f ′(x) < 0 on an interval, then f is

decreasing on that interval.

First Derivative Test

Suppose that c is a critical number of a

function f that is continuous at c.

(a) If f ′(x) changes from positive to

negative at c, then f (x) has a local

maximum at c.

(b) If f ′(x) changes from negative to

positive at c, then f (x) has a local

minimum at c.

(c) If f ′(x) does not change sign at c, then

f (x) has no local max. or min. at c.

Concavity Test

(a) If f ′′(x) > 0 for all x ∈ I , then the

graph of f (x) is concave upward on

the interval I .

(b) If f ′′(x) < 0 for all x ∈ I , then the

graph of f (x) is concave downward on

the interval I .

Definition: a point P on a curve y = f (x)

is called an inflection point if f is

continuous there and the curve changes

from concave upward to concave

downward or from concave downward to

concave upward at P .

Second Derivative Test

Suppose that f ′′ is continuous near c

(a) If f ′(c) = 0 and f ′′(c) > 0, then f has a

local minimum at c.

(b) If f ′(c) = 0 and f ′′(c) < 0, then f has a

local maximum at c.

From last time:

Sketch the graph of

f (x) = x2/3(6− x)1/3.

Use the fact that

f ′(x) =4− x

x1/3(6− x)2/3

and

f ′′(x) =−8

x4/3(6− x)5/3.

Example: sketch the graph of

f (x) = x2/3(6− x)1/3.

Next we will sketch:

f (x) = e1/x

Before we attempt to sketch this curve, let

us look at the guidelines for curve sketching.

Sketching guidelines

(A) Domain

(B) Intercepts

(C) Symmetry

(D) Asymptotes (Horizontal/Vertical/Slant)

(E) Intervals of increase/decrease

(F) Local maxima and minima

(G) Concavity & Inflection points

(H) Sketch!

More about symmetry

More about symmetry

Periodic functions: f (x + p) = f (x) for

some p > 0 and all x ∈ D.

A periodic function has translational

symmetry.

More about asymptotes

For horizontal asymptotes, calculate

limx→∞

f (x) and limx→−∞

f (x)

A function f (x) has a vertical asmyptote at

x = a if any of the following are true:

limx→a+

f (x) = ±∞ limx→a−

f (x) = ±∞

Slant asymptotes: next lecture.

Sketching guidelines

(A) Domain

(B) Intercepts

(C) Symmetry

(D) Asymptotes (Horizontal/Vertical/Slant)

(E) Intervals of increase/decrease

(F) Local maxima and minima

(G) Concavity & Inflection points

(H) Sketch!

Sketching guidelines applied to e1/x

(A) Domain x ∈ (−∞, 0) ∪ (0,∞)

(B) Intercepts no y-int, no x-int

(C) Symmetry no symmetry

(D) Asymptotes (Horizontal/Vertical/Slant)

limx→∞

e1/x = e0 = 1 = limx→−∞

e1/x

limx→0+

e1/x = limt→∞

et =∞

limx→0−

e1/x = limt→−∞

et = 0

Sketching guidelines applied to e1/x

(E) Intervals of increase/decrease

(F) Local maxima and minima

(G) Concavity & Inflection points

(H) Sketch!

Sketch of f (x) = e1/x

Example:

Use the guidelines to sketch the curve of

y =2x2

x2 − 1

Sketch:

Example:

Use the guidelines to sketch the curve of

f (x) =x2√x + 1

Sketch: