MAT01B1: Curve Sketching
Dr Craig
22 August 2018
My details:
I Consulting hours (this week):
Wednesday 14h40 – 16h15
Thursday 11h20 – 12h55
Friday 11h20 – 12h55 & 14h00 – 16h00
I Office C-Ring 508
https://andrewcraigmaths.wordpress.com/
(Or, just google ‘Andrew Craig maths’.)
Semester Test 1
I Saturday 25 August
I D1 Lab 308
I Starts at 09h00. Be seated by 08h45.
I Scope: Ch 7.1–7.5, 7.8, 4.1, 4.2, 4.3
I Also examinable: Proofs of Fermat’s
Theorem, Rolle’s Theorem, Mean Value
Theorem
Tips for test preparation
I Read the textbook, do the tut problems
I Do the past paper (memo will go on BB)
I Do the Sat. worksheets (memos on BB)
I Ask questions at the tutor centre
I Consult with lecturers
I Study your theorems
I Pace yourself (50 marks in 90min)
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:
Given the curve f (x) = x4 − 4x3, find the
following:
I intervals of concavity
I points of inflection
I local maxima and minima
Use the above information to sketch the
curve.
Another example from 4.3:
Sketch the graph of
f (x) = x2/3(6− x)1/3.
We have
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:
Next time:
Example:
Use the guidelines to sketch the curve of
f (x) =x2√x + 1
.
Example:
Use the guidelines to sketch the curve of
f (x) = x(ex).
