MAT01B1: Curve Sketching - WordPress.com · De nition: a point P on a curve y = f(x) is called an...
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MAT01B1: Curve Sketching
Dr Craig
28 August 2017
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My details:
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’.)
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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.
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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.
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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.
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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.
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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 .
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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 .
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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.
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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.
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Example: sketch the graph of
f (x) = x2/3(6− x)1/3.
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Next we will sketch:
f (x) = e1/x
Before we attempt to sketch this curve, let
us look at the guidelines for curve sketching.
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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!
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More about symmetry
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More about symmetry
Periodic functions: f (x + p) = f (x) for
some p > 0 and all x ∈ D.
A periodic function has translational
symmetry.
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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.
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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!
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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
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Sketching guidelines applied to e1/x
(E) Intervals of increase/decrease
(F) Local maxima and minima
(G) Concavity & Inflection points
(H) Sketch!
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Sketch of f (x) = e1/x
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Example:
Use the guidelines to sketch the curve of
y =2x2
x2 − 1
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Sketch:
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Example:
Use the guidelines to sketch the curve of
f (x) =x2√x + 1
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Sketch: