mat1a01: derivatives and differentiable functions · mat1a01: derivatives and di erentiable...
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MAT1A01: Derivatives and differentiablefunctions
Dr Craig
16 March 2016
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Announcements:
I Assignment 3 will not be marked. Please
collect your scripts and complete the
questions that you have not yet
attempted. A memo will be posted on
Monday next week.I Saturday class this week: D-Les 101 from
11h00 – 13h30. The focus will be on
Continuity and Infinite Limits.I The Sick Test will take place on Tuesday
22 March, 15h30 – 17h00. The venue will
be announced on Blackboard.
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Announcements:
I As stated in the Learning Guide, everyone
who scored < 40% for Semester Test 1
will be required to attend compulsorySaturday morning classes.
I The compulsory classes will start from
Saturday 09 April.
I This Saturday’s class is voluntary but is
highly recommended for all students
wanting to improve their understanding of
the material covered so far.
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Tangents
Consider a curve y = f (x) and a point
P (a, f (a)). If we want to find the tangent
line to the curve at the point P , we consider
a nearby point Q(x, f (x)), where x 6= a, and
compute the slope of the line PQ:
mPQ =f (x)− f (a)
x− a
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Consider a curve y = f (x) and a point
P (a, f (a)). If we want to find the tangent
line to the curve at the point P , we consider
a nearby point Q(x, f (x)), where x 6= a, and
compute the slope of the line PQ:
mPQ =f (x)− f (a)
x− aWe then let Q approach P by letting x
approach a. If mPQ approaches a value m,
then we say that m is the slope of the
tangent to the curve at P .
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Definition: The tangent line to the
curve y = f (x) at the point P (a, f (a)) is
the line through P with slope
m = limx→a
f (x)− f (a)
x− a
provided that this limits exists.
Example: Find the equation of the tangent
line to y = x2 at the point P (1, 1).
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We can also write
m = limh→0
f (a + h)− f (a)
h
Example: find an equation of the tangent line
to the hyperbola y = 3/x at the point (3, 1).
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Definition: The derivative of afunction f at a number a, denoted by
f ′(a) is
f ′(a) = limh→0
f (a + h)− f (a)
h
if this limit exists
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If we write x = a + h , then we have
h = x− a and so h approaches 0 if and only
if x approaches a. Thus we can write:
f ′(a) = limx→a
f (x)− f (a)
x− a
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The Derivative as a Function
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The following formula should be familiar:
f ′(x) = limh→0
f (x + h)− f (x)
hAt school you would have used this to
calculate the derivative (from first principles)
of a polynomial or a combination of power
functions.
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Example:
f (x) = x3 − x
Find f ′(x) and sketch both functions.
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Example:
Find f ′(x) if
f (x) =1− x
2 + x
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Other notations for the derivative:
Let y = f (x). Then
f ′(x) = y′ =dy
dx=
df
dx=
d
dxf (x)
= Df (x) = Dxf (x)
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Definition: A function f is
differentiable at a if f ′(a) exists. It is
differentiable on an open interval(a, b) [ or (a,∞) or (−∞, a) or
(−∞,∞)] if it is differentiable at every
number in the interval.
Example: where is the function f (x) = |x|differentiable?
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We like functions whose behaviour obeys
certain rules. We have already seen that
most functions that we work with regularly
are “well-behaved” in the sense that they are
continuous.
(Remember: functions continuous
everywhere on their domains included
rational functions, trig functions, inverse trig
functions, log functions, etc.)
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Being differentiable is also a nice property
for a function to have. If a function is
differentiable at a point then we know
whether the function values are increasing or
decreasing at that point.
Given a function f (x), is there a connection
between differentiability of a function at a
point a and continuity at a point a?
Yes!
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Theorem: If a function f is differentiable
at a point a, then f is continuous at a.
How do we prove this?
We need to prove that
limx→a
f (x) = f (a)
(and all of the conditions that are implied by
the above equality!)
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Some logic:
We have just shown that the following
implication is true:
f differentiable at a → f continuous at a
Is the converse true?
That is, does the implication
f continuous at a → f differentiable at a
hold for any function f?
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Some logic:
Consider the statement
“If Andrew is a human, then Andrew is a
mammal”
We can right this in the form H →M .
The converse of this statement would be
“If Andrew is a mammal, then Andrew is a
human”
Clearly H →M is true, but M → H is
false.
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Some logic:
We have just shown that the following
implication is true:
f differentiable at a → f continuous at a
Is the converse true?
NO!
However, the contrapositive is true:
f discontinuous at a→ f not differentiable at a
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Some logic:
We have just shown that the following
implication is true:
f differentiable at a → f continuous at a
Is the converse true?
NO!
However, the contrapositive is true:
f discontinuous at a→ f not differentiable at a
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There are three ways that a function can fail
to be differentiable:
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Vertical tangent example
Calculate
limx→0
3√x
You will find that
limx→0−
3√x = lim
x→0+
3√x =∞
That is, limx→0
3√x does not exist and hence
the function f (x) = 3√x is not differentiable
at x = 0.
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Higher derivatives
d
dx
(dy
dx
)=
d2y
dx2
f ′′ is called the second derivative of f .