MAT1A01: Derivatives and differentiablefunctions

Dr Craig

16 March 2016

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.

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.

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

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 .

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).

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).

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

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

The Derivative as a Function

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.

Example:

f (x) = x3 − x

Find f ′(x) and sketch both functions.

Example:

Find f ′(x) if

f (x) =1− x

2 + x

Other notations for the derivative:

Let y = f (x). Then

f ′(x) = y′ =dy

dx=

df

dx=

d

dxf (x)

= Df (x) = Dxf (x)

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?

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.)

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!

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!)

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?

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.

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

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

There are three ways that a function can fail

to be differentiable:

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.

Higher derivatives

d

dx

(dy

dx

)=

d2y

dx2

f ′′ is called the second derivative of f .