Chapter 3 Introduction to the Derivative Sections 3'5, 3'6, 3'7 and 3'8

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Chapter 3Introduction to the DerivativeSections
3.5, 3.6, 3.7 and 3.8
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Introduction to the Derivative

• Average Rate of Change
• The Derivative
• Derivatives of Powers, Sums and Constant
  asMultiples
• Marginal Analysis

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Average Rate of Change
The change of f (x) over the interval a,b is
The average rate of change of f (x) over the
interval a , b is
Difference Quotient
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Average Rate of Change
Is equal to the slope of the secant line through
the points (a , f (a)) and (b , f (b)) on the
graph of f (x)
secant line has slope mS
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Average Rate of Change as h?0
The average rate of change of f over the interval
a , b can be written in two different ways
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Average Rate of Change as h?0
We now look at the behavior of the average rate
of change of f (x) as b? a ? h gets smaller and
smaller, that is, we will let h tend to 0 (h?0)
and look for a geometric interpretation of the
result. For this, we consider an example of
values of and their corresponding secant lines.
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Average Rate of Change as h?0
Secant line tends to become the Tangent line
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Average Rate of Change as h?0
Secant line tends to become the Tangent line
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Average Rate of Change as h?0
Secant line tends to become the Tangent line
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Average Rate of Change as h?0
Secant line tends to become the Tangent line
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Average Rate of Change as h?0
Secant line tends to become the Tangent line
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Average Rate of Change as h?0
Secant line tends to become the Tangent line
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Average Rate of Change as h?0
Secant line tends to become the Tangent line
Zoom in
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Average Rate of Change as h?0
Secant line tends to become the Tangent line
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Average Rate of Change as h?0
Secant line tends to become the Tangent line
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Average Rate of Change as h?0

• We observe that as h approaches zero,
• The secant line through P and Q approaches the
  tangent line to the point P on the graph of f.
• and consequently, the slope mS of the secant line
  approaches the slope mT of the tangent line to
  the point P on the graph of f.

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Instantaneous Rate of Change of f at x a
That is, the limiting value, as h gets
increasingly small, of the difference
quotient is the slope mT of the tangent line
to the graph of the function f (x) at x a.
(slope of secant line)
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Instantaneous Rate of Change of f at x a
This is usually written as
That is, mS approaches mT as h tends to 0.
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Instantaneous Rate of Change of f at x a
We summarize the previous idea in symbols by
writing
and read
as the limit as h approaches 0 of
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Instantaneous Rate of Change of f at x a
that is, the limit symbol indicates that the
value of can be made arbitrarily close to mT by
taking h to be a sufficiently small number.
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Instantaneous Rate of Change of f at x a
Definition The instantaneous rate of change of f
(x) at x a is defined to be the slope of the
tangent line to the graph of the function f (x)
at x a. That is,
Remark The slope mT gives a precise indication
of how fast the graph of f (x) is increasing or
decreasing at x a.
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A Concrete Example
Consider the function f (x) ? (x2 9)/4, whose
graph is given below, at the points a ? 3 and a
? 1
At which of these two points is the function
increasing faster?
Intuition says at x ? 3 because we notice the
graph is steeper at the point x ? 3. Why?
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A Concrete Example
Consider the function f (x) ? (x2 9)/4, whose
graph is given below, at the points a ? 3 and a
? 1
Because our brain makes no distinction between
the graph of f and the tangent line to the graph
at the point in question.
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A Concrete Example
Consider the function f (x) ? (x2 9)/4, whose
graph is given below, at the points a ? 3 and a
? 1
We see how the tangent line basically coincides
with the graph of f near the point of contact.
What is the slope of each line?
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A Concrete Example
Consider the function f (x) ? (x2 9)/4, whose
graph is given below, at the points a ? 3 and a
? 1
That is, at the points of contact, the function
is increasing as fast as its corresponding
tangent line.
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A Concrete Example
Consider the function f (x) ? (x2 9)/4, whose
graph is given below, at the points a ? 3 and a
? 1
We now verify the claims by explicitly computing
for this function, the limit,
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A Concrete Example
For the function f (x) ? (x2 9)/4 at any
point a we have
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A Concrete Example
Thus, for the function f (x) ? (x2 9)/4 at
any point a we have
Therefore, as h tends to 0, mS approaches a/2
? mT .
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A Concrete Example
For the function f (x) ? (x2 9)/4 at any
point a we have
Thus,
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A Concrete Example
Let us try some other values of a for the same
function.
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Remark
The example clearly indicates that the slope mT
of the tangent line is itself a function of the
point a we choose.
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The Derivative
The instantaneous rate of change mT at a is also
called the derivative of f at x a and it is
denoted by f '(a). That is,
f '(a) is read f prime of a
mT f '(a) slope of the tangent line to f
'(x) at a instantaneous
rate of change of f at a
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The Derivative
The instantaneous rate of change mT at a is also
called the derivative of f at x a and it is
denoted by f '(a). That is,
In our previous example, the derivative of the
function f (x) ? (x2 9)/4 at any point a is
given by
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Rates of Change
Average rate of change of f over the interval a,
ah is
Slope of the secant line through the points (a ,
f (a)) and (a , f (ah))
Instantaneous rate of change of f at xa is
Slope of the tangent line at the point (a , f (a))
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Tangent Line and Secant Line
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Equation of Tangent Line at x a
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Terminology
Finding the derivative of the function f is
called differentiating f.
If f '(a) exists, then we say that f is
differentiable at x a.
For some functions f , f '(a) may not exist. In
this case we say that the function f is not
differentiable at x a.
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Quick Approximation of the Derivative
Recall that f '(a) is the limiting value of the
expression as we make h increasingly smaller.
Therefore, we can approximate the numerical value
of the derivative using small values of h. h
0.001 often works. In this case,
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Quick Approximation - Example
Demand The demand for an old brand of TV is
given by where p is the price per TV set, in
dollars, and q is the number of TV sets that can
be sold at price p . Find q (190) and estimate q
'(190) . Interpret your answers.
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Solution
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Geometric Interpretation
At a ? 190, q(p) decreases as fast as the tangent
line does, that is, at the rate mT ? q '(190) ?
?2.5 TVs/ What does this means?
It means that at the price of p ? 190, the
demand will decrease by 2.5 TV sets per dollar we
increase the price.
If we now set the price at 195, how many TV sets
do we expect to sell?
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Geometric Interpretation
At a ? 190, the equation of the tangent line is
y ? ?2.5( p-190 )500 Thus, when we set the
price at p ? 195, the line shows that we expect
to sell y ? 487.5 TVs
The actual number we expect to sell according to
the demand function is q (195) ? 487.8 ? 488
TVs which is very close to the prediction given
by the tangent line.
p ? 195
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Now Recall The Example
The slope mT ? f '(a) of the tangent line depends
on the point a we choose on the graph of f .
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The Derivative as a Function
If f is a function, its derivative function is
the function whose value at x is the derivative
of f at x. That is,
Notice that all we have done is substituted x for
a in the definition of f '(a). This way, when we
are done with the algebra, the answer will be
given in terms of x.
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The Derivative as a Function
Example Given the function
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Geometric Verification
At any point on the graph of f , the tangent line
agrees with the graph of which already is a
straight line of slope 1
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The Derivative as a Function
Example Given the function
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Geometric Verification
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The Derivative as a Function
Example Given the function
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The Power Rule
Example
Example
Example
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Verification when f (x) ? x1/2
Example Given the function
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Leibniz d Notation For Derivatives
Average rate of change
Instantaneous rate of change
means the same as
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Differential Notation Differentiation
means the derivative with respect to x.
The derivative of y ? f (x) with respect to x is
written
or as
Example
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Differential Notation Differentiation
Example
To find f '(?2), the derivative of y ? f (x) at x
? ?2, we need to find the function f '(x).
In Leibniz notation f '(?2) is denoted by the
symbol
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Differentiation Rules
Example
Example
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Differentiation Rules
Example
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Differentiation Rules
Example
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More Examples
Example Find the derivative of
First write f (x) as constant times a power
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Example Find the derivative of
First write f (x) as
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Functions Not Differentiable at a Point
Vertical tangent
Cusp
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Example The Absolute Value of x.
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Example The cube root of x.
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Introduction to the Derivative

• Average Rate of Change
• The Derivative
• Derivatives of Powers, Sums and Constant
  asMultiples
• Marginal Analysis

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Cost Functions

• A cost function specifies the total cost C as a
  function of the number of items x produced. Thus,
  C(x) is the cost of x items. The cost functions
  is made up of two parts
• C(x) variable costs fixed costs
• The marginal cost function is the derivative
  C'(x) of C(x). It measures the instantaneous
  rate of change of cost with respect to x.

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By definition the Marginal Cost Function is
C'(x). The units of marginal cost are units of
cost (say, ) per item.
Interpretation The marginal cost function C'(x)
approximates the change in actual cost of
producing an additional unit when x items are
already being produced.
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Interpretation of Marginal Cost
The Marginal Cost Function is C'(x). Thus, for
small h we have the quick approximation given by
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Interpretation of Marginal Cost
That is, for h1, we have
Notice that C(x1) - C(x) is the actual (exact)
cost of producing one additional item when the
production level is x. In practice, h1 is
considered to be small in relation to production
level x, which in general is a large number.
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Average Cost Functions
Given a cost function C(x) , the average cost
function given by
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Example
The total monthly cost function C, in dollars, of
producing x computer games is given by

• Find the marginal cost function and use it to
  estimate the cost of producing the 1,001st game.
• Find the actual cost of producing the 1,001st
  game.

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Solution
The marginal cost function is given by
An estimate for the cost of producing the 1,001st
game is
That is, it costs an additional 18 to produce
one more game when the production level is at
1,000 games a month
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Solution
The the actual cost of producing the 1,001st game
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Example Continued
The total monthly cost function C, in dollars, of
producing x computer games is given by

• Find the average cost per game if 1,000 games are
  manufactured.
• Find the average cost function.

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Solution
The average cost function
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More Marginal Functions
The Marginal Revenue Function measures the rate
of change of the revenue function. It
approximates the revenue from the sale of an
additional unit.
The Marginal Profit Function measures the rate of
change of the profit function. It approximates
the profit from the sale of an additional unit.
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More Marginal Functions
Given a revenue function, R(q), The Marginal
Revenue Function is
Given a profit function, P(q), The Marginal
Profit Function is
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Marginal Functions - Examples
The monthly demand for T-shirts is given by
where p denotes the wholesale unit price in
dollars and q denotes the quantity demanded. The
monthly cost for these T-shirts is 8 per shirt.

• Find the revenue and profit functions.
• Find the marginal revenue and marginal profit
  functions.

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Solution

• Find the revenue and profit functions.

Revenue qp
Profit revenue cost
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Solution

• Find the marginal revenue and marginal profit
  functions.

Marginal revenue
Marginal profit