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Introduction to the Derivative

chapter3• Average Rate of Change

• The Derivative

• Derivatives of Powers, Sums and Constant Multiples• Marginal Analysis• Limits and Continuity

Lecture 6

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The Derivative Function The Derivative Function

0

( ) ( )( ) lim

h

f x h f xf x

h

If f is a function, its derivative function is the function whose value is the derivative of f at x.

Enough all ready!!!

Clap if you don’t want to ever calculate another Difference Quotient…Ok…let’s find a shortcut…

and ENJOY Calculus for a change!

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Shortcut: The Power RuleShortcut: The Power Rule

( ) nf x x ( n is any constant)

2( )g x x

1/ 33

1 ( )f x x

x Ex.

Ex.

If

Then1( ) nf x nx

1( ) 2g x x

4 / 31 ( )

3f x x

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Differential Notation: Differentiationd

dxmeans “the derivative with respect to x.

The derivative of y with respect to x is written

( )d dy

ydx dx

Ex. If y = x5 then 45 .dy

xdx

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Differentiation RulesDifferentiation Rules

Ex.

( ) ( ) a constantd d

cf x c f x cdx dx

8( ) 3f x x

Ex.

( ) ( ) d d d

f x g x f x g xdx dx dx

2 12( ) 7f x x x

7 7( ) 3 8 24f x x x

11( ) 14 12f x x x

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Differentiation RulesDifferentiation Rules

Ex.

a constantd

cx c cdx

( ) 7f x x

Ex. ( ) 14f x

( ) 7f x

( ) 0f x

0 a constantd

c cdx

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Functions Not Differentiable at a Point

Vertical tangent Cusp

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The Derivative of the Absolute Value of x.

xdx

dx x

Note the derivative doesn’t exist when x = 0.

Ex. 5 2 5 2xd

x xdx x

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Position versus VelocityPosition versus Velocity

A position function s(t) specifies the position (height) as a function of time t. The velocity function is the derivative of s(t). It measures the rate of change of position with respect to t.

The Velocity Function:

)(ts

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Cost FunctionsCost Functions

A cost function specifies the total cost C as a function of the number of items x. The marginal cost function is the derivative of C(x). It measures the rate of change of cost with respect to x.

The Marginal Cost Function:

( )C x

The marginal cost function approximates the change in actual cost of producing an additional unit.

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Cost FunctionsCost Functions

Given a cost function C, the average cost function is given by:

( )( )

C xC x

x

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Cost FunctionsCost Functions

Ex. The monthly cost function C, for x items is given by:

2( ) 0.001 2 200 C x x x

Find the marginal cost and the average cost functions.

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Cost FunctionsCost Functions

The average cost function:

The marginal cost function:

20.001 2 200x x

x

( )( )

C xC x

x

10.001 2 200x x

( ) 0.002 2C x x

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Marginal FunctionsMarginal Functions

The Marginal Profit Function measures the rate of change of the profit function. Approximates the profit from the sale of an additional unit.

The Marginal RevenueFunction measures the rate of change of the revenue function. Approximates the revenue from the sale of an additional unit.

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Marginal FunctionsMarginal FunctionsGiven a revenue function, R(q),

The Marginal Revenue Function is:

dR

dq

Given a profit function, P(q),

The Marginal Profit Function is:

dP

dq

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Marginal Revenue and ProfitMarginal Revenue and ProfitThe monthly demand for T-shirts is given by

0.05 25 0 400p q q

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.

1. Find the revenue and profit functions.

2. Find the marginal revenue and marginal profit functions.

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Marginal FunctionsMarginal Functions

20.05 25 0.05 25 q q q q

20.05 25 8q q q

1. Find the revenue and profit functions.Revenue = qp

Profit = revenue – cost

qq. 17050 2

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Marginal FunctionsMarginal Functions2. Find the marginal revenue and marginal profit functions.

Marginal revenue = ( )R q

0.1 25q

0.1 17q

( )P qMarginal profit =