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The concept of derivatives through the slope of tangent lines and the behavior of functions. It covers the limit definition of derivatives, finding derivatives using the definition, and interpreting derivatives as rates of change. Examples are provided for various functions.
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Math 103 โ Rimmer 3.1/3.2 The Derivative
lim x a
f x f a m โ x a
The limit of the slopes of the secant lines is the slope of the tangent line.
the slope of the tangent line
to f x at x = a.
secant line slope
Math 103 โ Rimmer 3.1/3.2 The Derivative
0
lim h
f a h f a m โ h
Another expression for the slope of the tangent line.
the slope of the tangent line
to f x at x = a.
secant line slope
Math 103 โ Rimmer 3.1/3.2 The Derivative
If you zoom in on the point of tangency, the function is
"locally linear" there.
http://www.stewartcalculus.com/tec/
Module 3.
Math 103 โ Rimmer List the following numbers from smallest to largest. 3.1/3.2 The Derivative
โ
g โฒ (^) ( 0 )< 0 < g โฒ( 4 )< g โฒ( 2 )< g โฒ( โ (^2) )
Math 103 โ Rimmer 3.1/3.2 The Derivative
The derivative is the
of with respect to when.
f a
y f x x x a
instantaneous rate of
change
2 1
y^ f^ x^ f^ x
x x x
โ โ
This is the of
with respect to over the interval ,.
y f x
x x x
average rate of change =
This is called a
difference quotient
( ) ( )
2 1
2 1
0 2 1
lim lim x x x
y^ f^ x^ f^ x
โ โ (^) x โ x x
โ โ
Math 103 โ Rimmer Interpreting the derivative as a rate of change. 3.1/3.2 The Derivative
The cost of producing x ounces of gold from a new gold mine is C (^) ( x )dollars.
What is the meaning of C โฒ ( x )? What are its units?
change in measures the ratio: change in
C C C x x x
โ โฒ = โ
( ) is the rate of change of production cost with respect to
the number of ounces produced, this is called.
C โฒ x
marginal cost
The units for C โฒ ( x )are dollars per ounce.
What does C โฒ (^) ( 800 ) =17 mean?
C โฒ ( 800 )is a ratio so let's turn 17 into a fraction.
( )
17 800 1
C โฒ =
When you are producing 800 ounces of gold and you
increase production by 1 to 801 ounces, cost will increase by $17.
Math 103 โ Rimmer 3.1/3.2 The Derivative
0
lim h
f x h f x f x โ h
Let the number a vary.
is called , if it is differentiable
for all numbers in ,.
f a b
a b
differentiable on
Math 103 โ Rimmer 3.1/3.2 The Derivative
Find the derivative of the function using the definition of the derivative.
2 f x = 4 x โ 7 x
2 f x + h = 4 x + h โ 7 x + h
0
lim h
f x h f x f x โ h
( )
2 2 = 4 x + 4 h โ 7 x + 2 xh + h
2 2 f x + h = 4 x + 4 h โ 7 x โ 14 xh โ 7 h
2 โ f x = โ 4 x + 7 x
2
0
lim h
f x h f x f x โ h
0
lim h
h x h
โ h
0
lim 4 14 7 h
x h โ
f โฒ^ ( x (^) ) = 4 โ 14 x
Math 103 โ Rimmer 3.1/3.2 The Derivative
Match the graph of each function in (a)-(d) with the graph of its derivative in I-IV.
The main connection: function: sign of the slope of the tangent line derivative: + above axis, below axis 0 "touches" axis
x x x
โ โ โ โ โ โ โ
(a): sign of the slope of the tangent line โ โ 0 โ + โ 0 โ โ
(a) โII
deriv.: below,then 0, then above, then 0, then below
(b): sign of the slope of the tangent line
(b) โIV
deriv.: above, then jump to below, then jump to above
(c): sign of the slope of the tangent line โ โ 0 โ +
(c) โI
deriv.: below,then 0, then above
(d): sign of the slope of the tangent line
(d) โIII
deriv.: above, then 0, then below, then 0, then above, then 0, then below
Math 103 โ Rimmer 3.1/3.2 The Derivative Animation of the graph of the derivative function
http://www.stewartcalculus.com/tec/ Module 3.