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The Derivative and theTangent Line Problem
Lesson 3.1
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Definition of Tan-gent
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Tangent Definition
• From geometry– a line in the plane of a circle– intersects in exactly one point
• We wish to enlarge on the idea to include tangency to any function, f(x)
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Slope of Line Tangent to a Curve
• Approximated by secants– two points of
intersection
• Let second point get closerand closer to desiredpoint of tangency
•• •
View spreadsheet simulation
View spreadsheet simulation
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Animated Tangent
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Slope of Line Tangent to a Curve
• Recall the concept of a limit from previous chapter
• Use the limit in this context ••
0 0
0
( ) ( )limx
f x x f xm
x
x
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Definition ofa Tangent
0 0
0
( ) ( )limx
f x x f xm
x
• Let Δx shrinkfrom the left
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Definition ofa Tangent
• Let Δx shrinkfrom the right
0 0
0
( ) ( )limx
f x x f xm
x
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The Slope Is a Limit
• Consider f(x) = x3 Find the tangent at x0= 2
• Now finish …
0
3 3
0
2 3
0
(2 ) (2)lim
(2 ) 2lim
8 12 6( ) ( ) 8lim
x
x
x
f x fm
x
xm
x
x x xm
x
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Animated Secant Line
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Calculator Capabilities
• Able to draw tangent line
Steps• Specify function on Y= screen• F5-math, A-tangent• Specify an x (where to
place tangent line)
•Note results
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Difference Function
• Creating a difference function on your calculator– store the desired function in f(x)
x^3 -> f(x)– Then specify the difference function
(f(x + dx) – f(x))/dx -> difq(x,dx)– Call the function
difq(2, .001)•Use some small value for dx
•Result is close to actual slope
•Use some small value for dx
•Result is close to actual slope
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Definition of Derivative
• The derivative is the formula which gives the slope of the tangent line at any point x for f(x)
• Note: the limit must exist– no hole– no jump– no pole– no sharp corner
0 0
0
( ) ( )'( ) lim
x
f x x f xf x
x
A derivative is a limit !A derivative is a limit !
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Finding the Derivative
• We will (for now) manipulate the difference quotient algebraically
• View end result of the limit• Note possible use of calculator
limit ((f(x + dx) – f(x)) /dx, dx, 0)
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Related Line – the Normal
• The line perpendicular to the function at a point– called the “normal”
• Find the slope of the function
• Normal will have slope of negative reciprocal to tangent
• Use y = m(x – h) + k
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Using the Derivative
• Consider that you are given the graph of the derivative …
• What might theslope of the original function look like?
• Consider …– what do x-intercepts show?– what do max and mins show?– f `(x) <0 or f `(x) > 0 means what?
To actually find f(x), we need a specific
point it contains
To actually find f(x), we need a specific
point it contains
f `(x)
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Derivative Notation
• For the function y = f(x)
• Derivative may be expressed as …
'( ) "f prime of x"
"the derivative of y with respect to x"
f x
dy
dx
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Assignment
• Lesson 3.1
• Page 123
• Exercises: 1 – 41, 63 – 65 odd