lesson3.1 the derivative and the tangent line
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The Derivative and theTangent Line Problem
Lesson 3.1
Definition of Tan-gent
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)
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
Animated Tangent
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
Definition ofa Tangent
0 0
0
( ) ( )limx
f x x f xm
x
• Let Δx shrinkfrom the left
Definition ofa Tangent
• Let Δx shrinkfrom the right
0 0
0
( ) ( )limx
f x x f xm
x
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
Animated Secant Line
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
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
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 !
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)
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
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)
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
Assignment
• Lesson 3.1
• Page 123
• Exercises: 1 – 41, 63 – 65 odd