Download 1.1The Derivative and theTangent Line

The Derivative and the
Tangent 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 closer
and closer to desired
View spreadsheet
point of tangency
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
f ( x0  x)  f ( x0 )
m  lim
x 0
x
••
x
Definition of
a Tangent
• Let Δx shrink
from the left
f ( x0  x)  f ( x0 )
m  lim
x 0
x
Definition of
a Tangent
• Let Δx shrink
from the right
f ( x0  x)  f ( x0 )
m  lim
x 0
x
The Slope Is a Limit
• Consider f(x) = x3
x 0= 2
Find the tangent at
f (2  x)  f (2)
m  lim
x 0
x
(2  x)3  23
m  lim
x 0
x
2
3
8  12x  6(x)  (x)  8
m  lim
x 0
x
• Now finish …
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
Definition of Derivative
• The derivative is the formula which
gives the slope of the tangent line at
any point x for f(x)
f ( x0  x)  f ( x0 )
f '( x)  lim
x 0
x
• Note: the limit must exist
– no hole
– no jump
– no pole
– no sharp corner
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 the
f `(x)
slope of the original
function look like?
• Consider …
– what do x-intercepts show? To actually find f(x), we
– what do max and mins show?need a specific point it
contains
– f `(x) <0 or f `(x) > 0 means what?
Derivative Notation
• For the function y = f(x)
• Derivative may be expressed as …
f '( x) "f prime of x"
dy
"the derivative of y with respect to x"
dx
Assignment
• Lesson 3.1
• Page 123
• Exercises: 1 – 41, 63 – 65 odd