Transcript The Derivative and the Tangent Line Problem

The Derivative and the Tangent
Line Problem
Section 2.1
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After this lesson, you should be able to:
find the slope of the tangent line to a
curve at a point
use the limit definition of a derivative to
find the derivative of a function
understand the relationship between
differentiability and continuity
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Tangent Line
Remember our tangent line problem from section 1.1?
Just in case you’ve forgotten, here are the cliff notes:
•Finding the slope of a curve at x = c is equivalent to finding
the slope of the tangent line to the curve at x = c.
•To approximate the slope of the tangent line at c we can use
the secant line through (c, f(c)) and another point on the curve
really close to c.
•If we keep moving that second point closer and closer to the
point of tangency, c, we get this very slick definition of the
slope of the tangent line (See next slide).
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Definition of Tangent Line with Slope m
If f is defined on an open interval containing c, and if the limit
lim
x  0
f (c   x )  f (c )
x
m
exists, then the line passing through  c , f ( c )  with slope m is the
tangent line to the graph of f at  c , f ( c ) 
We can also say the m is the slope of the graph of f at x = c.
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Let's see how this applies to some basic functions.
Example involving a linear function:
Find the slope of the graph of f(x) = 5x - 7 at the point (-1, -12).
So we need to find the slope of the graph of f at c = -1.
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For non-linear functions the slope is not constant.
Example involving a quadratic function.
Find the slope of the tangent line to the graph of f(x) = x2 + 6 at any
point  c , f ( c )  on the graph.
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Quadratic Example continued
From this, find the slope of f at:
a) (0, 6)
b) (-3, 15)
Now pay attention…here comes the important part 
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Definition of the derivative of f
The derivative of f at x is given by
f  ( x )  lim
x  0
f ( x  x)  f ( x)
x
for all x for which the limit exists.
There are many notations for indicating the derivative of f at x:
"f prime of x"
f ( x )
dy
dx
y
d
dx
 f ( x)
"the derivative of y with respect to x"
"y prime"
"the derivative of f with respect to x"
___________________________ is the process of finding the derivative of a
function.
A function is ______________________________ at x if its derivative exists at 8x.
Using the definition of the derivative
1) Given f(x) = x3 - 12x,
a) find f '(x) using the definition of the derivative.
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Using the definition of the derivative
(continued)
b) What is the slope of f at (-2, 16)?
Check this out on the calculator.
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Using the definition of the derivative
2) Given
f ( x) 
x 1,
a) find f '(x) using the definition of the derivative.
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Using the definition of the derivative
(continued)
b) What can you say about the slope of the graph of f at (1, 0)?
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Using the definition of the derivative
3) Find the equation of the tangent line to the graph of
f ( x) 
1
x 1
at x = -2.
a) First find the _________ of the graph of f at any point x in the domain.
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Using the definition of the derivative
(continued)
3 b) Next, find the slope of f when x = _____ .
3 c) Write the equation of the line with m = _______ that contains the point
(-2, ____).
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Using your calculator
3 d) Using your calculator, graph f and its tangent line at x = -2.
1 Graph the function
f ( x) 
on your calculator.
1
x 1
3
4
2
Select 5: Tangent(
Type the x value, which in this
case is -2, and then hit 
Now, hit  DRAW
Here’s the equation of the tangent
line…notice the slope…it’s approximately
what we found
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Differentiability Implies Continuity
If f is differentiable at x, then f is continuous at x.
The converse is not true; i.e. continuity does not imply differentiability!
Some things which destroy differentiability:
1. A discontinuity (a hole or break or asymptote)
e.g. For
f ( x) 
1
x3
,
f '(3)
does not exist because f is not continuous at x = 3.
2. A sharp corner (continuous but not differentiable)
e.g. For
f ( x)  x ,
f '(0 )
does not exist because there is a sharp corner at x = 0.
3. A vertical tangent line (continuous but not differentiable)
e.g. For
f ( x) 
3
x,
f '(0)
does not exist because there is a vertical tangent at x = 0.
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Homework
Section 2.1 page 103 #1, 5-27 odd, 33, 37-40 all, 81-85 odd, 93
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