Section 2.1: The Derivative and the Tangent Line Problem

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Section 2.1 The Derivative and the Tangent Line
Problem
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Section 2.1 Classwork 1
TANGENT LINE
Slope -16
These can be considered average slopes or average
rates of change.
Slope -17.6
Slope -24
Secant Lines
Slope -32
Slope -48
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Secant Line

• A line that passes through two points on a curve.

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Tangent Line

• Most people believe that a tangent line only
  intersects a curve once. For instance, the first
  time most students see a tangent line is with a
  circle
• Although this is true for circles, it is not true
  for every curve

Every blue line intersects the pink curve only
once. Yet none are tangents.
The blue line intersects the pink curve twice.
Yet it is a tangent.
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Tangent Line
As two points of a secant line are brought
together, a tangent line is formed. The slope of
which is the instantaneous rate of change
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Tangent Line
As two points of a secant line are brought
together, a tangent line is formed. The slope of
which is the instantaneous rate of change
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Tangent Line
As two points of a secant line are brought
together, a tangent line is formed. The slope of
which is the instantaneous rate of change
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Tangent Line

• As two points of a secant line are brought
  together, a tangent line is formed. The slope of
  which is the instantaneous rate of change

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Tangent Line
As two points of a secant line are brought
together, a tangent line is formed. The slope of
which is the instantaneous rate of change
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Tangent Line
As two points of a secant line are brought
together, a tangent line is formed. The slope of
which is the instantaneous rate of change
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Slope of a Tangent Line

• In order to find a formula for the slope of a
  tangent line, first look at the slope of a secant
  line that contains (x1,y1) and (x2,y2)

(x2,y2)
?x
(x1,y1)
In order to find the slope of the tangent line,
the change in x needs to be as small as possible.
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Instantaneous Rate of Change

• Tangent Line with Slope m
• If f is defined on an open interval containing c,
  and if the limit
• exists, then the line passing through (c, f(c))
  with slope m is the tangent line to the graph of
  f at the point (c, f(c)).

f(x)
m
(c, f(c))
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Example 1

• Determine the best way to describe the slope of
  the tangent line at each point.

A.
Since the curve is decreasing, the slope will
also be decreasing. Thus, the slope is negative.
A
B.
The vertex is where the curve goes from
increasing to decreasing. Thus, the slope must
be zero.
C
B
C.
Since the curve is Increasing, the slope will
also be increasing. Thus the slope is Positive.
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Example 2
Find the instantaneous rate of change to
at (3,-6).
Substitute into the function
Simplify in order to cancel the denominator
c is the x-coordinate of the point on the curve
Direct substitution
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Example 3
Find the equation of the tangent line to
at (2,10).
Substitute into the function
Simplify in order to cancel the denominator
c is the x-coordinate of the point on the curve
Just the slope. Now use the point-slope formula
to find the equation
Direct substitution
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A Function to Describe Slope

• In the preceding notes, we considered the slope
  of a tangent line of a function f at a number c.
  Now, we change our point of view and let the
  number c vary by replacing it with x.

A constant.
A variable.
A function whose output is the slope of a tangent
line at any x.
The slope of a tangent line at the point x c.
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Example
Derive a formula for the slope of the tangent
line to the graph of .
Multiply by a common denominator
Simplify in order to cancel the denominator
Substitute into the function
A formula to find the slope of any tangent line
at x.
Direct substitution
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The Derivative of a Function

• The limit used to define the slope of a tangent
  line is also used to define one of the two
  fundamental operations of calculus
• The derivative of f at x is given by
• Provided the limit exists. For all x for which
  this limit exists, f is a function of x.

READ f prime of x.
Other Notations for a Derivative
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Example 1
Differentiate .
Simplify in order to cancel the denominator
Substitute into the function
Make the problem easier by factoring out common
constants
Direct substitution
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Example 2
Find the tangent line equation(s) for
such that the tangent line has a slope of
12.
Find when the derivative equals 12
Find the derivative first since the derivative
finds the slope for an x value
Find the output of the function for every input
Use the point-slope formula to find the equations
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How Do the Function and Derivative Function
compare?
f is not differentiable at x -½
Domain
Domain
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Differentiability Justification 1
In order to prove that a function is
differentiable at x c, you must show the
following In other words, the derivative from
the left side MUST EQUAL the derivative from the
right side. Common Example of a way for a
derivative to fail
Other common examples Corners or Cusps
Not differentiable at x -4
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Differentiability Justification 2
In order to prove that a function is
differentiable at x c, you must show the
following In other words, the function must
be continuous. Common Example of a way for a
derivative to fail
Other common examples Gaps, Jumps, Asymptotes
Not differentiable at x 0
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Differentiability Justification 3
In order to prove that a function is
differentiable at x c, you must show the
following In other words, the tangent line
can not be a vertical line. An Example of a
Vertical tangent where the derivative to Fails
to exist
Not differentiable at x 0
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Example

• Determine whether the following derivatives exist
  for the graph of the function.

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Example 2

• Sketch a graph of the function with the following
  characteristics
• The derivative does not exist at x -2.
• The function is continuous on (-6,3)
• The Range is (-7,-1

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Example 3

• Show that does not exist if
  .

First rewrite the absolute value function as a
piecewise function
Find the Left Hand Derivative
Find the Right Hand Derivative
Since the one-sided limits are not equal, the
derivative does not exist
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Function v Derivative
Compare and contrast the function and its
derivative.
Positive Slopes
x-intercept
-5
Vertex
Negative Slopes
Decreasing
Increasing
-5
FUNCTION
DERIVATIVE
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Function v Derivative
Compare and contrast the function and its
derivative.
Local Max
Decreasing
Positive Derivatives
Positive Derivatives
Increasing
Increasing
x-intercept
x-intercept
-5
5
5
-5
Negative Derivatives
Local Min
FUNCTION
DERIVATIVE
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Example 1

• Accurately graph the derivative of the function
  graphed below at left.

Make sure the x-value does not have a derivative
The Derivative does not exist at a corner.
The slope from -8 to -7 is -2
The slope from 2 to 8 is 1
The slope from -7 to 2 is 0
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Example 2

• Sketch a graph of the derivative of the function
  graphed below at left.

Negative
Negative
Increasing
Increasing
Increasing
Decreasing
Decreasing
Positive
Positive
Positive
1. Find the x values where the slope of the
tangent line is zero (max, mins, twists)
2. Determine whether the function is increasing
or decreasing on each interval
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Example 3

• Sketch a graph of the derivative of the function
  graphed below at left.

Positive
Negative
Increasing
Decreasing
Increasing
Positive
1. Find the x values where the slope of the
tangent line is zero (max, mins, twists)
2. Determine whether the function is increasing
or decreasing on each interval
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Example 4

• Sketch a graph of the derivative of the function
  graphed below.

(-3,0)
(3,0)
(0,0)
(-2.2,-4)
(2.2,-4)
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Example 5

• Sketch a graph of the function with the following
  characteristics
• The derivative is only positive for -6ltxlt-3 and
  5ltxlt10.
• The function is differentiable on (-6,10)

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Differentiability Implies Continuity

• If f is differentiable at x c, then f is
  continuous at x c.
• The contrapositive of this statement is true If
  f is NOT continuous at x c, then f is NOT
  differentiable at x c.
• The converse of this statement is not always true
  (be careful) If f is continuous at x c, then f
  is differentiable at x c.
• The inverse of this statement is not always true
  If f is NOT differentiable at x c, then f is
  NOT continuous at x c.

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Example

• Sketch a graph of the function with the following
  characteristics
• The derivative does NOT exist at x -2.
• The derivative equals 0 at x 1.
• The derivative does NOT exist at x 4.
• The function is continuous on -4,-2)U(-2,5