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Chapter 3 - Derivatives
3.1 Intro to the Derivatives
3.2 Rules for Derivatives
3.3 The Product and the Quotient Rules
3.4 Derivatives of Trigonometric Functions
3.5 Derivatives as Rates of Change
3.6 The Chain Rule
3.7 Related Rates
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Chapter 3.1 Intro to the Derivative
Recall: The Average Rate of Change of f(x) over [a,x]. The average
rate of change is a measure of the slope of the secant line
connecting the two points.
Definition: A tangent line intersects the curve at one point, say P,
such that the direction of the tangent line and the curve are the same
at point P.
Average rate of change
over [a,x]
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Definition: Instantaneous rate of change of f(x) at a point (a, f(a)) is
the slope of the tangent line at that point and is given by:
Instantaneous rate at
x = a.
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Example: Find the instantaneous rate of change of
at the point (1,5).
Note: The direction of the function at x = 1
is the same as the direction of the tangent line.
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Example:
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Find the equation of the tangent line at (1,5)
Equation of tangent line
at (a, f(a))
Note: The direction of the function at x = 1
is the same as the direction of the tangent line.
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Alternative Definition: Instantaneous rate of change at a point P.
Ex: Instantaneous rate of change of
at x=1 of
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Definition: Instantaneous rate of change at a point P.
Ex: Instantaneous rate of change of at x=0 of
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Definition: Instantaneous rate of change at a point P.
Ex: Instantaneous rate of change of at x= -1 of
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The Derivative as a function.
The slope of all tangent lines as x moves along the domain of f(x) is
itself a new function called the Derivative function.
Definition: The Derivative. The derivative of f(x) is the function
provided the limit exists and x is in the domain of f(x)!!
If the limit exists then we say f(x) is differentiable at x. If f(x) is
differentiable at every point of an open interval I, we say that f(x) is
differentiable on I.
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Example: Find the derivative (function which gives the slope of the
tangent line at ALL values of x in the domain of f) of the function
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Example: Verify that the derivative gives the slope of the tangent
line when x = 1, x = 0, and x = -1.
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Example: Find the derivative of the function
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Example: Find the derivative of the function
Example: Find the SLOPE of the tangent line
at x = 4, at x = 1, at x = 1/8. What is the slope
tending towards as x approaches 0+?
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Example: Find the derivative of the function
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Example: Find the derivative of the function
Example: Find the SLOPE of the tangent line
(derivative) at x = 1, x = 1/2, x = 1/4. What is the
slope tending towards as x approaches 0+?
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The Derivative as a function.
The slope of all tangent lines as x moves along the domain of f(x) is
itself a new function called the Derivative function.
Definition: The Derivative. The derivative of f(x) is the function
provided the limit exists and x is in the domain of f(x)!!
If the limit exists then we say f(x) is differentiable at x. If f(x) is
differentiable at every point of an open interval I, we say that f(x) is
differentiable on I.
Other notation:
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Example: Given the graph of f(x) below, graph the derivative f '(x).
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Example: Graph the derivative function for the function f(x) in the
graph below:
Questions:
1. Is f(x) continuous over [-5,5]?
2. Is f '(x) continuous over [-5,5]?
3. Where is f(x) NOT differentiable?
(i.e. where is the derivative undefined?)
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Derivative: does not exist at sharp corners where the slope on
either side are different. We say the derivative is undefined at that
point so the derivative function is not continuous at that point.
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Derivative: does not exist where it is undefined. It is undefined if it
is infinite.
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Example: A slingshot launches a stone vertically with an initial
velocity of 300 ft/s from an initial height of 6 ft. Its vertical motion is
given by
a) For which times is the velocity positive, negative, and zero?
b) What is the maximum height and when does it reach it?
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Example: Identify regions on the graph of f(x) where the
derivative, f '(x), is positive, negative, zero, or undefined.
NOTE: These are exactly the regions where f(x) is increasing,
decreasing, or flat.
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Example: Graph the derivative g '(x) for the function g(x) in the
graph below: Note if the derivative is defined everywhere.
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Example: Graph the derivative g '(x) for the function g(x) in the
graph below:
Is g'(x) continuous everywhere?
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Theorem 3.1 Differentiable Implies Continuous.
If f(x) is differentiable at x=a then f(x) is continuous at x = a.
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Theorem 3.1 Differentiable Implies Continuous.
If f(x) is differentiable at x=a then f(x) is continuous at x = a.
Proof: Differentiable Implies Continuous.
Assume differentiable which means
exists.
Show that f(x) is then continuous by verifying the 3 steps:
1. f (x) is defined at x = a.
2. Show
exists.
3. Show that f(a) =
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Theorem 3.1 Differentiable Implies Continuous.
If f(x) is differentiable at x=a then f(x) is continuous at x = a.
Proof:
Show that f(x) is then continuous means to show the 3 steps:
1. Let f(x) be defined as:
2.
2. Limit exists
3.
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Theorem 3.1 (Alternative) Not Continuous Implies Not Differentiable.
If f(x) is not continuous at x = a then f(x) is not differentiable at x = a.
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Theorem 3.1 (Alternative) Not Continuous Implies Not Differentiable.
If f(x) is not continuous at x = a then f(x) is not differentiable at x = a.
Recall: differentiable which means
exists.
which means left-hand and right-hand limits (slopes) are the same:
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Theorem 3.1 (Alternative) Not Continuous Implies Not Differentiable.
If f(x) is not continuous at x = a then f(x) is not differentiable at x = a.
Recall: differentiable which means
exists.
which means left-hand and right-hand limits (slopes) are the same:
Examples:
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Theorem 3.1 (Corollary) Continuous does NOT imply Differentiable.
If f(x) is continuous at x = a then f(x) may or may not be differentiable
at x = a.
Recall: differentiable which means
exists.
which means left-hand and right-hand limits (slopes) are the same:
Examples:
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