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ECE: Math
Chapter Three
DIFFERENTIATION
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OUTLINE:
 Steepness and the Tangent Line
 The Derivative as a Function--------- Sec. 3.1/ p.147
 Differentiation Rules------------------- Sec. 3.2/ p.159
 The Derivative as a Rate of Change-------------- Sec. 3.3/ p.171
 Derivatives of Trigonometric Functions--------- Sec. 3.4/ p.183
 The Chain Rule and Parametric Equations---- Sec. 3.5/ p.190
 Implicit Differentiation----------------------------- Sec. 3.6/ p.205
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The Steepness
• Slope represents 'Steepness‘
• The rate of change in a graph is represented as 'Slope'.
• I have two lines in a graph as shown below. If I ask 'Which
of the lines is steeper ?', everybody would come out with
answer right away. Line (1) is steeper.
Then what if I ask 'How do you represent the steepness in a
number ?'.
We use the slope to indicate the changes in the horizontal and
vertical directions.
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The Tangent Line
• Suppose you are told to figure out the slope at a point on a
curve as
• You learned only about getting the slope on a straight line.
What is the meaning of getting the slope on a curve ?".
• The slope on a curve is defined as the slope of the straight
line which is tangential to the point on the curve
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The Tangent Line
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The Tangent Line
• There are mainly two methods to find the tangent line. One
is a kind of Geometrical/Graphical method and the other
method is Algebraic method.
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The Tangent Line
• About the value of h, we have to try to make h as small as
possible as shown below
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The Tangent Line
Application of a Tangent Line ‐ Finding Min/Max
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The Tangent Line
Application of a Tangent Line ‐ Finding Min/Max
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The Tangent Line
Application of a Tangent Line ‐ Finding Min/Max
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The Derivative as a Function
• Differentiation is a method to represent the rate at which one
variable (dependent variable) changes with respect to the
changes in another variable (independent variable).
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The Derivative as a Function
• Now let's think of the meaning of "limit" in this expression. It says
"h approaches to 0". It means "it get closer to 0". What would the
graph look like when h gets closer to 0. You would get the meaning
of this mathematical expression at the graph shown next.
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The Derivative as a Function
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The Derivative as a Function
• If ƒ` exists at a particular x, we say that ƒ is differentiable (has
a derivative) at x. If ƒ` exists at every point in the domain of ƒ,
we call ƒ differentiable.
• The calculation of a derivative
is called differentiation.
Example 1, 2, p. 149
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The Derivative as a Function
• If the limit in the above definition exists, the function f is
said to be differentiable at x, and the process of calculating f`
is called differentiation of f.
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The Derivative as a Function
• There are many ways to denote the derivative of a function
• To indicate the value of a derivative at a specified number
x = a, we use the notation
Differentiable on an Interval
• A function is differentiable on an open interval (finite or
infinite) if it has a derivative at each point of the interval.
• It is differentiable on a closed interval [a, b] if it is
differentiable on the interior (a, b) and if the limits exist at
the endpoints.
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The Derivative as a Function
• Right-hand and left-hand derivatives may be defined at any
point of a function’s domain.
• The usual relation between one-sided and two-sided limits
holds for these derivatives.
• The function has a derivative at a point if and only if it has
left-hand and right-hand derivatives there, and these onesided derivatives are equal.
Example 5, 6 p. 152
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The Derivative as a Function
When Does a Function Not Have a Derivative at a Point?
• The derivative of a function at a point is the slope of
the function at that point. Hence, if at a
certain point a function does not have a clearly defined slope,
then it does not have a derivative at that point.
• So, if you can’t draw a tangent line, there is no derivative.
• A function has a derivative at a point if the slopes of the
secant lines through and a nearby point Q on the graph
approaches P(xo, f(x0))a limit as Q approaches P.
• Whenever the secants fail to take up a limiting position or
become vertical (slope is infinity) as Q approaches P, the
derivative does not exist.
• Thus differentiability is a “smoothness” condition on the
graph of ƒ. A function whose graph is otherwise smooth will
See the four cases P 153
fail to have a derivative at a point.
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The Derivative as a Function
Differentiable Functions Are Continuous
• A function is continuous at every point where it has a
derivative.
See Exercise 3.1, p. 155
All the questions except (33-34, 45-66)
• For the questions with
, understand the idea and ignore the graphing
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Differentiation Rules
Example 1, P. 159
Proof of Rule 1, P. 159
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Differentiation Rules
Binomial Theorem
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Differentiation Rules
Binomial Theorem
Example 2, P. 160
Proof of Rule 2, P. 160
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Differentiation Rules
Binomial Theorem
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Differentiation Rules
Example 3, P. 161
Proof of Rule 3, P. 161
Example 4 ,5, 6, P. 162
Proof of Rule 4, P. 162
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Differentiation Rules
Example 7, 8, 9, P. 164
Proof of Rule 5, P. 164
Example 10, P. 165
Proof of Rule 6, P. 166
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Differentiation Rules
Second‐ and Higher‐Order Derivatives
Example 11, 12, 13, 14, P. 166
Proof of Rule 7, P. 167
See Exercise 3.2, p. 169
All the questions
For the questions with
, understand the idea and ignore the graphing
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Rate of Change
Example 1, p. 172
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Rate of Change
Example 3, p. 174
• Speed is the time rate at which an object is moving along a
path, while velocity is the rate and direction of an object's
movement.
• Speed and velocity both measure an object's rate of motion.
Acceleration and Jerk
• The rate at which a body’s velocity changes is the body’s
acceleration.
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Rate of Change
• The acceleration measures how quickly the body picks up or
loses speed.
• A sudden change in acceleration is called a jerk.
Example 4, 5, p. 175
See Exercise 3.3, p. 179
(1-22)
• For the questions with
, understand the idea and ignore the graphing
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Derivatives of Trigonometric Functions
Example 1, p. 184
Proof, p. 184
Proof, p. 183
Example 2, 3, 4 p. 185
Example 5, 6, 7 p. 187
See Exercise 3.4, p. 188
(1-26, 37-50)
• For the questions with , understand the
idea and ignore the graphing
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Chain Rule & Parametric Equations
• Chain Rule says that the derivative of the composite of two differentiable functions is
the product of their derivatives evaluated at appropriate points.
• The Chain Rule is one of the most important and widely used rules of differentiation.
“Outside‐Inside” Rule
It sometimes helps to think about the Chain Rule this way differentiate the “outside”
function ƒ and evaluate it, then multiply by the derivative of the “inside function.”
See Exercise 3.5, p. 201
(1-66)
Example 1‐7 p. 191
• For the questions with , understand the
idea and ignore the graphing
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Implicit Differentiation
• Most of the functions we have dealt with so far have been
described by an equation of the form y = ƒ(x) that expresses y
explicitly in terms of the variable x .
• When we cannot put an equation in the form y = ƒ(x) to
differentiate it in the usual way;
then we may still be able to find dy/dx by implicit differentiation.
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Implicit Differentiation
• We shall content ourselves with learning a method for finding
the derivatives of functions determined implicitly by equations.
• This consists of differentiating both sides of the equation with
respect to x and then solving the resulting equation.
Example 1, 2, 3, 4, 5, 6, 7 p. 206
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Implicit Differentiation
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Chain Rule & Parametric Equations
See Exercise 3.6, p. 211
(1-72)
• For the questions with , understand the idea and
ignore the graphing
•
•
Explicit: the dependent variable is expressed in terms of independent
variable (traditional, default form). y=x+1
Implicit: to describe the relationship between dependent and independent
variables. No differences can be made between x and y. x and y can't be
separated.
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