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C. Further Reading: Continuity
Let us return briefly to the concept of continuity. Informally we discussed continuity in terms of a graph
with no breaks. We later formalized this using limits and said that a function f is continuous at a point a if
.
111
C. Further Reading: More Examples
Question 1 Construct the graph of
.
Solution We begin by finding the x and y intercepts. Since
one x intercept and it is also the y intercept.
, we see that there is only
Now we need to find and simplify the first and
107
B. Class Activities
In the previous lesson, we saw how to find the derivative of polynomial (and similar) functions using
short cut methods which avoid the use of the limit definition of the derivative. In this lesson (and the next
one) we look at fur
93
Lesson 8: Basic Rules for Derivatives
A. Reading: Finding Derivatives of Polynomial Functions
We have seen that the derivative
of a function f can be interpreted as:
the instantaneous rate of change of the function with respect to the independent varia
100
C. Further Reading: Some more examples
Question 1 Find an equation for the line tangent to the graph of
which has y intercept 15.
Solution We see that there are two conditions that must be met: the line must pass through the point
(0,15) (note that th
96
B. Class Activities
Example: 1. Use the definition of the derivative to calculate the derivative of the following functions:
a)
b)
c)
d)
e)
f)
g)
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Theorem (Sum rule for derivatives.) Let
Theorem (Constant rule for derivatives.) If
Theorem (Power rule
89
C. Further Reading: More about the Derivative
Since this material is central to the course, we will review the details of the definitions:
The slope of the secant line is called the difference quotient:
.
When we let the a + h approach a we are letting
79
Lesson 7: Definition of the Derivative
A. Reading: Local Linearity
If a function is to have a derivative at a point, then it must have a
clearly defined rate of change at that point or, in other words, it must
have a tangent line at that point. Conside
83
B. Class Activities
Example: 1. Suppose
. It can be shown that
.
a) Sketch the graph of and its tangent line at
on your calculator.
b) Zoom in on the point of tangency until the graph no longer changes. What do you observe?
The above property is known
105
Lesson 9: More Rules for Derivatives
A. Reading: Derivatives of Exponential Functions, Product and Quotient Rules
What is the derivative of
The
? Using the definition of the derivative:
can be factored out of the limit because the active variable in t
117
B. Class Activities
Our goal in this lesson is to look at derivatives of trigonometric functions and derivatives of
compositions of functions.
Graph of
(by calculator)
Graph of
(by calculator)
Graph of
(by calculator)
Graph of
(by calculator)
Formulas
149
Lesson 13: Optimization
A. Reading: Finding Global Extrema of a Function
One important application of calculus is to solve problems where the goal is to find the best (i.e. largest or
smallest) value of something.
Example:
Solution 1:
Find the global
139
Lesson 12: Derivatives of Implicitly Defined Functions
A. Reading: Implicit Functions
Consider the two equations:
and
. They are just rearrangements of each
other, so define the same relationship. We say that the first equation defines a function expl
141
B. Class Activities
Example: 1. Find an equation of the tangent line to
Solution 1: (solve for
)
Solution 2: (differentiate first)
Further examples: Find
2.
at
at the indicated points:
at
.
142
3.
5.
at
at
4.
6.
at
at
and at
.
143
7.
, if
8.
at
, if
N
135
C. Further Reading
We have found the derivatives of the inverses of a number of transcendental functions:
f (x)
f (x)
There are some interesting things to notice about these functions and derivatives:
The derivatives of the trigonometric functions are
145
C. Further Reading: More Examples
Question 1 Let
. Find
and
and then evaluate each at (1,1) and (1, 2).
Solution Differentiate both sides of the equation with respect to x.
Now to find the second derivative we differentiate again:
Question 2 Find the
131
B. Class Activities
Definition: The inverse of a function f is a function denoted by
for which:
Note that f needs to be one-to-one for the inverse to exist. In this lesson we will look at how to calculate
the derivative of an inverse function.
Theorem
122
C. Further Reading: The derivative of the sine function
In the derivation of the formula for the derivative of sin(x) we used the following limits:
and
We justified these results graphically. Now we will provide more rigorous proofs. To do this we nee
115
Lesson 10: Still More Rules for Derivatives
A. Reading: Derivatives of Trigonometric Functions and the Chain Rule
Imagine an object moving around a circle of radius 1 in
such a way that it s position is
at time t.
Its velocity is then tangent to the c
62
C. Further Reading: Critical Points
If we know that the derivative of a function is 0 at a particular point, we can use the first derivative test to
determine whether that point is a local maximum, a local minimum or neither of these. However, if we
wa
57
B. Class Activities
Example: 1. a) Sketch the graph of a continuous function
2
2
3
b) Sketch the graph of a continuous function
2
that is consistent with these data:
1
1
2
1
1
2
0
1
2
that is consistent with these data:
1
2
1
1
3
1
1
1
2
2
1
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Example
219
Lesson 20: Newton's Method; Parametric Equations
A. Reading: Intermediate Value Theorem and Approximating Roots
Consider the polynomial function
, whose graph
is shown. Note three facts about p: (i) p is continuous everywhere,
(ii)
and
and (iii) p has
241
B. Class Activities
Recall the definition of the derivative:
As this limit may or may not exist, we say that
is differentiable at
when the limit does exist.
Example: 1. Determine where the following functions are continuous, and where they are differe
238
Further Exercises
*
1. Evaluate the following:
a)
*
b)
c)
2. Evaluate the following:
a)
b)
c)
3. Evaluate the following: a)
*
b)
c)
b)
c)
d)
5. Evaluate:
a)
b)
c)
6. Evaluate the following: a)
*
d)
4. Evaluate the following:
a)
*
d)
d)
b)
c)
7. Find e
209
Lesson 19: Linear and Quadratic Approximation
A. Reading: Approximations using Tangent Lines
If a function is complicated or difficult to work with, it is often useful to be able to approximate the
function with a simpler function. We know that a func
215
C. Further Reading: Accuracy in Linear Approximations
Whenever we use one function approximate another, it is reasonable to ask about the accuracy of the
approximation. When we use a linearization l for a function f based at a point a, there will be s
222
B. Class Activities
Theorem 4 (Intermediate Value Theorem):
Let be continuous on the interval
, and let
any number between
and
. Then
Example: 1. Show that
be
has a solution.
While we can solve equations like
by using the quadratic formula, few people
195
C. Further Reading: Polynomials, rational functions and asymptotes.
Do polynomials ever have horizontal or vertical asymptotes?
Recall that we have said that polynomials are continuous everywhere. Thus for any a
.
Since for and a,
has a finite value,
205
C. Further Reading: Technology and Theory
In practice, constructing a detailed and accurate graph of a function is usually an interaction between
technology (calculator or computers) and theory. We may construct a preliminary graph using technology
an
244
C. Further Reading: Review
Now that we have completed the final lesson in this course, it is time to review what has been learned.
Certainly, a lot of new concepts, new calculations and new applications have been studied. Sometimes it
is hard to get a
2.2 2.3
Limits
Limit is the foundation of Calculus. In differential Calculus, we use limit to define the derivative of a
function. In integral calculus, we use limit to define the definite integral of a function. To understand what
limit is, let us look a
10.1 Curves Defined by Parametric Equations
If a particle moves on the plane along a path shown in
the figure on the right, it is rather difficult to describe
the path by a function of the form y f x because the
path is not the graph of a function. But if
The test is scheduled for Friday, Nov. 21.
There are four questions worth a total of 19 marks.
The test covers Sections 4.1 and 4.2.
Can you draw a possible graph of a function given information about its local and absolute
extrema (or lack thereof)?
Do
Math 1120-S10 Test #3 Information
Your test has four questions worth a total of 30 marks.
The test covers Sections 3.1-3.5* (*not including inverse trigonometric functions and their
derivatives)
One question requires you to calculate derivatives of variou
Math 1120-S10 Test #4 Information
The test is scheduled for Friday, Nov. 7.
There are four questions worth a total of 23 marks.
The test covers the Inverse Trig. Supplement and Sections 3.6, 3.7, 3.9 and 3.10 (including the
supplement on Quadratic Approxi