TMT1171 Mathematical Techniques I, Trimester 1 2008/2009
TOPIC 3: Limits, Continuity and Differentiation
A. 1. LIMITS OF A FUNCTION LIMIT DEFINITION
Definition: For a function f = f (x) defined on an open interval ( a , b ) about a number c , except possi
Certificate Mathematics in Action Full Solutions 4A
4 More about Equations
Follow-up Exercise
p. 162
1. By substituting x2 = u into the equation x4 10x2 + 9 = 0, we have u2 10u + 9 = 0 (u 1)(u 9) = 0 u = 1 or u = 9 x2 = u x2 = 1 or x2 = 9 x = 1 or x = 3
Math 135 Class Notes
Business Calculus
Spring 2009
1.8
Higher-Order Derivatives
Given a function y = f (x), its derivative f 0 (x) is itself a function. This means we can take its derivative. The derivative of the derivative is called the second derivativ
Math 135 Class Notes
Business Calculus
Spring 2009
1.6
Differentiation Techniques: The Product and Quotient Rules
In Section 1.5, we saw how to differentiate the sum of two functions. This section shows how to differentiate the product or quotient of two
Math 135 Class Notes
Business Calculus
Spring 2009
1.3
Average Rates of Change
Consider a function y = f (x) and two input values x1 and x2 . The change in input, or the change in x, is x2 - x1 . The corresponding change in output, or the change in y, is
Math 135 Business Calculus Exam 1 Solutions 1. (20 points) The graph of a function f is shown. (a) Determine the value of each of the following limits, if it exists. If it does not exist, explain why not. (i) lim f (x)
x-2
Spring 2009
y
5 4 3 2 1 5 4 3 2
Math 135 Project: Monopoly Pricing Due Monday, April 20
Business Calculus
Spring 2009
Introduction When there is only one firm that produces a certain product, that one firm is called a monopoly. Monopolies are familiar to most consumers. Local phone serv
Math 135 Class Notes
Business Calculus
Spring 2009
1.7
The Chain Rule
THE EXTENDED POWER RULE According to the Power Rule, the derivative of the power function y = xk is given by d k (x ) = kxk-1 . dx For a power function y = xk , we have a variable x rai
Math 135 Class Notes
Business Calculus
Spring 2009
1.4
In the previous section, we saw that a secant line joining the points x, f (x) and x + h, f (x + h) on the graph of a function y = f (x) has slope msecant =
Differentiation Using Limits of Difference
Math 135 Class Notes
Business Calculus
Spring 2009
1.1
Limits: A Numerical and Graphical Approach
The limit of a function is the fundamental concept in calculus and is used to define the derivative of a function, the subject of this first chapter. In this