MATH 1462
Pre-Calculus
Summer I 2011
Dr. Raftkah
Limits: A Numerical and Graphical Approach
ASSIGNMENT
Limit Numerically
Let x approach 1 from the right:
Thus, it appears that
Since,
a)
and
b)
Then,
does not exist.
Limit Numerically
First, let x approach
MATH 1462
Pre-Calculus
Summer I 2011
Dr. Raftkah
Differentiation
OBJECTIVE
Find derivatives and values of derivatives
Find equations of tangent lines
DEFINITION:
The slope of the tangent
line at (x, f(x) is
This limit is also the
instantaneous rate of
cha
MATH 1462
Pre-Calculus
Summer I 2011
Dr. Raftkah
Average Rates of Change
OBJECTIVE
Compute an average rate of change.
Find a simplified difference quotient.
1.3 Average Rates of Change
DEFINITION:
As x approaches a, the limit of f (x) is L, written
if all
MATH 1462
Pre-Calculus
Summer I 2011
Dr. Raftkah
Higher Order Derivatives
OBJECTIVE
Find derivatives of higher order.
Given a formula for distance, find velocity and acceleration.
Higher-Order Derivatives:
Consider the function given by
Its derivative f i
MATH 1462
Pre-Calculus
Summer I 2011
Dr. Raftkah
Product & Quotient
OBJECTIVE
Differentiate using the Product and the Quotient Rules.
Use the Quotient Rule to differentiate the average cost, revenue, and profit
functions.
THEOREM 5: The Product Rule
Use t
MATH 1462
Pre-Calculus
Summer I 2011
Dr. Raftkah
Differentiation Techniques
OBJECTIVE
Differentiate using the Power Rule or the Sum-Difference Rule.
Differentiate a constant or a constant times a function.
Determine points at which a tangent line has a sp
MATH 1462
Pre-Calculus
Summer I 2011
Dr. Raftkah
Algebraic Limits
OBJECTIVE
Develop and use the Limit Principles to calculate limits.
Determine whether a function is
continuous at a point.
LIMIT PROPERTIES:
The limit of a constant is the constant.
LIMIT P
MATH 1462
Pre-Calculus
Summer I 2011
Dr. Raftkah
Limits: A Numerical and Graphical Approach
OBJECTIVE
Find limits of functions, if they exist, using numerical or graphical methods.
1.1 Limits: A Numerical and Graphical Approach
DEFINITION:
As x approaches
MATH 1462
Pre-Calculus
Summer I 2011
Dr. Raftkah
Limits: A Numerical and Graphical Approach
ASSIGNMENT
Quick Check 1 Solution
First let X approach 3 from the left:
Thus it appears that
Next let x
is 6.
approach 3 from the right:
Thus it appears that
is 6
MATH 1462
Pre-Calculus
Summer I 2011
Dr. Raftkah
First Derivatives
OBJECTIVE
Find relative extrema of a continuous function using the First-Derivative Test.
Sketch graphs of continuous functions.
DEFINITIONS:
A function f is increasing over I if, for ever