Sec 1.3
The Limit of a Function
A. Limits
lim f x L
DEFN:
xa
The limit of f(x) as x approaches a, equals L.
(Where is the functions value headed as x is "on its way" to a?)
lim f x
xa
lim f x
xa
The limit of f(x) as x approaches a from the LEFT
The li
MATH 1241 -011
Spring 2014
HW #2
Sections 2.1-2.8
Name:_
(PRINT Last Name First)
Part I Multiple Choice - You must show ALL your work in the space provided to receive credit
1.) Which of the following is the definition of the derivative?
(
a)
)
( )
( )
c)
Math 1241-011
Dr. Taylor
Homework #1
Chapters 1.1 2.2
Name:_
(Print Last Name First)
Part I - Multiple Choice You must show ALL your work in the space provided to receive credit
1.)
Find the asymptotes of the function f x
3x
2x 2
2
, Vertical Asymptote a
Linear Approximation and Differentials
Sec 2.8
A. Differentials
y
Slope = x
Slope of the tangent line =
Also,
dy
dx
dy
f x
dx
y dy
x dx
dy f x dx
Example: Find the differential of y given that:
1.) y 5x 3
2.) y
3x 2 4 x 5
5 sin x 2 x
Note:
x dx and y d
Math 1241
Project 2 - Difference quotient and Derivative
Name:
In this project, numerical values of the derivative will be calculated using the limit of the difference
quotient and a calculator program will be created to compute the slope of the secant li
Math 1241
Project 1 - Limits and Asymptotes
Name:
In this project, Limits will be calculated and the asymptotic behavior of rational functions will be studied
using graphs and tables.
x3 x2 3
x3 x2 3
.
x 2 x 2 ( x 1)( x 2)
It is known that this function i
Chain Rule
Sec 2.5
A. The Chain Rule
hg x hg x g x
Alternative Notation:
f x h g x hg x then f x hg x g x
hg x d hg dh dg
dx
dg dx
In Plain English: First, identify which function is on the outside and which is on the inside. (For the composition
Product and Quotient Rules
Sec 2.4
A. Product Rule
f x g x hx then f x g x hx g x hx
d g h dg
dg
h g
dx
dx
dx
Alternative Notation:
In Plain English: The derivative of the product of two functions (which we will call the first function and the
se
Sec 2.2
The Derivative of a Function
A. Definition of the Derivative
For a function
f x the derivative is f x lim
h0
f x h f x
h
Examples: Using the definition of the derivative, find the derivative of the following functions:
1.)
f x x 2 3x 2
2.) g x
x
Related Rates
Sec 2.7
Before getting started with Related Rates, let us re-visit the following items first: Notation, Implicit Differentiation and
Geometric Formulas
A. Notation
f x y
dy
dx
Although all of the above notations are equivalent, we will use
Basic Differentiation
Sec 2.3
A. Properties and Formulas (The short way Yeah!)
1. Basic Functions
Function
Derivative
f x c (Constant)
f x 0
f x x
f x 1
f x cx
f x c
f x x n
f x nx n1
f x cx n
f x cnx n1
f x c g x
f x c g x
f x g x hx
f x gx hx
f x g x
Differentiation of Implicit Functions
Sec 2.6
Recall from the previous week, that when we take the derivative of y f x then y f x where f x is a
function in terms of x (i.e. the only variable in the function is x)
Example: If y x 3 2 x then y
dy
3x 2 2
Sec 1.1 and 1.2
Functions: Review of Algebra and Trigonometry
A. Functions and Relations
DEFN
Relation: A set of ordered pairs.
(x,y) (domain, range)
DEFN
Function:
A correspondence from one set (the domain) to anther set (the range) such that each
elemen
Sec 1.5
Continuity
A. Definition of Continuity
DEFN: A function f is continuous at a number a if:
(i) f a exists
(ii)
lim f x exists
xa
(iii) lim f x f a
x a
A function is defined as continuous only if it is continuous at every point in the domain of the
Sec 2.1
Derivatives and Rates of Change
A. Slope of Secant Functions
Recall: Slope = m =
rise y y 2 y1
=
=
. From this we are able to derive:
run x x2 x1
Slope of the Secant Line to a Function: m
y 2 y1
x 2 x1
or m
f x2 f x1
x2 x1
Examples:
1. a.) Find
Sec 1.4
Calculating Limits
A. Limit Laws
Assume that f and g are functions and c is a constant.
1.)
lim f x g x lim f x lim g x
2.)
lim f x g x lim f x lim g x
3.)
lim c f x c lim f x
xa
xa
4.)
5.)
xa
xa
xa
xa
xa
xa
lim f x
f x x a
lim
x a g x
li
Sec 1.6
Limits Involving Infinity
A. Infinity vs. DNE
1
DNE since the function value kept increasing. Now we will be more
x 0 x 2
descriptive; any value that keeps increasing is said to approach infinity ( ), and any value that keeps
decreasing is said to
Math 1241-011 Calculus I Fall 2014
Tuesday and Thursday 2:00-3:15pm
Classroom: Fretwell 121
Professor: Dr. John R. Taylor
E-mail: [email protected]
Office: Fretwell 335A
Phone: 704-687-5611
Office Hours: Monday, Tuesday and Thursday 12-1:30pm, on Wednesda