International Business
University of Business Technology
UBT – CEIT
CALCULUS I – MATH 101
FALL – 2012
Instructor : Abdulraheem Zabadi
STUDY GUIDE
Table of Contents
Limits
Differential Calculus
Integral Calculus
SOME USEFUL FORMULAS Chapter One : Limits
Properties of Limits
If b and c are real numbers, n is a positive integer, and the functions ƒ and g have limits as x → c , then the following properties are true.
Scalar Multiple : limx→c (b f(x))=b limx→c fx
Sum or difference : limx→c ( fx± g(x)) = limx→c fx ± limx→c ( gx)
Product : limx→c (fx. g(x)) = limx→c fx . limx→c gx
Quotient: limx→c fxgx = limx→cf(x)limx→cg(x) , gx≠0
One-Sided Limits limx → a+fx x approaches c from the right limx → a-fx x approaches c from the left
Limits at Infinity limx →∞f(x) = L or limx → -∞f(x) = L The value of ƒ(x) approaches L as x increases/decreases without bound. y = L is the horizontal asymptote of the graph of ƒ.
Some Nonexistent Limits limx→01x2 limx→0 xx limx→0sin1x Some Infinite Limits
limx→01x2= ∞ limx→ 0+ ln x= - ∞
Exercise: What limx→0sinxx ?
a. 1 b. 0 c. ∞ d. DNE
The answer is a ( you should memorize this limit)
Continuity
Definition : A function ƒ is continuous at c if:
1. ƒ(c) is defined 2. limx→cfxexists 3. limx→cfx= fc
Graphically, the function is continuous at c if a pencil can be moved along the graph of ƒ(x) through (c, ƒ(c)) without lifting it off the graph.
Exercise: If f(x) = 3 x2+x2x , for x x ≠0 f(0) = k and f is continuous at x = 0 , find the value of k ? a. -32 b. 32 c. -1 d. 0 e.1 , answer is b
Intermediate Value
UBT – CEIT
CALCULUS I – MATH 101
FALL – 2012
Instructor : Abdulraheem Zabadi
STUDY GUIDE
Table of Contents
Limits
Differential Calculus
Integral Calculus
SOME USEFUL FORMULAS Chapter One : Limits
Properties of Limits
If b and c are real numbers, n is a positive integer, and the functions ƒ and g have limits as x → c , then the following properties are true.
Scalar Multiple : limx→c (b f(x))=b limx→c fx
Sum or difference : limx→c ( fx± g(x)) = limx→c fx ± limx→c ( gx)
Product : limx→c (fx. g(x)) = limx→c fx . limx→c gx
Quotient: limx→c fxgx = limx→cf(x)limx→cg(x) , gx≠0
One-Sided Limits limx → a+fx x approaches c from the right limx → a-fx x approaches c from the left
Limits at Infinity limx →∞f(x) = L or limx → -∞f(x) = L The value of ƒ(x) approaches L as x increases/decreases without bound. y = L is the horizontal asymptote of the graph of ƒ.
Some Nonexistent Limits limx→01x2 limx→0 xx limx→0sin1x Some Infinite Limits
limx→01x2= ∞ limx→ 0+ ln x= - ∞
Exercise: What limx→0sinxx ?
a. 1 b. 0 c. ∞ d. DNE
The answer is a ( you should memorize this limit)
Continuity
Definition : A function ƒ is continuous at c if:
1. ƒ(c) is defined 2. limx→cfxexists 3. limx→cfx= fc
Graphically, the function is continuous at c if a pencil can be moved along the graph of ƒ(x) through (c, ƒ(c)) without lifting it off the graph.
Exercise: If f(x) = 3 x2+x2x , for x x ≠0 f(0) = k and f is continuous at x = 0 , find the value of k ? a. -32 b. 32 c. -1 d. 0 e.1 , answer is b
Intermediate Value