Calculus

Evaluating Limits

In this text we’ll just introduce a few simple techniques for evaluating limits and show you some examples. The more formal ways of finding limits will be left for...

Defining a Limit

A limit of a function is the value that function approaches as the independent variable of the function approaches a given value. The equation f (x) = t is...

Limits and Continuity

The concept of the limit is one of the most crucial things to understand in order to prepare for calculus. A limit is a number that a function approaches...

Sequence:Basic Definitions

Consider the sequence . It is clear that we have The numbers are getting bigger and bigger. Now consider the sequence . In this case, we have Notice that...

Sequences

Sequences are to calculus what at calculator is to a scientist. There are many ways to introduce sequences. Here we will follow a somewhat unorthodox way. Indeed, consider a...

Sequences and Series

Sequences and Series In this chapter we’ll be taking a look at sequences and (infinite) series.  Actually, this chapter will deal almost exclusively with series. However, we also need...

Taylor and Maclaurin Series

Definition of Taylor series: f(x)=f(a)+f′(a)(x−a)+f′′(a)(x−a)22!+⋯+f(n−1)(a)(x−a)n−1(n−1)!+Rn Rn=f(n)(ξ)(x−a)nn! where a≤ξ≤x, ( Lagrangue’s form ) Rn=f(n)(ξ)(x−ξ)n−1(x−a)(n−1)! where a≤ξ≤x, ( Cauch’s form ) This result holds if f(x) has continuous derivatives of order n at last. If limn→+∞Rn=0, the...

Power Series

Powers of Natural Numbers ∑k=1nk=12n(n+1) ∑k=1nk2=16n(n+1)(2n+1) ∑k=1nk3=14n2(n+1)2 Special Power Series 11−x=1+x+x2+x3+⋯(for −1<x<1) 11+x=1−x+x2−x3+⋯(for −1<x<1) ex=1+x+x22!+x33!+⋯ ln(1+x)=x−x22+x33−x44+⋯(for −1<x<1) sinx=x−x33!+x55!−x77!+⋯ cosx=1−x22!+x44!−x66!+⋯ tanx=x−x33+2x515−17x7315+⋯(for −π2<x<π2) sinhx=x+x33!+x55!+x77!+⋯ coshx=1+x22!+x44!+x66!+⋯

Arithematic and Geometric Series

Notation: Number of terms in the series: n First term: a1 Nth term: an Sum of the first n terms: Sn Difference between successive terms: d Common ratio: q...

Limits Formulas

The General Limit Formulas If limx→af(x)=l and limx→ag(x)=m ,then limx→a [f(x)±g(x)]=l±m limx→a [f(x)⋅g(x)]=l⋅m limx→af(x)g(x)=lm limx→ac⋅f(x)=c⋅l limx→a1f(x)=1l The Common Limits limx→∞ (1+1n)n=e limx→∞ (1+n)1/n=e limx→0 sinxx=1 limx→0 tanxx=1 limx→0 cosx−1x=0 limx→a xn−anx−a=nan−1