What is The Squeeze (or Sandwich) Theorem?

<正>Let's do an exercise as warming up.Ex:Find lim x→0sin1/x.lim x→0sin1/x=lim1/x→∞sin1/x=lim t→∞sint(Let t=1/x).We know lim t→∞sint does not exist,so lim x→0sin1/x DNE.We can use graphing calculator to show the result is DNE(from figure 1to figure 8).     

What is The Mean Value Theorem？ （1）

All the textbooks about calculus are shown the Rolle＇s theorem at the first, then the Mean Value Theorem （MVT）. In some persons＇ opinion, the Rollers theorem is the special case of the MVT.     

What is the differential equation？

An equation involving a derivative is called a differential equation.Such as,dy/dx=2x,and the function y=f（x）satisfies this equation.When we know the additional condition that y=2 when x=-1,the function y=f（x）will be find exactly.The additional condition is called the initial condition.It is used to evaluate constant of integration.     

What is the Substitution Rule？

When I learned the following basic formula of indefinite integrals in the calculus, at that time I was a teenager, the strange ideas had jump out my head.     

What is The Rolle＇s Theorem？ （2）

1.2 Why is the Rolle＇s theorem correct？
To answer the question we need a proof.
Proof of Rolle＇s theorem：     

What is the limit of an infinite sequence？

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What is The Mean Value Theorem？ （4）

Example 7. PROOF tan ^-1x ＋ cot^-1 x=π/2.     

What is The Mean Value Theorem？ （2）

2. How does the MVT work？Examplel： Iff（x） = x^2,F（x） = x^3 ,Find ξ on the interval [0, 1] using MVT.     

What does Part 2 of the Fundamental Theorem of Calculus mean?

We defined the indefinite integral as an anti-derivative,and defined the definite integral as the limit of Riemann sums.Both of them are very different and seem to be little in common.Part 1of the Fundamental Theorem of Calculus shows how indefinite integration and definite integration are related.In the other words,it shows how anti-derivative and the area are related.Today,we’ll learn Part 2 of the Fundamental Theorem of Calculus.     

What is the basic integration rule？

In this paper you will learn the basic integration rule from the basic differentiation rule.
Last time we have learned anti-derivative, indefinite integral, and the power rule in integration.     

What is the exponential gronth （or decay）？

We know a differential equation contains derivatives. A derivative gives information a- bout the rate of change of a function. For ex- ample：