1.Does a continuous function mean differentiable function ? If a function f(x) is continuous at a number a , then , f(x) has a derivative at a number a.Is it true ? For instance , we know the function f (x)=|x|is continuous at a number 0 , is it true that f(x) has a derivative at a number a ? Example 1 : Where is the function f(x)=|x|differentiable ? Solution : We know if x>0 , then f(x)=|x|=x.And we choose hsmall enough that x+h>0 , hence f(x+h)=x+h=x+h. Therefore , for x>0 , we
Since we know the derivative of the function,so it is the thinking way in math to find a function of F whose derivative is a known function f.If such a function Fexists,we can call it an anti-derivative of f.Let us think about it.For instance,let f(x)=x2.We can find an anti-derivative of f,if we use the Power Rule on it.What F(x)=1/3x1/3 is the one could be discovered,since it is satisfied with.Is there anyone else? Yes,you are right.More
In general,it is not easy to find the exact sum of a series.We could find geometric series with a simple formula for the n-th partial sum S_n.And we are also lucky can find the n-th partial sum S_n of the series ∞∑(n=1)1/(n(n+1)).
At the last time,we’ve learned the First Part of the Fundamental Theorem of Calculus.It shows how indefinite integration and definite integration are related.In the other words,it shows how anti-derivative and the area are related.Today,you’ll deeply understand the First FTC by using it.
Now we＇ll learn what the alternating se-ries estimation theorem is？ A partial sum S,, of any convergent seriescan be used as an approximation to the totalsum. If we want estimate the accuracy of ap-proximation, then the error and remaindershould be considered. The error involved inusing S≈S,, is the remainder.
Give a series∑a_n.Let’s assume that the series terms are all positive and is decreasing.So we can use the integral test to show that it converges.Write down the remainder for
