lecture 28 1 what is the fundamental theorem of calculus write a short

Question

Lecture 28 1. What is the fundamental theorem of calculus? Write a short history of this important theorem. 2. Use the fundamental theorem of calculus to find derivative \frac{df}{\differentialD x}^^20if^^20f(x)=\int ^x_0\cos tdt. 3. The binomial theorem, as stated by Newton in his letter to Oldenburg, is equivalent to the more familiar form (1+x)^r=1+rx+\frac{r(r-1)}{2 !}x^2+\frac{r(r-1)(r-2)}{3!}x^3+\ldots where r is an arbitrary integral or fractional exponent. The necessary condition |2| < 1 for convergence was not stated by Newton. Use the binomial theorem to obtain the following series expansion: (1+x)^-1/3 up to the x^3 term.