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Know the Definition of Derivative and Integral!

A question from a sample AP Calculus B/C by College Board is given in a booklet,

 
Q.   If F' is a  continuous function for all real x, the  limh→0 (1/h)∫a+h F'(x)dx   is 
 
         (a) 0    (b) F(0)    (C) F(a)    (D) F'(0)     (E) F'(a)
 
 
A.   (1) Definition of Definite Integral
               If a function f(x) is continuous on the interval [a, b] and divided by n equal subintervals, then
               the definite integral of f(x) from a to b is 
 
                      ∫b f(x) = limn→∞ i=0 [f(xi)((b-a)/n)] = F(b) - F(a)
 
                      
       (2) Definition of Derivative
               If the function f '(x) is the derivative of f(x) with respect to x is defined as,  
 
                      f'(x) = limh→0 (f(x+h)-f(x))/h 
                     
 
       Therefore, using (1) definition of definite integral, we can rewrite the integral part as,
 
                     ∫a a+h F'(x)dx = F(a+h) - F(a)
                  
                     
          and then, using (2) definition of derivative, we can take a limit as,
 
                     limh→0 (F(a+h)-F(a))/h = F'(a)
 
                   
Ans. (E) 
 
 
 
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