Question

If the function f has a continuous derivative on [0,c], the the integral(o to c) of f'(x)dx=

a)f(c)-f(0)
b)absolute value (f(c)- f(0))
c) f(c)
d)f'(x)=c
e)f"(c)-f"(0)

My work: so the the answer to the integral is f(x) and when find the answer from o t0 c, it is f(c)-f(0).
is that the right answer? i'm confused because is there anything i have to do with the point [0,c]. or is that unneccessary info.


problem #2:

let f be a polynomial function with degree greater than 2. if a does not equal b and f(a)=f(b)=1, which of the following must be true for atleast one value of x between a and b?

I)f(x)=0
II)f'(x)=0
III)f"(x)=0

you can choose more than one choice in the choices mentioned of I, II, III

i'm having trouble coming up with the equation and choosing a and b



The correct answer to (#1) is f(c)-f(0).
That is because the definite integral of a function (f') is the difference between the indefinite integral (f) evaluated at the two limits of integration.

The correct answer to (#2) is f'(x) = 0. Imagine all possible continuous curves you can draw from a to b, going though f = 1 at both points. The curve MUST have zero slope somewhere. There is no requirement that f or f'' be zero at intermediate points. you don't need an equation to prove this. You just need to invoke the Mean Value Theorem

http://archives.math.utk.edu/visual.calculus/3/mvt.3/index.html

Answer 1

1. The answer is actually f'(c)-f'(0)! I'm assuming this was just a typo on your part.