# 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

## For problem #1, your answer is correct. The integral of f'(x) from 0 to c is equal to f(c) - f(0). The information about [0, c] is indeed necessary for calculating the definite integral.

For problem #2, the correct answer is II) f'(x) = 0. According to the Mean Value Theorem, if a function f is continuous on [a, b] and differentiable on (a, b), then there exists at least one value c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a). In this case, f(a) = f(b) = 1, so (f(b) - f(a))/(b - a) = 0/ (b - a) = 0. Hence, f'(c) = 0 for some value c between a and b. There is no requirement that f or f" be zero at intermediate points.

## For problem #1, you are correct! The integral from 0 to c of f'(x)dx is equal to f(c) - f(0). This is known as the Fundamental Theorem of Calculus. The endpoint values [0,c] are necessary information because they define the limits of integration.

For problem #2, the correct answer is II) f'(x) = 0. Since f(a) = f(b) = 1 and f(x) is a polynomial of degree greater than 2, by the Mean Value Theorem, there must be at least one point, call it c, between a and b where the derivative (f'(x)) is equal to zero. This means that the function has a point of zero slope between a and b.

## For problem #1, the correct answer is indeed f(c) - f(0). To understand why, we can use the Fundamental Theorem of Calculus. According to this theorem, if f(x) has a continuous derivative on an interval [a, b], then the integral of f'(x) from a to b is equal to f(b) - f(a).

In this case, we have the interval [0, c], and we know that f(x) has a continuous derivative on this interval. So, the integral of f'(x) from 0 to c is equal to f(c) - f(0).

Therefore, the correct answer is a) f(c) - f(0).

For problem #2, we are given that f(a) = f(b) = 1 and f is a polynomial function with degree greater than 2. We need to determine which statements must be true for at least one value of x between a and b.

Since f(a) = f(b) = 1, we know that the polynomial function intersects the y = 1 line at both a and b. Now, since the degree of the polynomial is greater than 2, there must be at least one critical point (a point where the derivative is zero) between a and b. This is because polynomial functions of degree greater than 2 can have multiple turning points.

Therefore, statement II) f'(x) = 0 must be true for at least one value of x between a and b. The polynomial function must have a critical point within this interval.

Statements I) f(x) = 0 and III) f"(x) = 0 are not necessarily true. The function does not need to have any roots (zeroes) between a and b, and it also does not need to have zero second derivative at any point between a and b.

So, the correct answer is II) f'(x) = 0.