That's not supposed to be ... between -4 and x, is it?josh3189 said:Homework Statement
Use the The Fundamental Theorm of Calculus to Find the following answers
f(x) =
0 IF x < -4
2 IF -4<=x<0
-3 IF 0<=x<5
0 IF x>= 5
g(x) = Integral of f(t) dt Between 4 and x
Yes, the derivative of g(x) is f(x). How can you use that information ?Determine the value of each of the following:
(a) g(−6)=
(b) g(−3)=
(c) g(1)=
(d) g(6)=
(e) The absolute maximum of g(x) occurs when x= and is the value =
I get that the derivative of g(x) will be f(x) but I'm stuck after that?
SammyS said:That's not supposed to be ... between -4 and x, is it?
Yes, the derivative of g(x) is f(x). How can you use that information ?
What do you suppose g(4) is ?
Can you state The Fundamental Theorem of Calculus ?
With the above correction, I would have asked you, 'What is g(-4) ? ' ... with the answer being g(-4) = 0 .josh3189 said:Yes, sorry my bad its between -4 and x
g(4) would just be the integral of f(x) and using x = 4 as a substitution
That should say, g is differentiable on the open interval(a,b).In this case though, I'm not really sure what f(x) is exactly since it has multiple values or I would just integrate it and sub directly.
The Fund Theorem states that if f be a continuous real-valued function defined on a closed interval [a, b] then f is differentiable on the open interval(a,b).
Just in case someone comes across this later :In this case, I realize that it simple g'(x) = f(x)
EDIT : I made a graph of f(x) and got it from there, I forgot that the integral of f(x) would simply be the area of the function as well so it became quite clear on the graph. Thanks, your questions helped me realize it on my own.!
The Fundamental Theorem of Calculus 1 states that the derivative and the integral are inverse operations of each other. This means that if we take the derivative of a function and then integrate it, we will end up with the original function (up to a constant). Conversely, if we integrate a function and then take the derivative of the resulting expression, we will also end up with the original function.
The Fundamental Theorem of Calculus 1 is an essential tool in calculus. It allows us to evaluate definite integrals by finding the antiderivative of a function and plugging in the upper and lower limits of integration. It also helps us to find the derivative of a function using the Fundamental Theorem of Calculus 2, which states that the derivative of an integral of a function is just the original function.
The first part of the Fundamental Theorem of Calculus 1 states that if we have a continuous function f(x) on the interval [a, b], then the function g(x) defined as the integral of f(x) from a to x is continuous on [a, b] and differentiable on (a, b). The second part states that the derivative of g(x) is equal to f(x).
The Fundamental Theorem of Calculus 1 deals with the relationship between derivatives and integrals, while the Fundamental Theorem of Calculus 2 deals with finding derivatives of definite integrals. In other words, the first theorem relates the two operations, while the second theorem uses integration to find the derivative of a function.
The Fundamental Theorem of Calculus 1 is important because it provides a powerful connection between differentiation and integration. It allows us to solve problems involving both operations, and it also forms the basis for many other important theorems and techniques in calculus. Without the Fundamental Theorem of Calculus 1, many concepts and applications in calculus would not be possible.