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dessyj1
 one year ago
Calculus 1
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dessyj1
 one year ago
Calculus 1 Question is attached.

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dessyj1
 one year ago
Best ResponseYou've already chosen the best response.1Sorry, my internet connection suddenly slows down when i try to upload a picture.

freckles
 one year ago
Best ResponseYou've already chosen the best response.2lol oh that is a choice \[\lim_{h \rightarrow 0}\frac{1}{k}\ln(\frac{2+h}{2})\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.2I didn't realize it was a multiple choice thingy

freckles
 one year ago
Best ResponseYou've already chosen the best response.2\[F'(x)=f(x) \\ \int\limits_a^b f(x) dx=?\] What does F'=f mean? I will give you a hint that F is the ____derivative of f.

freckles
 one year ago
Best ResponseYou've already chosen the best response.2I will also give you another hint: fundamental theorem of calculus

freckles
 one year ago
Best ResponseYou've already chosen the best response.2for example: how do you evaluate this: \[\int\limits_{1}^{2}x^2 dx\]

dessyj1
 one year ago
Best ResponseYou've already chosen the best response.1Since i know that the derivative of F(x) is itself i can just switch them around in the integral equation right?

freckles
 one year ago
Best ResponseYou've already chosen the best response.2recall: \[\frac{d}{dx}(\frac{x^3}{3})=x^2 \text{ for all } x \\ \ \text{ so } \int\limits_1^2 x^2 dx=[\frac{x^3}{3}]_1^2 =\frac{2^3}{3}\frac{1^3}{3}\] you are given \[\frac{d}{dx}(F)=f \text{ for all } x \\ \int\limits_a^b f dx=[ ? ]_a^b\] and that f is continuous which is another important thing

dessyj1
 one year ago
Best ResponseYou've already chosen the best response.1but wouldnt, f(x) in your example be the same function as its derivative?

freckles
 one year ago
Best ResponseYou've already chosen the best response.2math is case sensitive so when they say F they don't mean f

freckles
 one year ago
Best ResponseYou've already chosen the best response.2so no we aren't given f'=f

freckles
 one year ago
Best ResponseYou've already chosen the best response.2do you know usually to integrate you need to find the antiderivative of the expression that is the integrand ?

freckles
 one year ago
Best ResponseYou've already chosen the best response.2so if we are given F'=f that means the antiderivative of f is F since F'=f

dessyj1
 one year ago
Best ResponseYou've already chosen the best response.1They do not give is the functions. but lets assume the function is e^x

freckles
 one year ago
Best ResponseYou've already chosen the best response.2did you not understand the example I gave above?

dessyj1
 one year ago
Best ResponseYou've already chosen the best response.1i did not understand it

freckles
 one year ago
Best ResponseYou've already chosen the best response.2\[\frac{d}{dx}(\frac{x^3}{3})=x^2 \text{ for all } x \\ \ \text{ so } \int\limits_1^2 x^2 dx=[\frac{x^3}{3}]_1^2 =\frac{2^3}{3}\frac{1^3}{3}\] I started off exactly as your question did

freckles
 one year ago
Best ResponseYou've already chosen the best response.2replace the x^2 with f and replace the x^3/3 with F you can do this since (x^3/3)'=x^2 and (F)'=f

dessyj1
 one year ago
Best ResponseYou've already chosen the best response.1the problem with that is the fact that F and f are not the same for all values of x like the question stated.

freckles
 one year ago
Best ResponseYou've already chosen the best response.2No it is saying F'(x)=f(x) for all x

freckles
 one year ago
Best ResponseYou've already chosen the best response.2Also why do F and f have to be the same? You are definitely not given that.

freckles
 one year ago
Best ResponseYou've already chosen the best response.2F' and f have to be the same for all x

freckles
 one year ago
Best ResponseYou've already chosen the best response.2which they are because when you differentiate (x^3/3) you do get x^2

freckles
 one year ago
Best ResponseYou've already chosen the best response.2http://mathworld.wolfram.com/FundamentalTheoremsofCalculus.html This is just the fundamental theorem of calculus

dessyj1
 one year ago
Best ResponseYou've already chosen the best response.1Okay, I think I have a hard time understanding this because we never learned the fundamental principle of calculus and I am currently studying for the final, so that means my teacher never intended to teach that concept.

freckles
 one year ago
Best ResponseYou've already chosen the best response.2so you guys never cover definite integrals?

dessyj1
 one year ago
Best ResponseYou've already chosen the best response.1We did, but we were given the rules.

freckles
 one year ago
Best ResponseYou've already chosen the best response.2so have you ever done the one or know how to do the one I mentioned before: \[\int\limits_1^2 x^2 dx\]

dessyj1
 one year ago
Best ResponseYou've already chosen the best response.1We were not taught how to evaluate an integral using the definition.

freckles
 one year ago
Best ResponseYou've already chosen the best response.2like how would you tackle that one then?

dessyj1
 one year ago
Best ResponseYou've already chosen the best response.1I can do definite integrals. Why did you choose x^2? as one of the functions?

freckles
 one year ago
Best ResponseYou've already chosen the best response.2I can choose 1 or x is doesn't matter it is just an example

freckles
 one year ago
Best ResponseYou've already chosen the best response.2\[\int\limits_1^2 1 dx=?\] sure we can use whatever function that is continuous and has a continuous derivative

freckles
 one year ago
Best ResponseYou've already chosen the best response.2I just want to see what you do to evaluate something like that if you never been taught the fundamental theorem of calculus

freckles
 one year ago
Best ResponseYou've already chosen the best response.2Like do you not normally find the antiderivative of the integrand ?

freckles
 one year ago
Best ResponseYou've already chosen the best response.2I know but I want to know how you get there

freckles
 one year ago
Best ResponseYou've already chosen the best response.2like what steps do you take

freckles
 one year ago
Best ResponseYou've already chosen the best response.2ok good but isn't the derivative of x, 1?

freckles
 one year ago
Best ResponseYou've already chosen the best response.2by the way you use the fundamental of theorem of calculus above know it or not

freckles
 one year ago
Best ResponseYou've already chosen the best response.2(x)'=1 so you are given this

freckles
 one year ago
Best ResponseYou've already chosen the best response.2and then you said this \[\int\limits_1^2 1 dx=x_1^2 \]

dessyj1
 one year ago
Best ResponseYou've already chosen the best response.1we were that a set of rules to deal with different types of equations such as exponents, and natural logs

freckles
 one year ago
Best ResponseYou've already chosen the best response.2in our question we are given (F)'=f so \[\int\limits_1^2 f dx=F_1^2\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.2I replaced the lower and upper with 1 and 2

freckles
 one year ago
Best ResponseYou've already chosen the best response.2do you not see this yet \[\int\limits_a^b f(x) dx=F(x)_a^b=F(b)F(a) \\ \]? you know the fundamental theorem of calculus even though you are saying you don't know because you just applied it just a sec ago

freckles
 one year ago
Best ResponseYou've already chosen the best response.2you know given the other stuff such as f is continuous and F'=f

dessyj1
 one year ago
Best ResponseYou've already chosen the best response.1this works if you are telling me the integral of f(x) is equal to F(prime)(x)

freckles
 one year ago
Best ResponseYou've already chosen the best response.2no I'm telling you that at all

freckles
 one year ago
Best ResponseYou've already chosen the best response.2\[f(x)=1 \\ F(x)=x +C \text{ where } C \text{ is a constant } \\ \text{ do you not agree that } F'(x)=f(x) ?\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.2Since F'=f then F is the antiderivative of f

dessyj1
 one year ago
Best ResponseYou've already chosen the best response.1I do agree, but i feel like you are making the functions up now.

freckles
 one year ago
Best ResponseYou've already chosen the best response.2\[\int\limits_{a}^{b}f(x) dx \\ \text{ \to integrate this I need the antiderivative of } f \\ \text{ which is } F \\ \text{ since } F'=f \\ \int\limits_a^b f(x) dx=F(x)_a^b\] I'm giving you examples

dessyj1
 one year ago
Best ResponseYou've already chosen the best response.1If were were able to seamlessly communicate this would be easier for me to grasp. But i cannot understand, i will just have to ask my teacher tomorrow. Regardless, thank you for all your help and effort.

freckles
 one year ago
Best ResponseYou've already chosen the best response.2if f(x)=1 the antiderivative let's call it F is F(x)=x+c we will just use F(x)=x since we have a definite integral anyways that we will be working with \[\int\limits_a^b 1 dx=x_a^b=(ba) \\ \int\limits_a^b f(x) dx=F(x)_a^b=F(b)F(a)\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.2like in the example don't you see that (x)'=1 and in the question you are given (F)'=f

freckles
 one year ago
Best ResponseYou've already chosen the best response.2maybe @ganeshie8 can explain it better if you think maybe it is just me

dessyj1
 one year ago
Best ResponseYou've already chosen the best response.1No, i do not think it is just you.

dessyj1
 one year ago
Best ResponseYou've already chosen the best response.1is the antiderivate of F(prime)(x)= F(x) ?

freckles
 one year ago
Best ResponseYou've already chosen the best response.2\[F'=f \\ \int\limits_a^b f(x) dx=\int\limits_a^b F'(x) dx=F(x)_a^b\]

dessyj1
 one year ago
Best ResponseYou've already chosen the best response.1I did not know the mathematical notation for the derivative of a capital function like F(x) was f(x)

freckles
 one year ago
Best ResponseYou've already chosen the best response.2well it could have had a different name like for example maybe they said where g'=f g'=f still means that g is the antiderivative of f

freckles
 one year ago
Best ResponseYou've already chosen the best response.2\[\int\limits_a^b f(x) dx=g(x)_a^b=g(b)g(a) \text{ since } g'=f \\ \text{ aka since } g \text{ is the antiderivative of } f \]

freckles
 one year ago
Best ResponseYou've already chosen the best response.2or I think the way you understand it better was like this: \[\int\limits_a^b f(x)=\int\limits_a^b g'(x) dx \text{ since } f=g' \\ \text{ now } \int\limits_a^b f(x) dx=\int\limits_a^b g'(x)dx=g(x)_a^b =g(b)g(a)\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.2you know assuming g is continuous on [a,b] of course

dessyj1
 one year ago
Best ResponseYou've already chosen the best response.1so the answer to this question is D?

freckles
 one year ago
Best ResponseYou've already chosen the best response.2don't just say that if you don't believe it I will not be upset and @ganeshie8 is here he is awesome at explaining things if you do not feel I did the trick

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.1somehow i feel @dessyj1 you're confusing "anti derivatives" with "definite integrals"

dessyj1
 one year ago
Best ResponseYou've already chosen the best response.1I do understand the question now. The derivative of F(x) was equal to f(x). That means that the integral(indefinite integral, antiderivative) of f(x) is just like asking for the integral of F(prime)(x) which was F(x)

dessyj1
 one year ago
Best ResponseYou've already chosen the best response.1i think antiderivative and indefinite integrals are the same thing.
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