dessyj1
  • dessyj1
Calculus 1 Question is attached.
Mathematics
  • Stacey Warren - Expert brainly.com
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jamiebookeater
  • jamiebookeater
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dessyj1
  • dessyj1
Sorry, my internet connection suddenly slows down when i try to upload a picture.
dessyj1
  • dessyj1
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freckles
  • freckles
4d?

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dessyj1
  • dessyj1
sorry number 5
freckles
  • freckles
lol oh that is a choice \[\lim_{h \rightarrow 0}\frac{1}{k}\ln(\frac{2+h}{2})\]
freckles
  • freckles
I didn't realize it was a multiple choice thingy
freckles
  • freckles
\[F'(x)=f(x) \\ \int\limits_a^b f(x) dx=?\] What does F'=f mean? I will give you a hint that F is the ____-derivative of f.
freckles
  • freckles
I will also give you another hint: fundamental theorem of calculus
freckles
  • freckles
for example: how do you evaluate this: \[\int\limits_{1}^{2}x^2 dx\]
dessyj1
  • dessyj1
Since i know that the derivative of F(x) is itself i can just switch them around in the integral equation right?
freckles
  • freckles
recall: \[\frac{d}{dx}(\frac{x^3}{3})=x^2 \text{ for all } x \\ \ \text{ so } \int\limits_1^2 x^2 dx=[\frac{x^3}{3}]_1^2 =\frac{2^3}{3}-\frac{1^3}{3}\] 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
  • dessyj1
but wouldnt, f(x) in your example be the same function as its derivative?
freckles
  • freckles
are you saying f=f'?
freckles
  • freckles
math is case sensitive so when they say F they don't mean f
freckles
  • freckles
so no we aren't given f'=f
freckles
  • freckles
do you know usually to integrate you need to find the antiderivative of the expression that is the integrand ?
freckles
  • freckles
so if we are given F'=f that means the antiderivative of f is F since F'=f
dessyj1
  • dessyj1
They do not give is the functions. but lets assume the function is e^x
freckles
  • freckles
did you not understand the example I gave above?
dessyj1
  • dessyj1
i did not understand it
freckles
  • freckles
\[\frac{d}{dx}(\frac{x^3}{3})=x^2 \text{ for all } x \\ \ \text{ so } \int\limits_1^2 x^2 dx=[\frac{x^3}{3}]_1^2 =\frac{2^3}{3}-\frac{1^3}{3}\] I started off exactly as your question did
freckles
  • freckles
replace 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
  • dessyj1
the problem with that is the fact that F and f are not the same for all values of x like the question stated.
freckles
  • freckles
No it is saying F'(x)=f(x) for all x
freckles
  • freckles
Also why do F and f have to be the same? You are definitely not given that.
freckles
  • freckles
F' and f have to be the same for all x
freckles
  • freckles
which they are because when you differentiate (x^3/3) you do get x^2
freckles
  • freckles
x^2=x^2 for all x
freckles
  • freckles
http://mathworld.wolfram.com/FundamentalTheoremsofCalculus.html This is just the fundamental theorem of calculus
dessyj1
  • dessyj1
Okay, 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
  • freckles
so you guys never cover definite integrals?
freckles
  • freckles
covered*
dessyj1
  • dessyj1
We did, but we were given the rules.
freckles
  • freckles
so have you ever done the one or know how to do the one I mentioned before: \[\int\limits_1^2 x^2 dx\]
dessyj1
  • dessyj1
We were not taught how to evaluate an integral using the definition.
freckles
  • freckles
like how would you tackle that one then?
dessyj1
  • dessyj1
I can do definite integrals. Why did you choose x^2? as one of the functions?
freckles
  • freckles
I can choose 1 or x is doesn't matter it is just an example
dessyj1
  • dessyj1
would e^x work then?
freckles
  • freckles
\[\int\limits_1^2 1 dx=?\] sure we can use whatever function that is continuous and has a continuous derivative
freckles
  • freckles
I just want to see what you do to evaluate something like that if you never been taught the fundamental theorem of calculus
freckles
  • freckles
Like do you not normally find the antiderivative of the integrand ?
dessyj1
  • dessyj1
answer is 1
freckles
  • freckles
I know but I want to know how you get there
freckles
  • freckles
like what steps do you take
dessyj1
  • dessyj1
ill draw what i did
dessyj1
  • dessyj1
|dw:1434604206920:dw|
freckles
  • freckles
ok good but isn't the derivative of x, 1?
dessyj1
  • dessyj1
it is
freckles
  • freckles
by the way you use the fundamental of theorem of calculus above know it or not
freckles
  • freckles
(x)'=1 so you are given this
freckles
  • freckles
and then you said this \[\int\limits_1^2 1 dx=x|_1^2 \]
dessyj1
  • dessyj1
we were that a set of rules to deal with different types of equations such as exponents, and natural logs
freckles
  • freckles
in our question we are given (F)'=f so \[\int\limits_1^2 f dx=F|_1^2\]
dessyj1
  • dessyj1
Okay
freckles
  • freckles
I replaced the lower and upper with 1 and 2
freckles
  • freckles
do 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
  • freckles
you know given the other stuff such as f is continuous and F'=f
dessyj1
  • dessyj1
this works if you are telling me the integral of f(x) is equal to F(prime)(x)
freckles
  • freckles
no I'm telling you that at all
freckles
  • freckles
\[f(x)=1 \\ F(x)=x +C \text{ where } C \text{ is a constant } \\ \text{ do you not agree that } F'(x)=f(x) ?\]
freckles
  • freckles
Since F'=f then F is the antiderivative of f
dessyj1
  • dessyj1
I do agree, but i feel like you are making the functions up now.
freckles
  • freckles
\[\int\limits_{a}^{b}f(x) dx \\ \text{ \to integrate this I need the antiderivative of } f \\ \text{ which is } F \\ \text{ since } F'=f \\ \int\limits_a^b f(x) dx=F(x)|_a^b\] I'm giving you examples
dessyj1
  • dessyj1
If 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
  • freckles
if 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 \[\int\limits_a^b 1 dx=x|_a^b=(b-a) \\ \int\limits_a^b f(x) dx=F(x)|_a^b=F(b)-F(a)\]
freckles
  • freckles
like in the example don't you see that (x)'=1 and in the question you are given (F)'=f
freckles
  • freckles
maybe @ganeshie8 can explain it better if you think maybe it is just me
dessyj1
  • dessyj1
No, i do not think it is just you.
dessyj1
  • dessyj1
is the antiderivate of F(prime)(x)= F(x) ?
freckles
  • freckles
yes!!!
freckles
  • freckles
\[F'=f \\ \int\limits_a^b f(x) dx=\int\limits_a^b F'(x) dx=F(x)|_a^b\]
dessyj1
  • dessyj1
I did not know the mathematical notation for the derivative of a capital function like F(x) was f(x)
freckles
  • freckles
well 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
  • freckles
\[\int\limits_a^b f(x) dx=g(x)|_a^b=g(b)-g(a) \text{ since } g'=f \\ \text{ aka since } g \text{ is the antiderivative of } f \]
freckles
  • freckles
or 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
  • freckles
you know assuming g is continuous on [a,b] of course
freckles
  • freckles
I mean f also
dessyj1
  • dessyj1
so the answer to this question is D?
freckles
  • freckles
yep.
dessyj1
  • dessyj1
alright, i get it.
freckles
  • freckles
don'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
  • ganeshie8
somehow i feel @dessyj1 you're confusing "anti derivatives" with "definite integrals"
dessyj1
  • dessyj1
I 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
  • dessyj1
i think anti-derivative and indefinite integrals are the same thing.
ganeshie8
  • ganeshie8
looks perfect!
dessyj1
  • dessyj1
Thank you.

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