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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0\[\sf \frac{d}{dx}\int_1^{4x} \sqrt{t^2+1}dt = \frac{d}{dx}\left[F(4x)F(1)\right]\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Remember that \(\sf (F(t))' = f(t)\) and \(\sf f(t) =\sqrt{t^2+1}\) So what would \(\sf (F(x))'\) = ?

misty1212
 one year ago
Best ResponseYou've already chosen the best response.0replace the \(t\) in the integrand by \(4x\)

misty1212
 one year ago
Best ResponseYou've already chosen the best response.0then, via the chain rule, multiply the whole thing by 4

misty1212
 one year ago
Best ResponseYou've already chosen the best response.0\[\sqrt{\left(\text{ put 4\(x\) here}\right)^2+1}\times 4\]

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1Anytime, in general: \(\large\color{black}{ \displaystyle \frac{d}{dx}\left[\int_{\rm C}^{g(x)} F'(t)dt\right] }\) \(\large\color{black}{ \displaystyle \frac{d}{dx}\left[F(t)~{\Huge }^{g(x)}_{\rm C}~\right] }\) \(\large\color{black}{ \displaystyle \frac{d}{dx}\left[~F(g(x))F({\rm C})~\right] }\) \(\large\color{black}{ \displaystyle F'(g(x))\cdot g'(x)0=F'(g(x))\cdot g'(x) }\)  Conclusion: \(\large\color{blue}{ \displaystyle \frac{d}{dx}\left[\int_{\rm C}^{g(x)} F'(t)dt\right]=F'(g(x))\cdot g'(x) }\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Sorry for the late reply. So the answer would be C?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0@SolomonZelman How would I solve an integral for an absolute value?

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1You would split the function for x>0 and x<0

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1Oh, then you don't need splitting

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1The function from 0 to 3 is never positive. So you can just write \(\displaystyle \int_{3}^{0}(x+2)dx \)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1because absolute value function on the interval (0,3) is negative (at least not positive). So, it is just a line y=(x+2) at this interval.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1Did I make sense just now?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Yeah, I pretty much got it. Solving atm

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1No seriously, I am terrible at text language. I have hopefully managed to learn proper English, but text language is not for me. I guess it is too informal.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Oh well, lol meant laugh out loud, just in case you didn't know.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1I know "lol" "jk" and some others perhaps, but I think this is pretty much it.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1Anyway, what did you get for your integral ?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I got 3 for the final answer.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1yes, that is right

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Thanks. Wish I could give you another medal.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1I don't really care about medals, I got over 12K:)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0LOL, then you wouldn't mind helping me with another? :p

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1I guess not, if I will be able to:)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0This one should be easy, just can't remember what to do. https://gyazo.com/eb75236776b0079f4699d2e338fe6ec5

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1I refreshed the page, I have to do it again

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Ok, well anyways, if I recall properly the answer should be D.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1\(\large\color{black}{ \displaystyle e^0=1 }\)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1Proof: \(\large\color{black}{ \displaystyle a^0=a^{bb}=\frac{a^b}{a^b}=\frac{\cancel{a^b}}{\cancel{a^b}}=1 }\)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0By the way would I plug in 0,1, and 2 or just 0 and 2?

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1Just plug in 2, and then 0 F(2)F(0)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0This is what I did. \[\frac{ 1 }{ 4}e ^{4x}_{0}^{2}\]

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1yes, correct, that is what you had to do:)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1yes, it should be C.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0One last one please. Actually two but one I already solved. https://gyazo.com/f8561038cb4b1113268276fbaf824c2e https://gyazo.com/e7aea1e2a2b3ca9e81319fcbe8b70cb1

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Should be C for first and A for second?

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1yes, correct for both.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1But, you have to still know that as \(x\rightarrow0\), then \(\ln(x)\rightarrow \infty\).

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Appreciate the help. Just so you know I got the absolute value one wrong, but I see where my mistake is. I added instead of subtracted.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1just in case... and this you can easily demonstrate by defining zero as: \(\large\color{black}{ \displaystyle \lim_{n\rightarrow \infty}\left[10^{n} \right] }\) And then, \(\large\color{black}{ \displaystyle \lim_{n\rightarrow \infty}\left[~~\log_{10}\left(10^{n}\right) \right] = \lim_{n\rightarrow \infty} \left[n\log_{10}(10)\right]=\lim_{n\rightarrow \infty} \left[n\right]=\infty}\)

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1You added what for absolute value/

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0instead of F[0]F[3] I put F[0]+F[3] which...........now that I look at it I did do right no? That would give me 3. :/

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1yes, because F[0]+F[3]

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1I mean because   = +

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1I have to go right now. Be well:)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Bye. Thanks for the help.

SolomonZelman
 one year ago
Best ResponseYou've already chosen the best response.1You are welcome!
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