Question

MEDAL AND FAN

Answer 1

fan and medal

Answer 2

@Directrix, @SolomonZelman, @Nnesha, @Luigi0210, @Bibby, @agent0smith, @irishboy123

Answer 3

is it true i have to use ftoc 1?

Answer 4

the derivative of the integral returns the original function

but with some "details"

imagine we found the integral F(x) (whatever it happens to be)
and we put in 2 for the lower limit: F(2). that is now a constant (a fixed value)
so we can ignore the lower limit

Answer 5

I don't get it :(

Answer 6

I'm trying to think of a way to explain this clearly.

Answer 7

do i have to use substitution?

Answer 8

it will take me a few minutes to get my thoughts clear

Answer 9

This is a bit abstract
but by definition
\[ \frac{d}{dx} F(x) = F'(x) \ dx\]
that says the derivative of a function F(x) is F'(x) (i.e. its derivative) times the derivative of its argument x

for example
\[ \frac{d}{dx} \tan(x^2) = \frac{d\tan(x^2) }{dx}\frac{d}{dx} x^2 \]

Answer 10

but what do i have to do with the integration ? :(

Answer 11

Yes, indeed. That makes use of the Chain Rule. Nicely presented.

Answer 12

and by definition, the integral is
\[ \int_a^b F'(x) \ dx = F(b) - F(a) \]

Answer 13

the derivative discards information, as in it just returns the integrand without the limits

Answer 14

and if we take the derivative of the integral, we have your problem
\[ \frac{d}{dx} \int_a^b F'(x) \ dx = \frac{d}{dx} F(b) - \frac{d}{dx} F(a) \\
= F'(b) \ d b - F'(a) \ d a\]
in your case F'(x) matches up with tan(x^2)

and you should get
\[ \tan( (x^4)^2) \frac{d}{dx} x^4 - \tan(4) \frac{d}{dx} 4 \]

Answer 15

the second term is 0. you end up with
\[ 4 x^3 \tan(x^8) \]

Answer 16

it's a bit annoying that \(x\) is used everywhere (in \(\tan( x^2)\) too), but I think you must think of it as a \(u\), and \(dx\) as \(du\). Then @phi gave the solution.

Answer 17

@phi how do you have tan(4)d/dx4

Answer 18

it's \(\tan(2^2)\) : \(\tan(x^2)\) evaluated at \(x=2\).

Answer 19

oh okay i got it, thank you guys so much, can you please help me with another problem?

Answer 20

x=2y^2 is a parabola "to the right".
Draw the other curve on this graph.