Question

Can anyone explain me the fundamental theorem of calculus? (#2)

Answer 1

I understand the simple one of F(b)-F(a) but i cldnt understand the second one

Answer 2

Hmm
\[\int_a^b f(x) = \left[ F(x) \right]_a^b = F(b) - F(a)\]
Hmm That is it Right?

Answer 3

ya but not that one i need nteh other one

Answer 4

It basically just explains that integration is the opposite of differentiation, hence the term antiderivative. There's not much to it really.

Answer 5

well can u give me an example?

Answer 6

Literally any integration you did at say, A level shows it. To be perfectly honest, I'm sure you just understand it, just the formal representation is throwing you. Have a go with a simple linear expression on a simple interval.

Answer 7

ok thanks

Answer 8

I dont get the theorem at all. I am totally confused

Answer 9

One of them is easier than the other. Or maybe one is expalined better in my book!!

Answer 10

They didnt provide any examples so i cldnt understand it

Answer 11

here's an example that might help. Suppose a rocket takes off and accelerates at a constant rate for 60 seconds of 100 m/s^2. So its velocity at time t is
v(t) = 100t. How far does the rocket travel in 60 seconds?

Answer 12

The answer is the integral of the velocity
\[ \int_0^{60} v(t) \ dt = \int_0^{60} 100t \ dt \]

Answer 13

ok

Answer 14

If you draw the graph of v(t) vs. times t, then this integral is the area under the curve of course from t=0 to 60.

Answer 15

Now, what is the value of that integral? How do you evaluate it?

Answer 16

F(t)=50t^2+C

Answer 17

right. and how did you get 50t^2 ?

Answer 18

F(60)-F(0)

Answer 19

well i just did it in my head

Answer 20

on what basis?

Answer 21

2x=100

Answer 22

in other words, how do you know that is the antiderivative?

Answer 23

well i add one to the power and i just do it inmy head

Answer 24

I need to see wait a sec

Answer 25

i add one to the power and play around with the coefficient.
I was never taught a way so i just play around

Answer 26

but how do you know you have the right coefficient? what property are you checking or relying on to find it?

Answer 27

I take the power of the antiderivative and multiply it by x and it equals the coefficient of the derivative so for example
2x=100
x=50 so the x is the coefficient

Answer 28

Ok. Well, the reason for that is, if we write as you have
\[ F(x) = \int_0^x 100t \ dt \]
then what you're relying on in fact--even if you don't explicitly realize it--is that
\[ \frac{dF}{dx} = 100x \]

Answer 29

In other words, \( F(t) = 50t^2 + C \) is exactly right because F has the property that
\[ dF/dt = 100t \]

Answer 30

ya i seee

Answer 31

This is the fundamental theorem of calculus, FTC.

Answer 32

ya but then there is another part to it

Answer 33

I HATE MY MATH BOOK

Answer 34

The FTC says that if
\[ F(x) = \int^x_c f(t) \ dt \]
then
\[ \frac{dF}{dx} = f(x) \]
This is how we find antiderivatives and solve integrals.

Answer 35

What is the second part now you're referring to?

Answer 36

oh yes that is what i was referring to wait let me just process that

Answer 37

can i give you an example and you will help me solve it???

Answer 38

sure

Answer 39

k wait a sec i am just gonna upload it

Answer 40

does it need to be a pdf?

Answer 41

or can it be a doc document

Answer 42

take a screen shot of it and upload the picture

Answer 43

I need help with number 6

Answer 44

long story, but I can't read .docx documents on this computer

Answer 45

k wait i will save it as pdf then

Answer 46

\[ \frac{d \ }{dx} \int_2^x \ln(t^2+1) \ dt \] yes?

Answer 47

the upper limit is x and the lower limit is 2

Answer 48

yup :D

Answer 49

Well, define F(x) as before:
\[ F(x) = \int_2^x \ln(t^2 + 1) \ dt \]

Answer 50

By the Fundamental Theorem of Calculus, what is
\[ \frac{dF}{dx} \]
equal to?

Answer 51

what isi the anti derivative of ln(t^2+1)

Answer 52

Ah, that's the beauty of this. You don't actually need to explicitly calculate F at all. But scroll, up and see what \( dF/dx \) is equal to.

Answer 53

it is equal to to ln(t^2+1)

Answer 54

Careful with variables. \[ dF/dx = \ln(x^2 + 1) \] yes.

Answer 55

the variable t in the integral is a 'dummy' variable here in as much as it could have been anything as it doesn't appear in the final answer; it's just a piece of notational convenience that we call it something.

Answer 56

oh but here the x refers to all true values

Answer 57

oh man my comp is becoming slower

Answer 58

I have no idea what 'true values' means and it's not a term in calculus. But notice we defined a function F with a variable x. That's why we're differentiating wrt x and why dF/dx in turn is also a function of x.

Answer 59

ya i make up my own words so just forget abt that!! LOL

Answer 60

OHHHHHHHHH i see

Answer 61

but what abt 2 the lower limit

Answer 62

Here's another for you. What is
\[ \frac{d \ }{dx} \int_{15\pi}^x \sin(t^2)/t \ dt \]

Answer 63

sinx/x

Answer 64

y are we ignoring the limits?

Answer 65

before we do the other example I just gave you, the #6 integral is equal to F(x) - F(2)

Hence when you differentiate it, you have dF/dx minus d(F(2))/dx. But F(2) is a number, a constant and hence its derivative is zero.

Answer 66

So to be super explicit, define a function \( i(x) \) for integral, by
\[ i(x) = \int_2^x \ln(t^2+1) \ dt \]

Answer 67

can we start a new thread it is way too

Answer 68

can we start a new thread it is way too

Answer 69

Then i(x) = F(x) - F(2), where F is an antiderivative of ln(t^2+1)

Answer 70

slow

Answer 71

Hence the derivative of the integral wrt x is
\[ \frac{di}{dx} = \frac{d \ }{dx} \left( F(x) - F(2) \right) = \frac{d \ }{dx} F(x) - \frac{d \ }{dx} F(2) \]

Answer 72

Now, dF/dx is by the FTC the integrand, ln(x^2 +1). But d/dx of F(2) is zero because F(2) is a constant.

Answer 73

oh i see

Answer 74

Therefore
\[ \frac{d\ }{dx} \int_2^x \ln(t^2+1) \ dt = \ln(x^2+1) \]

Answer 75

Now go back to the new question I gave you. What's the answer and be careful.

Answer 76

ok but can we do it on a new postcuz it takes forever for me to type??

Answer 77

sure