Solving for a derivative of an integral

[dnt]

the question is find a function, f, and a value, a, such that:

2 times the integral from a to x of f(t)dt = 2sinx - 1

(sorry i don't know how to make it look nice)

well the first thing i did was divide by 2, then take the derivative of each side:

f(x) - f(a) = cos x

and now I am stuck. am i correct so far? where do i go from here? thanks.

 

[Pseudo Statistic]

You made a mistake.
If you took the derivative of both sides dx, why would F(a) be differentiated to f(a)? Wouldn't it be 0?

 

[d_leet]

the question is find a function, f, and a value, a, such that:

2 times the integral from a to x of f(t)dt = 2sinx - 1

(sorry i don't know how to make it look nice)

well the first thing i did was divide by 2, then take the derivative of each side:

f(x) - f(a) = cos x

and now I am stuck. am i correct so far? where do i go from here? thanks.

Why did you divide by 2, and what happened to that factor of 2 when you took the derivative? You also might want to recheck your work taking the derivative, do remember what the second fundamental theorem of calculus says?

 

[dnt]

You made a mistake.
If you took the derivative of both sides dx, why would F(a) be differentiated to f(a)? Wouldn't it be 0?

i thought it would be the derivative of f at a (whatever value that may be...i don't know because i don't know what a is)

 

[dnt]

Why did you divide by 2, and what happened to that factor of 2 when you took the derivative? You also might want to recheck your work taking the derivative, do remember what the second fundamental theorem of calculus says?

i thought i should divide by 2 just to get rid of it. on the right side it becomes sin x - 1/2. after i take the derivative of that i get cos x.

 

[d_leet]

i thought i should divide by 2 just to get rid of it. on the right side it becomes sin x - 1/2. after i take the derivative of that i get cos x.

Yes, but what you do to one side of the equation you must do to the other side as well and so I asked what happened to the factor of 1/2 that would have been multiplying the integral?

$$\frac{d}{dx}\int_{a}^{x} f(x) dx\ = \ f(x)$$

That is very basically what the second fundamental theorem of calculus says and is what you need to use in this problem.

 

[Pseudo Statistic]

i thought it would be the derivative of f at a (whatever value that may be...i don't know because i don't know what a is)

Nope.
Think about it this way...
A definite integral yields a numerical answer... UNLESS one of the limits of integration is a variable that you're dealing with-- could be x OR a.
In this case, you have taken the derivative of both sides with respect to x-- thus treating x as the variable, not a...
In any case, F(a) does not have anything to do with x... differentiating it just kills it; thus, it disppears...

 

[dnt]

i guess i don't get how you take the derivative of an integral. the integral from say, a to x, is the area under the curve, right? what does the derivative of an area mean?

 

[Pseudo Statistic]

Well, according to the fundamental theorem of calculus, derivative and integrals are opposites..
Say you have a function f..
Take the integral, you have a function F.
Differentiate F, you get f again!
That's it, really...

 

[dnt]

Well, according to the fundamental theorem of calculus, derivative and integrals are opposites..
Say you have a function f..
Take the integral, you have a function F.
Differentiate F, you get f again!
That's it, really...

right, so then when i take the derivative of the integral from a to x of f(t) dt, shouldn't it just be f(x) - f(a)?

 

[dnt]

and i still don't get how I am suppose to take the derivative of a function when i don't even know what it is.

FTC has always been a very tough topic for me to comprehend. still trying to understand all of this.

 

[d_leet]

right, so then when i take the derivative of the integral from a to x of f(t) dt, shouldn't it just be f(x) - f(a)?

If we define the integral of f(x) to be F(x) then if we take the integral from a to x we get F(x) - F(a), but when we differentiate with respect to x we treat a as a constant so the derivative of that integral will be F'(x) which is equal to f(x) because of how we defined F(x), F(a) will just be some consatnt that when we differentiate just disappears since it will be zero. Take a look at my previous post for a rather informal statement of the second fundamental theorem of calculus.

 

[a_ramage_1989]

A worked solution

The attached document has a worked solution to your problem. The fundamental theorem of calculus makes more sense if you think of integrals as anti-derivatives, as they were first taught to me.