Anti-derivative question - Calculus I

[qwertyuiop]

This is the only other question i do not get on practice sheet.
Find the function 'F' for the family of functions that satisfy the conditions given.

1.) f ' (x) = 4 cos x; point (0, 3)

2.) f ' (x) = 3e^x + x; point (0, 4)

honestly here I have no clue on what to ask or on how to even begin these types of problems, like I did on the last post. I do not understand this at all. Help^2
please? and thank you for taking the time to look at my post.

 

[stapel]

Note: This is my second posting of this reply. The other one wasn't displaying in the "Replies" count in the forum listing. Dunno why not.

Find the function 'F' for the family of functions that satisfy the conditions given.

1.) f ' (x) = 4 cos x; point (0, 3)

2.) f ' (x) = 3e^x + x; point (0, 4)

honestly here I have no clue...

Have you done "anti-derivatives" at all? If so, then that is what you're looking for here. If not, then I can't think how they're expecting you to do this. :shock:

 

[Ishuda]

This is the only other question i do not get on practice sheet.
Find the function 'F' for the family of functions that satisfy the conditions given.

1.) f ' (x) = 4 cos x; point (0, 3)

2.) f ' (x) = 3e^x + x; point (0, 4)

honestly here I have no clue on what to ask or on how to even begin these types of problems, like I did on the last post. I do not understand this at all. Help^2
please? and thank you for taking the time to look at my post.

Same thing stapel said in different words: The way I would take this is shown by the following example:
Find the function 'F' for the family of functions that satisfy the conditions given
f'(x) = 1; point (0,3)
The family of functions that satisfy f'(x) = 1 is the family of functions f(x) = \(\displaystyle \int 1 dx\) = x + c, where c is a constant. The function F belongs to that family and F(0) = 3 so c = 3 and F(x) = x + 3.

 

[markraz]

This is the only other question i do not get on practice sheet.
Find the function 'F' for the family of functions that satisfy the conditions given.

1.) f ' (x) = 4 cos x; point (0, 3)

2.) f ' (x) = 3e^x + x; point (0, 4)

honestly here I have no clue on what to ask or on how to even begin these types of problems, like I did on the last post. I do not understand this at all. Help^2
please? and thank you for taking the time to look at my post.

These are called "velocity to position" problems. Basically the problem is giving you the derivative of the position which is the velocity (That's why it has the little apostrophe) before the f ().

So you have to go backwards by finding the integral to get back to the position. P(x)

Look at the given point (0,3). The 0 is the X value , the 3 is the Y.

So when you do the anti-derivative/integral=

4sin(x)+C

Your anti-derivative/integral is just a function of Y.

So rewrite:

Y=4sin(x)+C


Then plug in the points.

3=4sin(0)+C

solve

sin(0) =0

3= (4)(0)+C

3 = (0+ C)

3= C

so

C=3

and

4sin(x)+C is the anti/derivative

so finally you can write it as follows plugging in the constant:

4sin(x)+3


Steps:
1. Integrate 4 cos x (to go from velocity v(x) back to position p(x)) => Y=4sin(x)+C
2. Plugin X,Y points (0,3) in the solved integral Y=4sin(x)+C (which is now the position p(x) function.)
3. Solve 3= (4)(0)+C
4. Plugin constant for you final answer. 4sin(x)+3

done


Y=3e^x+(x^2/2)+C
4=3e^0+(0^2/2)+C
solve

 

[HallsofIvy]

These are called "velocity to position" problems. Basically the problem is giving you the derivative of the position which is the velocity (That's why it has the little apostrophe) before the f ().

Physicists may call them that! Others (mathematicians, chemists, economists, etc.) use them for a wide variety of problems that have nothing to do with "position" and "velocity"!

 

[markraz]

Physicists may call them that! Others (mathematicians, chemists, economists, etc.) use them for a wide variety of problems that have nothing to do with "position" and "velocity"!

In my Stewart calculus book the problems are specifically labeled "velocity to position". In my AP calculus book they are coincidentally referred to as "velocity to position". See a pattern? Regardless, they don't seem to be much use for economists since they can't get anything right, maybe economists are still utilizing the same 2 sets of constants from these problems?? not sure. Hopefully they are more useful in chemistry or for "mathematicians" with what ever it is they do.

 

[qwertyuiop]

Thank you so much guys for helping me......I talked to my professor and he said that I was supposed to skip these 2 problems because obviously we had not covered them in class (sorry I must not have heard him when he said that, that was my fault and me being the OCD fiend that I am, panicked...hehehehe). We only covered derivatives but, im glad i posted this because now i can be a step ahead of everyone else. Especially now that we are covering this topic in class.
Thanks again for your help.

 

[stapel]

...my professor...said that I was supposed to skip these 2 problems....

It's good that you asked and tried to learn!