Integration: Finding a Function that satisfies given f(t) conditions

[rcoggin]

Hi all,

This is my first time posting in here, so I'm hoping I can find some help with this one problem that has been giving me trouble. I am currently wrapping up a Calculus I class for college, and have completed all required assignments (even the ones that are due two weeks from now), yet I am completely stuck on this problem. I'm hoping that I am missing something simple here, and that my brain has just decided to lock itself out of a common sense evaluation of what I need to do, but here goes.

It is simpler to just post a screenshot of my work thus far - we have to show all work via MathType as opposed to written out on paper. I am good with taking the integrals of the function, that's not the problem.. the problem is that I end up with two variables of the constant C (C1t and C2) in the final f(t) function and neither of the values they give eliminates either of the C's so I can solve for the other. So this has really become an algebra problem I suppose..

Please disregard my work past Step 3, part a, as this is where I made errors prior.. everything up to that point should be correct.

The examples in the book are far more straightforward and give at least one f(0) condition that allows you to easily solve for one of the C's (as it sets all other terms to zero) but in this case this does not work. I end up with two variables of the Constant C and I am unable to figure out how to solve for them to get the right value for both f(t) and f'(t).

Any help is greatly appreciated!!

Rob

 

[stapel]

I was with you through this line:

. . . . .\(\displaystyle \mbox{3.a: }\, 2(1)^2\, +\, (1)^3\, +\, C_1 (1)\, +\, C_2\, =\, 3\)

This simplifies as:

. . . . .\(\displaystyle 2(1)\, +\, (1)\, +\, C_1\, +\, C_2\, =\, 3\)

. . . . .\(\displaystyle 2\, +\, 1\, +\, C_1\, +\, C_2\, =\, 3\)

. . . . .\(\displaystyle 3\, +\, C_1\, +\, C_2\, =\, 3\)

Then subtract from either side to get:

. . . . .\(\displaystyle C_1\, +\, C_2\, =\, 0\)

How did you eliminate the other variable?

Note: Evaluating the other initial condition will create another equation in the two variables, thus setting up a system of linear equations which can be solved for the values of the two integration constants (that is, "the two variables").

 

[rcoggin]

Hi Stapel, thank you for the response. I didn't eliminate the other variable correctly past Step 3, part a... beyond that is where it fell apart. When I did the problem originally I incorrectly assumed it was giving me two different function values that could be used to solve for the C in the f(t) function; however as I was reviewing my work before turning it in I noticed that in fact they are asking to find the values of C that satisfies both given f(t) conditions. So the work beyond that part is not valid, and is the part I am trying to redo.

I did (on a separate piece of scratch paper) arrive at the same conclusion of C1 + C2 = 0, but this doesn't help me get either of the actual values of C.. they could be +/- 3, they could be +/- 2, they could be 1 and -1...

Not sure I follow on your last note... sorry. I feel like I understand the calculus much better than some of the algebra at times! lol

 

[rcoggin]

I evaluated the other condition (-1) and arrived at C2 - C1 = 0.

So.. C2 - C1 = C1 + C2?

 

[Deleted member 4993]

I evaluated the other condition (-1) and arrived at C2 - C1 = 0.

So.. C2 - C1 = C1 + C2?

So you have two equations:

C1 + C2 = 0 .........................(1)

C1 - C2 = 0 .........................(2)

If you add (1) and (2) - what do you get?

 

[rcoggin]

2C2, and the C1 goes away?

 

[Deleted member 4993]

I evaluated the other condition (-1) and arrived at C2 - C1 = 0.

So.. C2 - C1 = C1 + C2?

How did you get that?

That is incorrect - please show work.

 

[Deleted member 4993]

2C2, and the C1 goes away?

Anyway, your conclusion of C1 - C2 = 0 was incorrect! So start before that.

 

[rcoggin]

Whoops, sorry - thank you for spotting that. I input the wrong value in on the other side of the equation. This is what I got for evaluating f(-1) (correctly this time)

f(-1) = 2(-1)^2 + (-1)^3 + C1(-1) + C2
-2 = 2 - 1 - C1 + C2
-2 = 1 - C1 + C2
-3 = C2 - C1

Well.. that's assuming C1 is a positive integer.. if it's negative then it would be + C1... hence my confusion here. Too many unknowns.

 

[Deleted member 4993]

Whoops, sorry - thank you for spotting that. I input the wrong value in on the other side of the equation. This is what I got for evaluating f(-1) (correctly this time)

f(-1) = 2(-1)^2 + (-1)^3 + C1(-1) + C2
-2 = 2 - 1 - C1 + C2
-2 = 1 - C1 + C2
-3 = C2 - C1 .................................(2)

Correct...

Now what do you get when you add (1) and (2)?

 

[rcoggin]

C2 + C1 = 0
+C2 - C1 = -3

I apologize, I'm hoping I am understanding what you are asking me to do... add the resulting equations together? Or the results? 0 + -3 = -3?

 

[rcoggin]

C₂-C₁+C₂+C₁ = -3 + 0
C₂ + C₂ = -3
2C₂ = -3
C₂ = -3/2

This does not seem correct... and this value of C₂ will not give me the correct function values as shown in the problem. I think I misunderstand what you're asking me to do. I apologize, and thank you very much for the help!

 

[Deleted member 4993]

C₂-C₁+C₂+C₁ = -3 + 0
C₂ + C₂ = -3
2C₂ = -3
C₂ = -3/2

This does not seem correct... and this value of C₂ will not give me the correct function values as shown in the problem. I think I misunderstand what you're asking me to do. I apologize, and thank you very much for the help!

What did you then get for C1?

Why do you think so?

 

[rcoggin]

When I solved for C₁ I got positive 3/2.

Using the value of C₂ of -3/2, if C₁ + C₂ = 0, then C₁ + -3/2 = 0, => C₁ is +3/2

Reason this does not work is:

f(-1)=-2 does not equal 2(-1)^2 + (-1)^3 - 3/2
f(1) = 3 does not equal 2(1)^2 + (1)^3 - 3/2

 

[Deleted member 4993]

When I solved for C₁ I got positive 3/2.

Using the value of C₂ of -3/2, if C₁ + C₂ = 0, then C₁ + -3/2 = 0, => C₁ is +3/2

Reason this does not work is:

f(-1)=-2 does not equal 2(-1)^2 + (-1)^3 - 3/2
f(1) = 3 does not equal 2(1)^2 + (1)^3 - 3/2

Where did you get those?

f(t) = t^3 + 2*t^2 + (3/2)*t - (3/2)

f(1) = 1^3 + 2*1^2 + (3/2)*1 - (3/2) = 1 + 2 + 3/2 - 3/2 = 3

f(-1) = (-1)^3 + 2*(-1)^2 + (3/2)*(-1) - (3/2) = -1 + 2 - 3/2 - 3/2 = - 2

Where is the problem????

I think you forgot to include C₁*t while evaluating f(1) and f(-1) [edit]