# Thread: Looking for help with my calculus exam practice question

1. ## Looking for help with my calculus exam practice question

Consider f(t) = -(2t + 3) sin( t )+ 5 cos(t)

find constants C1, C2, C3 so that the derivative of g(t) = (C1t + C2) cos(t), + C3 sin(t) is equal to f(t)

2. ## Re: Looking for help with my calculus exam practice question

Did you differentiate $\displaystyle (C_1t+ C_2)\cos(t)+ C_3\sin(t)$? The derivative of $\displaystyle C_1t+ C_2$ is $\displaystyle C_1$, the derivative of $\displaystyle \cos(t)$ is $\displaystyle -\sin(t)$, and the derivative of $\displaystyle \sin(t)$ is $\displaystyle cos(t)$. Of course you will need to use the "product rule" to differentiate $\displaystyle (C_1t+ C_2)\cos(t)$.

3. ## Re: Looking for help with my calculus exam practice question

Thanks for you response. I think I understand how to differentiate it.

So is my C1 = 2, C2 = 3 and C3 = 5?

4. ## Re: Looking for help with my calculus exam practice question

Originally Posted by khoile
Thanks for you response. I think I understand how to differentiate it.

So is my C1 = 2, C2 = 3 and C3 = 5?
No, you have the wrong $C_3$. My recommendation is exactly the same as that of HallsofIvy. I recommend differentiating $g(t)$ and see what you get. Once you do, equate coefficients.

5. ## Re: Looking for help with my calculus exam practice question

What did you get for the derivative?

6. ## Re: Looking for help with my calculus exam practice question

Not 100% sure its correct but this is what I got for the derivative:

g’(t) = (c1t + c2) -sin(t) + cos(t)C1 c3 cos(t)
= -(c1t + c2) sin(t) + (c1 + c3) cost (t)

7. ## Re: Looking for help with my calculus exam practice question

Originally Posted by khoile
Not 100% sure its correct but this is what I got for the derivative:

g’(t) = (c1t + c2) -sin(t) + cos(t)C1 c3 cos(t)
= -(c1t + c2)[sin(t)] + (c1 + c3) cost (t)
\begin{align*}g(x)&=(C_1t+C_2)\cos(t)+C_3\sin(t) \\g'(t)&=C_1\cos(t)-(C_1+C_2)\sin(t)+C_3cos(t)\\&=-(C_1+C_2)\sin(t)+(C_1+C_3)\cos(t) \end{align*}
So $C_1=2, ~C_2=3~\&~C_3=~?$