integral

**viet** · November 11th 2006, 09:17 PM

consider the function f(x) whose second derivative is f''(x) = 3x+2sin(x)
if f(0)=4, f'(x)=4. what is f(2) = ?

heres what i did
$f'(x) = \int (3x+2sin(x)$
$= \frac{3x^2}{2} - 2cos(x) + C1$

$f(x) = \int \frac{3x^2}{2} - 2cos(x) + C1$
$= \frac{x^3}{2} + C2 - 2sin(x) + C2$

**srinivas** · November 11th 2006, 09:54 PM

i thik something is wrong how can a function which has sin o cos i it give the derivative as 4 that is a constant?
it ought to be a straight line for that

**CaptainBlack** · November 12th 2006, 12:14 AM

Originally Posted by srinivas

i thik something is wrong how can a function which has sin o cos i it give the derivative as 4 that is a constant?
it ought to be a straight line for that

Because it is a typo in the intial conditions. Presumably they shout be:

f(0)=4, f'(0)=4.

RonL

**CaptainBlack** · November 12th 2006, 12:19 AM

Originally Posted by viet

consider the function f(x) whose second derivative is f''(x) = 3x+2sin(x)
if f(0)=4, f'(x)=4. what is f(2) = ?

heres what i did
$f'(x) = \int (3x+2sin(x)$
$= \frac{3x^2}{2} - 2cos(x) + C1$

$f(x) = \int \frac{3x^2}{2} - 2cos(x) + C1$
$= \frac{x^3}{2} + C2 - 2sin(x) + C2$

Last line should read:

$= \frac{x^3}{2} - 2\sin(x) + C1\ x + C2$

RonL

**Soroban** · November 12th 2006, 07:14 AM

Hello, viet!

I agree with the Captain: there must be a typo in the problem.
I'll assume it is supposed to be: $f'(0) = 4$

Consider the function $f(x)$ whose second derivative is: $f''(x) \:= \:3x+2\sin x$
If $f(0) = 4,\;f'(0) = 4$ , find $f(2)$

We have: . $f'(x)\;=\;\int \left(3x + 2\sin x\right)\,dx \;= \;\frac{3}{2}x^2 - 2\cos x + C_1$

Since $f'(0) = 4$ , we have: . $\frac{3}{2}(0^2) - 2\cos0 + C_1\:=\:4\quad\Rightarrow\quad C_1 = 6$

. . Hence: . $f'(x)\:=\:\frac{3}{2}x^2 - 2\cos x + 6$

We have: . $f(x)\;=\;\int\left(\frac{3}{2}x^2 - 2\cos x + 6\right)\,dx \;=\;\frac{1}{2}x^3 - 2\sin x + 6x + C_2$

Since $f(0) = 4$ , we have: . $\frac{1}{2}(0^3) - 2\sin0 + 6(0) + C_2\:=\:4\quad\Rightarrow\quad C_2 = 4$

. . Hence: . $\boxed{f(x) \;= \;\frac{1}{2}x^3 - 2\sin x + 6x + 4}$

Therefore: . $f(2) \;= \;\frac{1}{2}(2^3) = 2\sin2 + 6(2) + 4$ . . . etc.

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