Question

find the equation for y, given the derivative and the indicated point on the curve. dy/dx = 2(x-1) ; point (3,2)

Answer 1

Please help??

Answer 2

So is this a differential equation, or are you looking for some tangent line \(y\) to (3,2)?

Answer 3

Because you can really do either.

Answer 4

I am pretty sure it is a differential equation

Answer 5

\[\frac{dy}{dx}=2(x-1)\]
What we have here is called a "separable ordinary differential equation." By separable, we mean that, given an equation of the form \(\dfrac{dy}{dx}=f(x,y)\), we can split the function \(f\) into what I'll call component functions. These component functions are factors of \(f\), i.e. we can write \(\dfrac{dy}{dx}=g(x)h(y)\).

You don't have to know all that, though. Here's what you do:
\[dy=2(x-1)~dx\]
Integrate both sides:
\[\int dy=\int2(x-1)~dx\\
y\color{red}{+C_y}=2\left(\frac{x^2}{2}-x\color{blue}{+C_x}\right)\\
y=x^2-2x+C\]
Both C's are constants, so you can combine them into one general constant \(C\).

Now, given the point (3,2), you can solve for \(C\).

Answer 6

Ok so now I insert 3 into the x's right?

Answer 7

2 = 3^2 - 2(3) + C
2= 3 + C

so C= -1?

Answer 8

yes, correct