3.3.04 AP Calculus Exam 2nd derivative

[karush]

View attachment 9319

Ok not sure if I fully understand the steps but presume the first step would be divide both sides deriving$$\dfrac{dy}{dx}=\dfrac{2x-y}{x+2y}$$offhand don't know the correct answer
$\tiny{from College Board}$

 

[skeeter]

Re: 297 AP Calculus Exam 2nd direvative

$\dfrac{d}{dx} \left(\dfrac{dy}{dx} = \dfrac{2x-y}{x+2y}\right)$

$\dfrac{d^2y}{dx^2} = \dfrac{(x+2y) \left(2 - \frac{dy}{dx} \right) - (2x-y)\left(1+2\frac{dy}{dx}\right)}{(x+2y)^2}$note that $\dfrac{dy}{dx} \bigg|_{(3,0)} = 2$

finish the evaluation ...

 

[karush]

Re: 297 AP Calculus Exam 2nd direvative

are you suggesting a plug in of

$$\dfrac{(x+2y) \left(2 - (2) \right) - (2x-y)\left(1+2(2)\right)}{(x+2y)^2}$$

 

[topsquark]

Re: 297 AP Calculus Exam 2nd direvative

are you suggesting a plug in of

$$\dfrac{(x+2y) \left(2 - (2) \right) - (2x-y)\left(1+2(2)\right)}{(x+2y)^2}$$

... And we know what x and y are...

-Dan

 

[karush]

Re: 297 AP Calculus Exam 2nd direvative

... And we know what x and y are...

-Dan

so with $x=3$ and $y= 0$
$$\dfrac{(x+2y) \left(2 - (2) \right) - (2x-y)\left(1+2(2)\right)}{(x+2y)^2}
=\dfrac{(3+2(0)) \left(2 - (2) \right) - (2(3)-(0))\left(1+2(2)\right)}{((3)+2(0))^2}=-\frac{10}{3}$$

which is D of the selection hopefully

 

[skeeter]

Re: 297 AP Calculus Exam 2nd direvative

so with $x=3$ and $y= 0$
$$\dfrac{(x+2y) \left(2 - (2) \right) - (2x-y)\left(1+2(2)\right)}{(x+2y)^2}
=\dfrac{(3+2(0)) \left(2 - (2) \right) - (2(3)-(0))\left(1+2(2)\right)}{((3)+2(0))^2}=-\frac{10}{3}$$

which is D of the selection hopefully

not D ... take another look at the answer choices

 

[Greg]

\(\displaystyle (x+2y)\frac{dy}{dx}=2x-y\)

\(\displaystyle x\frac{dy}{dx}+2y\frac{dy}{dx}=2x-y\)

\(\displaystyle \frac{dy}{dx}+x\frac{d^2y}{dx^2}+2\left(\frac{dy}{dx}\right)^2+2y\frac{d^2y}{dx^2}=2-\frac{dy}{dx}\)

\(\displaystyle 2+3\frac{d^2y}{dx^2}+8=0\)

\(\displaystyle \frac{d^2y}{dx^2}=-\frac{10}{3}\)

 

[karush]

Re: 297 AP Calculus Exam 2nd direvative

not D ... take another look at the answer choices

then A

https://dl.orangedox.com/6rStfn4eMFHuHvAKuX