## anonymous one year ago I have a question about question 2A-11 from unit 2 Applications of Differentiation-Approximation and Curve Sketching. It is attached as a word document. Any help would be appreciated. Thanks!

1. anonymous

2. phi

Can you make your file a pdf or png? Meanwhile, I assume the attached is the problem? The "generic formula" for a 2nd order approximation is $f(x) \approx f(x_0) + f'(x_0) (x-x_0)+\frac{f''(x_0)}{2} (x-x_0)^2$ this tells us how to approximate f(x) when x is "near" x0. notice we only need values for f(x0) (and its derivatives). Generally we pick x0 where f(x0) is easy to find, and then use this formula to approximate f(x) nearby.

3. phi

In this problem, p is a function of v : $$p(v)= c v^{-k}$$ v plays the role of x. $$v= v_0 + \Delta v$$ and thus $$\Delta v= (v- v_0)$$ we will need $$\frac{dp}{dv}$$ and $$\frac{d^2 p}{dv^2}$$ $p= c v^{-k} \\ \frac{dp}{dv} = c \frac{d}{dv} v^{-k} = -k c v^{-(k+1)}$ and $\frac{d^2 p}{dv^2}= k(k+1) c v^{-(k+2)}$

4. phi

now plug these expressions into $f(x) \approx f(x_0) + f'(x_0) (x-x_0)+\frac{f''(x_0)}{2} (x-x_0)^2 \\ p(v) \approx c\ v_0^{-k} -k c v_0^{-(k+1)} \Delta v +\frac{k(k+1)}{2} v_0^{-(k+2)}\Delta v^2$ we can factor out the leading term, and write this as $p(v_0+\Delta v) \approx \frac{c}{v_0^k} \left( 1-k \frac{\Delta v}{v_0}+\frac{k(k+1)}{2} \left( \frac{\Delta v}{v_0}\right)^2\right)$

5. anonymous

I've attached the pdf. Let me know if you think your previous response answers my question. Thanks.

6. phi

Your solution is OK. if you divide (1+ax + bx^2) into 1 (where a and b are the coeffs) you get 1-ax +(a^2-b)x^2 plus higher order terms of x which we drop with a= k, b= k(k-1)/2 this becomes 1 -k x + k(k+1)/2 x^2 where x is $$\frac{\Delta v}{v_0}$$

7. anonymous

One final question. Can you explain the division to me: 1/(1+ax+bx^2). I tried to do it using the polynomial division algorithm but wasn't able to reach the solution. Thanks.

8. phi

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9. phi

the last term should have x^2 in there... but hopefully you get the idea...

10. phi

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