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Hard Maths HL Option Calc Questions

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Have had some trouble solving these questions, would appreciate any assistance or suggestions on how to approach the questions. I've tried the substitution y=vx and got to a stage where I am unable to integrate 1/f(y) and then have no idea how to further proceed (for question 7).

Question 8 I just have no idea how to approach.

 

 

hardquestions.jpg

Edited by IB`NOT`ez

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7 a) To solve any separable differential equation, get expressions of one variable on the same side and integrate: N9EhR4q.png. However the target equation is a definite integral so we need to know the bounds of integration. Because (x, v) = (1, k), (e, 2) are part of the solution, then the bounds are [k, 2] for v, and [1, e] for x. So simplify HbPNHRH.png to get 1 on right hand side. 
This works by a change of variables (u-substitution) of f(v) = x and {v = 2 <=> x= e, = k <=> x = 1}. Alternatively, suppose there is y = 1 / f(v) = 1/ x and you are finding the area under the curve 2 different ways. 

I believe it was a typo and this should be sufficient to answer the problem. However, with constant k defined and function f restrained, M0gQpY2.png and replace with any continuous variable it still hold correct.  

Regardless of v or y, no assumption of y = vx is necessary for this step. 

7 b) i) Assume v = y / x
nJWwiG1.png

I am trying to get the equation in form of 7a!

Xt10RzL.png

Finally, at x = e, v = 2 and this is 7a. Here I say that it was a typo because we have f(v), not f(y).

7 b) ii) Calculator

8. Start with

sA64JhQ.png

Edited by kw0573
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