Further Mathematics: Calculus Topical Past Paper Questions

Consider the differential equation $ \frac{dy}{dx}=\frac{y-x}{y+x} $, where x, y > 0. It is given that y=2 when x=1. (a) Solve the differential equation, giving your answer in the form $ f(x, y) = 0 $. (b) The graph of $y$ against $x$ has a local maximum between $x=2$ and $x=3$.…

[Maximum mark: 15] (a) Using the substitution $ u = x^{3} $, show that $$ \int\frac{x^{2}}{x^{6}+1}\mathrm{d}x=\frac{1}{3}\arctan\left(x^{3}\right)+\mathrm{constant}. $$ (b) The following diagram shows a sketch of part of the curve $ y = \frac{x^2}{x^6 + 1} $ together with line s…

[Maximum mark: 14] ) Draw slope fields for the following cases for $ -2 \leq x \leq 2 $, $ -2 \leq y \leq 2 $ (i) $ \frac{dy}{dx}=2 $; (ii) $ \frac{dy}{dx}=x+1 $; (iii) $ \frac{dy}{dx}=x-1. $ b) Explain what isoclines tell you about the slope field in each of the following cases,…