Paper 1
a) Evaluate \(\displaystyle \lim_{x\to4}\frac{\sqrt{x+5}-3}{x-4}\).
[4 marks]b) Let \(f(x)=ax+2\) for \(x<1\), and \(f(x)=x^2+b\) for \(x\ge1\). Find the relationship between the constants \(a\) and \(b\) required for \(f\) to be continuous at \(x=1\).
[4 marks]c) Find the vertical and horizontal asymptotes of \(g(x)=\frac{2x+1}{x-2}\).
[3 marks]d) Using first principles, find the derivative of \(f(x)=x^2-3x\).
[4 marks]a) Differentiate \(f(x)=(x^2+1)e^{-x}\).
[4 marks]b) For the implicit curve \(xy^2+x^2=5\), find \(\frac{dy}{dx}\).
[4 marks]c) For \(f(x)=x^4-6x^2+3\), find \(f''(2)\).
[3 marks]d) Use linear approximation about \(x=9\) to estimate \(\sqrt{9.3}\).
[4 marks]a) A rectangle has side lengths \(x\) and \(y\), with \(x+y=12\). Find the side lengths that maximise its area.
[4 marks]b) A sphere has radius \(r\) increasing at \(0.5\) units per second. How fast is the volume increasing when \(r=2\)?
[4 marks]c) For \(f(x)=x^3-6x^2+1\), find and classify the local stationary points.
[4 marks]d) Use L'Hopital's rule to evaluate \(\displaystyle \lim_{x\to0}\frac{e^{2x}-1}{x}\).
[3 marks]a) Evaluate \(\displaystyle \int 2x(x^2+4)^5\,dx\).
[3 marks]b) Evaluate \(\displaystyle \int x\cos x\,dx\).
[4 marks]c) Evaluate \(\displaystyle \int\frac{5x+6}{x^2-4}\,dx\), for \(x\ne\pm2\).
[4 marks]d) Differentiate \(\displaystyle A(x)=\int_1^{x^2}\ln(1+t^2)\,dt\).
[4 marks]a) Find the area enclosed by \(y=x\) and \(y=x^2\).
[4 marks]b) Find the volume formed by rotating the region under \(y=\sqrt{x}\) from \(x=0\) to \(x=4\) about the \(x\)-axis.
[4 marks]c) Determine whether \(\displaystyle \int_2^\infty \frac{1}{(x-1)^2}\,dx\) converges, and evaluate it if it does.
[4 marks]d) Use the ratio test to determine whether \(\displaystyle \sum_{n=1}^{\infty}\frac{3^n}{n^2}\) converges.
[3 marks]a) For \(f(x,y)=x^2y+3xy^2\), find \(\nabla f(1,2)\).
[4 marks]b) Using the function from part (a), find the directional derivative at \((1,2)\) in the unit direction \(\mathbf u=(3/5,4/5)\).
[3 marks]c) Let \(z=f(x,y)=x^2y+3xy^2\), where \(x=t\) and \(y=t^2\). Find \(\frac{dz}{dt}\) at \(t=1\).
[4 marks]d) Solve \(\frac{dy}{dx}=2xy\) with initial condition \(y(0)=3\).
[4 marks]