Differentiation and its applications – university Year 1 maths test to download as PDF

1. Differentiate \(f(x)=\dfrac{x}{e^{x}+1}\). (2 marks)
2. Find \(\dfrac{dy}{dx}\) for \(y=(\sin x)^x\) with \(0\lt x\lt \pi\). (2 marks)

A farmer has 120 m of fence and uses a straight river as one side of a rectangular pen. The sides at right angles to the river have length \(x\) m.

1. Show that the area is \(A=x(120-2x)\). (1 mark)
2. Find the value of \(x\) that gives the greatest area, and prove it is a maximum. (2 marks)
3. Find the greatest area. (1 mark)

Let \(f(x)=x^3\) on \([0,2]\).

1. State why the mean value theorem applies. (1 mark)
2. Find the value of \(c\) in \((0,2)\) guaranteed by the theorem. (3 marks)

1. Evaluate \(\displaystyle\lim_{x\to0}\dfrac{\sin x-x}{x^3}\). (3 marks)
2. Evaluate \(\displaystyle\lim_{x\to0}\dfrac{1-\cos x}{x^2}\) using the Maclaurin series. (1 mark)

(a) Find the Taylor polynomial of degree 2 of \(f(x)=\sqrt x\) about \(x=4\). (3 marks) (b) Use it to estimate \(\sqrt{4.4}\), and compare with the calculator value \(2.09762\). (1 mark)