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Maths
A Level

A tank is filled with water up to the height H0. At the bottom of the tank, there is a tap which is opened at t=0. How does the height of liquid change with time?(Hint: dH/dt is proportional to -H)

This is a typical C4 differential equation question. The same algorithm could be used for a lot of other problems.

First of all, from the hint we have that dH/dt = -kH (1).

We need to separa...

MB
Answered by Maxim B. • Maths tutor
7043 Views

Find the coordinate of the turning point of the curve y = x^2 - 10x + 7, by completing the square

First, we need to complete the square. We take the first part of the equation ignoring the constant ( + 7).  

y = x2 - 10x , we want to change the form of this equation from  x2

JP
Answered by James P. • Maths tutor
9733 Views

How do you integrate ln(x) with respect to x?

Rewrite ln(x) as 1ln(x) then integrate by parts.  The formula for integration by parts is ∫ uv' = uv - ∫ vu', here use u = ln(x) and v' = 1.  By differentiating u we get u' = 1/x, an...

AG
Answered by Anthony G. • Maths tutor
3052 Views

Differentiate the function f(x) = sin(x)/(x^2 +1) , giving your answer in the form of a single fraction. Is x=0 a stationary point of this curve?

The key concepts to apply in this question will be the product and chain rules, namely: if  f(x)=g(x)h(h), then f'(x)=g(x)h'(x) + g'(x)h(x), and if h(x)=u(v(x)), then h'(x)=u'(v(x))v'(x).

Equivalen...

BC
Answered by Bromlyn C. • Maths tutor
4964 Views

Show that: [sin(2a)] / [1+cos(2a)] = tan(a)

We start by expanding out the double trigonometric terms (sin(2a)) using the double angle formula, giving us: [2sin(a)cos(a)] / [1+cos^2(a) - sin^2(a)]. Next we spot that on the denominator (bottom half o...

GH
Answered by George H. • Maths tutor
8852 Views

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