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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 separate the variables ...
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Answered by Maxim B. • Maths tutor
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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 = x 2 - 10x , we want to change the form of this equation from x 2 + ax + (a/2) 2 into ( x + a/2 ...
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Answered by James P. • Maths tutor
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How do you integrate ln(x) with respect to x?

Rewrite ln(x) as 1 ln(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, and by integrating v' we get...
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Answered by Anthony G. • Maths tutor
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What is 2/3 + 1/4?

When adding or subtracting fractions, we must always start by putting the fractions over the same denominator , that is the number on the bottom half of the fraction. To do this, we need to find a number whi...
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Answered by Catherine C. • Maths tutor
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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). Equivalently, you ma...
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Answered by Bromlyn C. • Maths tutor
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