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Find the equation of the straight line passing through the origin that is tangent to the curve y = ln(x).

Firstly, recognise the relevant equations. The two functions are y = mx and y = ln(x). As the straight line is a tangent, we know that at a certain point x₀, the functions and ...

Answered by Paul S. • Maths tutor

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A stationary point of inflection implies a second derivative of 0, does this work the other way around?

No, and we may take a counterexample to see why. If y=x^5+5/3x^4, dy/dx=5x^4+20/3x^3, d2y/dx2=20x^3+20x*2=20x^2(x+1). Setting this to 0 will give us the candidates for a POI, but not all these numbers wil...

Answered by Toby R. • Maths tutor

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Integrate x*(5e^x)

Using integration by parts let U=x therefore du=dx.Let dv=5e^x therefore v=5e^x.answer = x*5e^x - integral(5e^x) = (x-1)(5e^x)

Answered by Maximillian S. • Maths tutor

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Differentiate y=(5x^4)cos(2x)

Using chain and product rule.dy/dx=(20x^3)cos(2x) - (10x^4)sin(2x)

Answered by Maximillian S. • Maths tutor

4350 Views

A matrix M has eigenvectors (3,1,0) (2,8,2) (1,1,6) with corresponding eigenvalues 1, 6, 2 respectively. Write an invertible matrix P and diagonal matrix D such that M=PD(P^-1), hence calculate M^5.

Without even knowing M, the candidate can calculate M^5. This will follow from the fact that P is the matrix consisting of the eigenvectors of M as columns, and D will have the eigenvalues (in matching co...

Answered by Cameron B. • Maths tutor

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