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Given that p≥ -1 , prove by induction that, for all integers n≥1 , (1+p)^k ≥ 1+k*p.

First of all, we need to show that the statement is true for the base case n=1. For this case the expression becomes: 1+p≥1+p, which is clearly true as both sides are equal and hence solve the inequality. Ne...
EB
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It is given that f(x) = 2sinhx+3coshx. Show that the curve y = f(x) has a stationary point at x =-½ ln(5) and find the value of y at this point. Solve the equation f(x) = 5, giving your answers exactly

1.Differentiating: f'(x)= 2cosh(x)+3sinh(x) At a stationary point, we know f'(x)=0. Therefore 2cosh(x)+3sinh(x)=0. (easy to forget that unlike nromal trig there is no change in sign) Rearranging gives tanh(x...
SB
4798 Views

Differentiate arctan of x with respect to x.

Say arctan of x is equal to a value y. Now take the tangent of both sides; x now equals tan of y! Easy from here, differentiate both sides wrt x. Now 1 equals sec^2y dy/dx, and you can rearrange to find dy/d...
AM
4104 Views

Find the vector equation of the line of intersection of the planes 2x+y-z=4 and 3x+5y+2z=13.

The line of intersection will be parallel to both planes. We can write the equations of the two planes in 'normal form' as r.(2,1,-1)=4 and r.(3,5,2)=13 respectively. We can then read off the normal vectors ...
RS
21421 Views

Prove that 1+4+9+...+n^2 = n(n+1)(2n+1)/6.

Consider the case n=1. Then 1(1+1)(2*1+1)/6 = 1 = 1^2 and so the claim is true for n=1. Suppose the claim is true for some positive integer n, so that 1+4+9+...+n^2 = n(n+1)(2n+1)/6. Then by the inductive hy...
JR
21404 Views