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A curve has equation y = x^2 - 7x. P is a point on the curve, and the tangent to the curve at P has gradient 1. Work out the coordinates of P.

The gradient of the tangent at P (=1) will be equal to the gradient of the curve at point P (=1). The gradient at a point on a curve is given by dy/dx (the differential). First, differentiate the equation of...
ST
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Give the general solution to the Ordinary Differential Equation: (dy/dx) + 2y/x = 3x+2

It can first be observed that this differential equation is linear, so we can solve it by multiplying the whole equation by the integrating factor. As there is no coefficient in front of the dy/dx term, we d...
VG
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Find and describe the stationary points of the curve y = x^2 + 2x - 8

Stationary points occur when the derivative is = 0Derivative: 2x + 2 = 0, so a stationary point occurs when x = -1y = 1 + 2 - 8 = -5Second derivative = 2Therefore, the stationary point (-1,2) is a minimum
MN
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Prove by induction that for all positive integers n , f(n) = 2^(3n+1) + 3*5^(2n+1) , is divisible by 17.

Prove the base caseFor n=0, f(0)= 2 + 15 = 17Therefore, when n=0, f(n) is divisible by 17, base case is true2. Assume true for any integerAssume for n=k, f(k) is divisible by 17f(k)= 2 3k+1 + 3(5 2k+1 ) ;3. ...
SE
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Solve x^(-1/4) = 0.2

Get both sides in similar forms to make it easier to solve.1/x 1/4 = 1/5x 1/4 = 5x = 5 4 = 625
SE
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