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A cubic curve has equation y x3 3x2 1. (i) Use calculus to find the coordinates of the turning points on this curve. Determine the nature of these turning points.

y′=3x2 −6x use of y′ = 0 0= 3x^2 - 6x 0= 3x(x-2) therefore either x=0 or x=2 when x=0 y=1, when x=2 y=-3 (0, 1) or (2, −3) y''=6x-6 when x=0 y''=6(0)-6 = -6 as y'' is negative this means at x=o the curve is ...
CB
Answered by Charlotte B. • Maths tutor
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How do I show that (cos^4x - sin^4x) / cos^2x = 1 - tan^2x

Start with the LHS: (cos^4x - sin^4x) / cos^2x Recognise the difference of two squares on the top line, which simplifies to (cos^2x - sin^2x)(cos^2x + sin^2x): (cos^2x - sin^2x)(cos^2x + sin^2x) / cos^2x Bec...
JM
Answered by Jack M. • Maths tutor
20309 Views

Why is the integral of a function the area?

Say you have two functions, A and B, and A is the gradient of B. That is to say, A is as high as B is steep (at any point). Equally, A is as high as the rate of change of B (at any point). Now let’s say we w...
MC
Answered by Madison C. • Maths tutor
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integrate cos^2(x)*sin(x)

integral of cos 2 (x)*sin(x) we notice that we can use the spot the ball method here: Guessed integral: cos 3 (x) Differentiated Guessed Integral: -3sin(x)cos 2 (x), therefore our solution is (-1/3)*cos 3 (x)+c
CP
Answered by Cezar P. • Maths tutor
9152 Views

Solve the equation: 5^(2x+1) = 7, giving your answer correct to four decimal places.

First, we take logs of both sides: log(5^(2x+1))=log(7) Now, using the 3rd law of logs (index rule; using the power as the coefficient), we get: (2x+1)log(5)=log(7) i.e. 2x+1 = (log(7))/(log(5)) = 1.20906......
MS
Answered by Mohamad S. • Maths tutor
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