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A curve has equation y = x^3 - 48x. The point A on the curve has x coordinate -4. Is A a stationary point?

A stationary point indicates that the gradient at that coordinate is 0. Hence we need to find the gradient of the line, set this to 0, and then substitute our value of x. Find dy/dx. Remember the equation fo...
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Answered by Oliver H. • Maths tutor
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Sketch the graph of x^2 - y^2 = 16

Before attempting to re-arrange the equation, we take an organic approach to looking at asymptotes:1) What happens to y when x=0 and when x -> infinity? What happens to x when y=0 and when y -> infinit...
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Q15 from Senior Mathematical Challenge 2018: A square is inscribed in a circle of radius 1. An isosceles triangle is inscribed in the square. What is the ratio of the area of this triangle to the area of the shaded region? (Requires Diagram))

Radius = 1, therefore diameter = 2Let x be the length of one side of the square.Using Pythagoras,x 2 + x 2 = 2 2 2x 2 = 4x = sqrt(2)Area of isosceles triangle = side of square * half side of square= sqrt(2) ...
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Answered by Thomas H. • Maths tutor
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Solve e^(2x) = 5e^(x) - 6, giving your answers in exact form

Solve e 2x = 5e x - 6 e 2x - 5e x + 6 = 0(e x ) 2 - 5e x + 6 = 0 (Let k = e x ) k 2 - 5k + 6 = 0(k-3)(k-2) = 0k = 3 k = 2e x = 3 e x = 2x = ln(3) x = ln(3)
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Answered by Thomas H. • Maths tutor
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A curve C is mapped by the equation ( 1+x)(4-x). The curve intersects the x-axis at x = –1 and x = 4. A region R is bounded by C and the x-axis. Use calculus to find the exact area of R.

First draw the curve. Figure out and write the integration problem. Integral 4 -1 (1+x)(4-x) dx.Expand integral 4 -1 4 + 3x - x 2 dx.= 4 -1 [4x + 3x 2 /2) -x 2 ] = 16 + 30 - 64/3 - (-4 + 3/2 + 1/3)= 125/6
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Answered by Vishesh D. • Maths tutor
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