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Evaluate gf(-5) for the functions f(x)=3x+7, g(x)=3x^2+6x-9

I like to think of gf(-5) as g ( f(-5) ), which helps visualise the order in which you need to calculate the output for each function. First you need to find the answer for f(-5), which you then use as the i...
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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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Find the derivative of e^3x

You keep e 3x and times it by the derivative of the power so 3e 3x .
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