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(Note this is the kind of exercise I would ask someone who is doing further maths and especially someone MAT/STEP) Sketch the graph of y=sin(1/x)

First of all we want to think about the general behaviour of the function. Does it grow as x grows? Does it oscillate? Does it tend to a limit? We also need to think about if there are any worrying points in...
JT
Answered by James T. • MAT tutor
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[based on MAT 2018 (G)] The curves y = x^2 + c and y^2 = x touch at a single point. Find c.

If helpful draw a diagram to see what is going on, but keep the time in mind! From your sketch it should be apparent that c must be positive and that the single point of contact of the curves is above the x-...
Answered by • MAT tutor
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Let f(x) = 2x^3 − kx^2 + 2x − k. For what values of the real number k does the graph y = f(x) have two distinct real stationary points? (MAT 2017 q1.A)

A point a is a stationary point if and only if f'(a) = 0. Let's compute the derivative: f'(x) = (2x^3 − kx^2 + 2x − k)' = 6x^2 - 2kx + 2. (-k is a constant so it cancells out) f'(x) = 0 iff 6x^2 -2kx + 2 = 0...
JG
Answered by Jan G. • MAT tutor
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If f(x) =x^2 - 5x + 7 what are the coordinates of the minimum of f(x-2)?

The easiest way to do this is to notice that f(x-2) is just f(x) translated by two in the positive x-direction - it is not necessary to calculate what f(x-2) is at any point! So we calculate f'(x) = 2x - 5Se...
JN
Answered by James N. • MAT tutor
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Solve 8^x + 4 = 4^x + 2^(x+2).

Firstly, we want to notice that this question only involves powers of 2^x, ie. (2^x)^3 + 4 = (2^x)^2 + 4(2^x) so we can use the substitution u = 2^x to give u^3 - u^2 - 4u + 4 = 0 The problem (at least at th...
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Answered by Sachin A. • MAT tutor
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