HIGHER MATHEMATICS
Differentiation
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Given that y  12 x 3  8 x , where x  0 , find
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dy
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Differentiate 2 x 2  sin 2 x with respect to x.
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Find the equation of the tangent to the curve y  x 3  x 2  6 x  2 at the point
A(1, 8).
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Find the equation of the tangent to the curve y  x 3  9 x at the point where x  2 .
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Find the equation of the tangent to the curve y 
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at the point P where x  1.
x
Find the equation of the tangent to the curve y  x 2  5 at the point where x  2 .
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x  8 x  34 .
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The gradient of the tangent to the parabola at point P is 4.
Find the coordinates of P.
A parabola has equation y 
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(a) Find the x-coordinates of the stationary points on the graph with equation y  f (x) ,
where f ( x)  x 3  3x 2  24 x .
(b) Hence determine the range of values of x for which the function f is strictly increasing.
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A function f is defined by f ( x)  (2 x  1) 5 .
Find the coordinates of the stationary point on the graph with equation y  f (x) and
determine its nature.
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A council is setting aside an area of land to create six fenced plots where local
residents can grow their own food.
Each plot will be a rectangle measuring x metres by y metres as shown in the diagram.
(a) The area of land being set aside is 108 m2.
Show that the total length of fencing, L metres, is given by
L( x)  9 x 
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(b) Find the value of x that minimises the length of fencing required.
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