Differentiation. f(x) = x 3 + 2x 2 – 3x + 5 f’(x) = 3x 2 + 4x - 3 f’(1) = 3 x 1 2 + 4 x 1 – 3 = 3 + 4 – 3 = 4 If f(x) = x 3 + 2x 2 –

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Presentation transcript:

Differentiation

f(x) = x 3 + 2x 2 – 3x + 5 f’(x) = 3x 2 + 4x - 3 f’(1) = 3 x x 1 – 3 = – 3 = 4 If f(x) = x 3 + 2x 2 – 3x + 5 find the value of the gradient when x = 1

Find the gradient of the curve f(x) = x 3 – x 2 +1 at the point where x = 2. f(x) = x 3 – 8x 2 + 5x + 14 What is the gradient of the curve f(x) = ( 2x – 1) 2 at the point ( -2, 25 ).

Find the equation of the tangent to the curve y = x 3 + 3x 2 + x - 5 at the point (-1, -4 ).

Find the equations of the tangents to the curve f(x) = x 2 + 3x + 2 at the points where the curve cuts the x-axis. The curve cuts the x-axis when f(x) = 0 x 2 + 3x + 2 = 0 ( x + 2) ( x +1) = 0 Either x = -2 or x = -1the curve cuts through at both points f(x) = x 2 + 3x + 2 f’(x) = 2x + 3differentiate to find the gradient At x = -2, f’(-2) = -1gradient of tangent at x=-2 is -1 At x = -1,f’(-1) = 1gradient of tangent at x=-1 is 1 Equations of the tangents using y – y 1 = m ( x – x 1 ) are ( -2, 0 )( -1, 0 ) y - 0 = -1 (x - -2)y - 0 = 1 ( x – -1 ) y = - x – 2y = x + 1 Are the equations of the tangents where the curve cuts the x-axis.

1.Find the equation of the tangent to the curve y = x 2 – 5x + 4 at the point (4, 0) 2.Find the equation of the tangent to the curve y = x 3 – 3x at the point (1, -1). 3.What is the equation of the tangent to the curve y = -x 3 + 2x 2 + x – 2 at the point where the curve cuts the y-axis? 4.Find the equations of the tangents to the curve y = x 2 + x – 12 at the points where the curve cuts the x-axis.

. Find a point, given the gradient.

Find the coordinates where the function (i) Derivative = (ii) Find x where f’(x) = 9. Find a point, given the gradient.

Second Derivative Differentiate twice. Use the second derivative to identify features of graphs: where the graph is increasing, decreasing, points of inflection and stationary points. Use calculus to find local maximum, local minimum, and points of inflection. Finding the nature of Turning points NOTE: determining the nature of turning points you can use any method: 1.The second derivative, 2.2. By inspection of the graph of the function or 3.3. By testing the value of the function on each side of the turning point

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