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Increasing

6. y. 5. 4. 3. 2. 1. x. -6. -5. -4. -3. -2. -1. 1. 2. 3. 4. 5. 6. -1. -2. -3. -4. -5. -6. Features of +x 3 Graphs . The original function is… f(x) is… y is … . Stationary. Increasing. Decreasing. Increasing. Stationary. 6. y. 5. 4. 3. 2. 1. x. -6.

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Increasing

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  1. 6 y 5 4 3 2 1 x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 Features of +x3 Graphs The original function is… f(x) is… y is… Stationary Increasing Decreasing Increasing Stationary

  2. 6 y 5 4 3 2 1 x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 Investigate the tangents of +x3 Graphs The slope function is… f’(x) is… dy/dx is… Slope values are decreasing • Point of Inflection = slopes stop decreasing and start • increasing Slope values are increasing

  3. 6 y 5 4 3 2 1 x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 Features of the Slope Function Graph Reading the features of the graph of the slope function from the original function Turning Point of the slope function: where slopes turn from decreasing to increasing = min slope function = 0 (cuts x-axis) dy/dx= 0 Slope values are increasing →slope function increasing Slope values are decreasing →slope function decreasing dy/dx= 0 slope function = 0 (cuts x-axis) Slope Function: U shaped (positive cubic graph will have positive derivative graph) Minimum point at same x value as the point of inflection Cuts x-axis at the x values of the turning points

  4. 6 6 y y 5 5 4 4 3 3 2 2 1 1 x x -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 1 1 2 2 3 3 4 4 5 5 6 6 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 -6 -6 The slope function is… f’(x) is… dy/dx is… ORIGINAL FUNCTION y = f(x) dy/dx= 0; slope function = 0 Turning Point: Decreasing to increasing = min pt Slope values are decreasing SLOPE FUNCTION y = f’(x) Slope values are increasing Slope values are increasing Slope values are decreasing dy/dx= 0; slope function = 0 x dy/dx= 0; slope function = 0 dy/dx= 0; slope function = 0 Turning Point: Decreasing to increasing = min pt

  5. 6 6 y y 5 5 4 4 3 3 2 2 1 1 x x -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 1 1 2 2 3 3 4 4 5 5 6 6 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 -6 -6 Also, we can read where the slope function is above and below the x-axis from the original function + + + + + + + 0 - - - - - - - - - - 0 + + + + + + + Slopes are negative Slopes are positive Slopes are positive Slope Function above x-axis Slope Function above x-axis Slope Function below x-axis

  6. 6 y 5 4 3 2 1 x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 At what rate is the slope function changing? f’’(x) is… d2y/dx2 is... How fast is the rate of decrease of the slopes? How fast is the rate of increase of the slopes? Finding the rate of change of the rate of change…. Finding the second derivative

  7. 6 6 y y 5 5 4 4 3 3 2 2 1 1 x x -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 1 1 2 2 3 3 4 4 5 5 6 6 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 -6 -6 A step further to investigate the tangents of the slope function. Second Derivative Function is… f’’(x) is… d2y/dx2 is... ORIGNAL FUNCTION y = f(x) dy/dx= 0; slope function = 0 Turning Point: Decreasing to increasing = min pt Slope values are decreasing SLOPE FUNCTION y = f’(x) Slope values are increasing Slope values are decreasing Slope values are increasing dy/dx= 0; slope function = 0 dy/dx= 0; slope function = 0 dy/dx= 0; slope function = 0 Turning Point: Decreasing to increasing = min pt

  8. 6 y 5 4 3 2 1 x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 SLOPE FUNCTION y = f’(x) Slope values are decreasing Slope values are increasing dy/dx= 0; slope function = 0 dy/dx= 0; slope function = 0 Turning Point: Decreasing to increasing = min pt

  9. 6 6 y y 5 5 4 4 3 3 2 2 1 1 x x -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 -1 1 1 2 2 3 3 4 4 5 5 6 6 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 -6 -6 SLOPE FUNCTION y = f’(x) Slope values are increasing →Second Derivative Function is increasing Slope values are increasing →Second Derivative Function is increasing Slope=0 (d2y/dx2 = 0) Second Derivative Function =0 (cuts x-axis) SECOND DERIVATIVE FUNCTION y = f’’(x)

  10. 6 6 6 y y y 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 x x x -6 -6 -6 -5 -5 -5 -4 -4 -4 -3 -3 -3 -2 -2 -2 -1 -1 -1 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 -1 -1 -1 -2 -2 -2 -3 -3 -3 -4 -4 -4 -5 -5 -5 -6 -6 -6 Original Function, First Derivative Function, Second Derivative Function y = f(x) y = f’(x) y = f’’(x)

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