3Gradients When is the gradient positive? negative? zero? Height of a child on a swingWhen is the gradientpositive?negative?zero?What is happening then?This activity shows how to find accurate values for the gradients of curves.
4Measuring gradients Straight lines Curves m = gradient = y y = mx + c xCurvesyxtangenty stepx stepPy stepx stepm = gradient =
5Gradient of y = x2 gives an approximate value for the gradient step x y stepx stepP (3, 9)It can be calculated more accurately
6Incremental changes y = x2 Gradient of PQ1 = 7 Gradient of PQ2 = 6.5 As Q ® Pgradient ® 6= 6.25
7Gradients of functions of the form y = xn Equation of curveGradient functiony = x22xy = x33x2y = x44x3y = x5Think aboutWhat do you think is the gradient function for y = x5? How can you prove it?What about y = x6?Can you suggest an expression for the gradient of the general function y = xn ?
8Gradients Reflect on your work Describe the way in which the gradient of a curve can be found using a spreadsheet.What advantages does this have on drawing a tangent to a hand-drawn graph?What is the gradient function of y = xn ?