08Grelatinggradient

= Gradient Functions =


 * Previously, we saw how we can get the gradient of a function at a point
 * by finding the gradient of the tangent at that point
 * recall this can be done more accurately using a CAS calculator


 * If we do this multiple times to find the gradient at many points, we get the ** gradient function **
 * the gradient function can be summarised in a table
 * the gradient function can be sketched as a gradient graph
 * the gradient function can be written as a rule
 * the gradient function can be written as a rule
 * Example 1 **

... ... Use a CAS calculator to find multiple values for the gradient of the curve y = x 3 – 2x 2 – 4

... ... Sketch the graph of the gradient function

... ... Obtain a rule for the gradient function


 * Solution:**

... ... ... By graphing the function on the CAS ... ... ... and adding the tangent line at various points, ... ... ... we get the following gradients



... ... ... Enter these values into the Statistics section of the ClassPad ... ... ... Graphing the result produces this graph ... ... ... ... ... ... ... ... The resulting graph looks like a parabola ... ... ... Using Quadratic Regression gives these results ....... .. ... ...... .. ... ... ... ... ...  .... .. ... We can see that the equation of the gradient function is y = 3x 2 – 4x ... ... ... Graphing the gradient function produces the parabola shown


 * Note: **


 * The gradient of the original function is 0 at x = 0 and x = 4/3
 * the gradient function has an __**x-intercept**__ at those points


 * The gradient of the original function is __**positive**__ for x < 0
 * The gradient function is __**above**__ the x-axis for those values


 * The gradient of the original function is __**negative**__ for 0 < x < 4/3
 * The gradient function is __**below**__ the x-axis for those values


 * The gradient of the original function is __**positive**__ for x > 4/3
 * The gradient function is __**above**__ the x-axis for those values


 * Velocity-Time Graphs and Position-Time Graphs **


 * Velocity is the rate of change of position, so
 * Velocity is the gradient of a Position-Time graph at that time


 * Therefore a Velocity-Time Graph is the gradient function of a Position-Time Graph

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