Calculus Introduction Information from graphs Gradients The Gradient Function.

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Presentation transcript:

Calculus Introduction Information from graphs Gradients The Gradient Function

This graph shows a car journey from Nelson to Motueka The journey consists of 4 fifteen minute stages. From the graph we can see that in each stage the distance travelled is different. Calculate the speed for each stage of the journey. Remember Speed = Distance/Time taken Stage 1 Stage 2 Stage 3 Stage 4

Gradient = Speed  The speed for each stage of the journey corresponds to the gradient of the line.   In any line graph of this type the gradient of the graph provides information of how fast the dependent variable (vertical axis) changes with respect to the independent variable (horizontal axis).

What about when the graph is a curve?  In this graph the gradient of the curve is changing.  The gradient is defined “at a point”. It is equal to the gradient of the tangent to the curve at that point.  Estimate the gradients at each of the points. X=1 X=2 X=3 X=4 X=5

Is there a pattern?  Let’s investigate the curve

The Gradient Function  Look again at the Motueka Road Trip graph.  Underneath this graph we can draw a second graph showing us the gradient (speed) for the different parts of the journey. Speed km/h

The Gradient Function 2  Look at the Gradient Function for the simple parabola  The gradient function for the parabola y=x 2 is the line y=2x

The Gradient Function 3  Lets look at the gradient function for a cubic.  Students need to be able to draw a gradient function given an original function.  The gradient function for a cubic is a quadratic (parabola).  What are the key points?