Differential Calculus 15 y 10 5 x -6 -4 -2 2 4 6 8 -5 Next Slide © Annie Patton

To introduce and see the meaning of Differential Calculus. Aim of Lesson To introduce and see the meaning of Differential Calculus. Next Slide © Annie Patton

Differential Calculus Based on the rate of change of one variable in comparison to another. For example if y=x3 , we talk about the rate of change of y in comparison to x. Another example if t=time and s= the distance travelled and s and t are related by the equation s=2t+4. We then talk about the rate of change of the distance in relation to time. Next Slide © Annie Patton

Slope of a Graph The graph y=x2 (red graph) demonstrates how y changes in relation to x for this particular function. 10 y In Co-ordinate Geometry, we learned that the slope of a graph is the rate of change of the graph. 5 x -4 -3 -2 -1 1 2 3 4 When a Tangent (PURPLE LINE) is drawn from a point on a graph. The slope of the graph at that point equals the slope of the tangent at that point. What is a tangent? Next Slide © Annie Patton

Slopes of tangents So the slope of a tangent at a point is equal to the slope of the curve at that point. y 15 10 5 x -6 -4 -2 2 4 6 8 Note however the slope of each tangent is different. -5 Next Slide © Annie Patton

To find the slope of the curve y=f (x) Take the point (x, f (x)) 4 y Take a new point (x + h, f (x+ h)), where h is a small change in x. 3 (x+ h, f (x+ h)) 2 (x, f (x)) 1 x -3 -2 -1 1 2 3 h -1 Next Slide © Annie Patton

Notice even, if written as a fraction it does not mean a fraction. Next Slide © Annie Patton

To find the slope of the curve y=x2 Take the point (x, x2) 4 y Take a new point x + h, where h is a small change in x. 3 (x+ h, (x+h)2) 2 (x, x2) 1 x -3 -2 -1 1 2 3 h -1 Next Slide © Annie Patton

Slope and derivative Click here to see how the slope and the derivative of a function are related. Move the T and see what happens. Make sure you click Next at the bottom of the screen to see another example or click next. Next Slide © Annie Patton

How to calculate the derivative Ignore the words Difference Quotient in next link. Click here to see how to calculate the derivative, but note ignore the reference to the Difference Quotient. Next Slide © Annie Patton

THE END In this brief introduction you have been introduced to a very important aspect of mathematics. So have fun over the next weeks. © Annie Patton