08Einstantaneousrate

= Instantaneous Rate of Change =


 * When the rate of change is varying,
 * it is often useful to know the rate at which the quantity is changing at a particular time
 * we call this amount the ** instantaneous rate of change ** at a point.


 * For example,
 * A car speedometer shows the speed of the car (in km/hr) at that particular moment
 * An ammeter measures the current in an electrical circuit at that instant


 * On a straight line graph, the instantaneous rate of change will be the same as the average rate of change
 * because the rate of change is constant
 * On a curving graph, to find the instantaneous rate of change at a point
 * we draw a ** tangent ** line which touches the curve at that point
 * we find the gradient of the tangent
 * the instantaneous rate of change at the point will be equal to the gradient of the tangent


 * On the graph shown here, the tangent touches the curve at A
 * The gradient of the tangent will give the instantaneous rate of change of the curve at A
 * The gradient of the tangent will give the instantaneous rate of change of the curve at A


 * The ** gradient ** of a point on a curve is defined as
 * the instantaneous rate of change at that point .. (or)
 * the gradient of a tangent to the curve at that point.


 * Your calculator can find the instantaneous rate of change (gradient) at a point


 * Later in this course, we will use calculus to find the instantaneous rate of change (gradient) at a point
 * calculus is the central idea behind a large part of mathematics

Find the instantaneous rate of change (gradient) of the point on the circle marked with an X.
 * Example 1 **


 * Solution:**


 * ** The tangent at X is marked in as the line CD **
 * ** The gradient of CD will be the gradient of the circle at X **

math . \qquad \qquad m\big( CD \big) = \dfrac{0-7}{9-6} \\.\\ . \qquad \qquad m\big( CD \big) = \dfrac{-7}{3} \\.\\ . \qquad \qquad m\big( CD\big) = -\dfrac{7}{3} math


 * Hence, the instantaneous rate of change of the circle at X is

math . \qquad \qquad m\big( X \big) = -\dfrac{7}{3} math


 * Finding Instantaneous Rate of Change on the ClassPad **


 * Go to the Graph/Table section of your calculator
 * Type in the rule and graph it
 * Go to the ANALYSIS menu, SKETCH submenu, and select Tangent
 * Either use the side arrows to select a point
 * or type in the x-coordinate of a point
 * Press EXE


 * The tangent will appear on the screen


 * The x-coordinate (xc) and y-coordinate (yc) will appear just above the display box at the bottom
 * The equation of the tangent will appear in the display box at the bottom
 * The gradient of the tangent will be the coefficient of x in the tangent equation
 * Round off to a sensible number of decimal places


 * In the example shown, the gradient of the tangent is 1.5
 * So the instantaneous rate of change of the curve at (3, 2.25) is 1.5

Note
 * To clear the tangent, go to ANALYSIS menu, SKETCH submenu, select Cls

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