2. Using Algebraic Operations to determine the Instantaneous Rate of Change

3. Remember…

4. An average speed can be calculated by using 2 points D(m) T(s) 012345012345 402 0 secant What if we moved that second point closer?...

6. We are free to move around the endpoints of the line segment, but it is more efficient to leave one stationary and just move the other. P(x,y) For our structure, we will place point P on our point of interest, and we will call the moving point Q

7. We understand how we can attempt to find the instantaneous rate of change graphically, but this approach (drawing a tangent line) carries the usual graph problems: 1. takes too long 2. not overly accurate

8. We demand precision… We need an algebraic approach….. Which is what we’ll start today…

9. Imagine the graph of y = x 2 (the simplest model we’ll ever use) Determine the slope of the tangent at the point (3,9) (try it on your sketch for fun!!) (3,9) X Y

10. Algebraically (finally…) Step 1: assign point P as (3,9) P Q Step 2: assign point Q as some other point on the curve…. (draw a line to create a secant) (3,9) We are going to use the slope formula: We already have one point (3,9). All we need are coordinates for the other….

11. Since the distance between the 2 variables will vary…..we will use a variable to represent this distance…. Let “h” represent the horizontal distance between the 2 points.

12. Therefore, the coordinates of point Q would be: [(3 + h), P Q h (3,9) Try some numbers for “h” in your head to see. (3 + h) 2 ]

13. This is a brilliant algebraic structure. The entire model is built around a single variable!

14. To determine the slope of the line we use the slope formula (from grade 9…..) P (3,9) and Q (3+h), (3+h) 2

15. AMAZING!!! A precise slope!! m = 6 So now, to reduce the distance between the 2 points, we simply make h = 0

16. IN GENERAL Let P(a,f(a)) be a fixed point on the graph of y =f(x) Let Q[(a + h), f(a + h)] represent any other point on the graph (thus creating the secant line) h is the horizontal distance from P to Q Then the slope of the secant PQ is

17. This quotient is fundamental to calculus and is referred to as The Difference Quotient Putting it all together……

19. pg 20 10,11 14,15,16,19