1. Chapter 14 Sections D - E Devil’s Tower, Wyoming

2. Making things difficult Remember the adventurous cat that jumped on its own accord out of the window? Of course, mathematicians have to make things fancy, so here’s the formula we used for average speed between 1 second and t seconds: Average Speed = Does this “format” look familiar?

3. Suppose you drive to grandma’s house 200 miles away and it takes you 4 hours. To get there Then your average speed is: If you look at your speedometer during this trip, it might read 95 mph. This is your instantaneous speed.

4. Remember Slope? Not this  This 

5. Tangent Line to a Graph We learned how the slope of a line indicates the rate at which a line rises or falls. For a line, this rate (or slope) is the same at every point on the line. For graphs other than lines, the rate at which the graph rises or falls changes from point to point.

6. For instance, the parabola is rising more quickly at the point (x 1, y 1 ) than it is at the point (x 2, y 2 ). At the vertex (x 3, y 3 ), the graph levels off, and at the point (x 4, y 4 ) the graph is falling.

7. To determine the rate at which a graph rises or falls at a single point, you can find the slope of the tangent line at that point. In simple terms, the tangent line to the graph of a function f at a point P (x 1, y 1 ) is the line that best approximates the slope of the graph at the point. The figure shows other examples of tangent lines.

8. Think back to geometry (when math was simple), remember that a line is tangent to a circle when the line intersects the circle at only one point Tangent lines to noncircular graphs can intersect the graph at more than one point. In the first graph on the previous slide, if the tangent line were extended, then it would intersect the graph at a point other than the point of tangency.

9. Think back to geometry (when math was simple), remember that a line is tangent to a circle when the line intersects the circle at only one point Tangent lines to noncircular graphs can intersect the graph at more than one point.

10. Visually Approximating the Slope of a Graph Use the graph to approximate the slope of the graph of f (x) = x 2 at the point (1, 1).

11. Solution From the graph of f (x) = x 2, you can see that the tangent line at (1, 1) rises approximately two units for each unit change in x. So, you can estimate the slope of the tangent line at (1, 1) to be = 2

12. Solution Because the tangent line at the point (1, 1) has a slope of about 2, you can conclude that the graph of f has a slope of about 2 at the point (1, 1). cont’d

13. Try This cont’d

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17. Slope and the Limit Process You have been approximating the slope of a graph at a point by creating a graph and then “eyeballing” the tangent line at the point of tangency. A better method of approximating tangent lines makes use of a secant line through the point of tangency and a second point on the graph.

18. A secant line (IB Calls it a Chord) is a line that touches the graph at 2 points. The average rate of change of a function equals the slope of the secant line. x 2 – x 1 f(x 2 ) - f(x 1 ) (x 2, f(x 2 )) (x 1, f(x 1 )) Secant Line y = f(x)

19. The slope of a line is given by: The slope at (1,1) can be approximated by the slope of the secant through (4,16). We could get a better approximation if we move the point closer to (1,1). ie: (3,9) Even better would be the point (2,4).

20. The slope of a line is given by: If we got really close to (1,1), say (1.1,1.21), the approximation would get better still How far can we go?

21. slope slope at The slope of the curve at the point is:

22. is called the difference quotient of f at a. If you are asked to find the slope using the definition or using the difference quotient, this is the technique you will use. It’s also called “Differentiation From First Principles”

23. “f prime x” or“the derivative of f with respect to x” “y prime” “dee why dee ecks”or “the derivative of y with respect to x” “dee eff dee ecks” or “the derivative of f with respect to x” “dee dee ecks uv eff uv ecks”or “the derivative of f of x”

24. Opportunity for Directed Unsupervised Activity Page 357 (1 – 6)