1. AP CALCULUS 1005: Secants and Tangents

2. Objectives SWBAT determine the tangent line by finding the limit of the secant lines of a function. SW use both informal and precise mathematical language to describe the tangent line of a function

3. Average Rates of Change The AVERAGE SPEED (average rate of change) of a quantity over a period of time is the amount of change divided by the time it takes. In general, the average rate of change of a function over an interval is the amount of change divided by the length of the interval. Therefore, the average rate of change can be thought of as the slope of a secant line to a curve.

4. Average Rate of Change Known Formula Average Rate of Change : (in function notation) ax Slope of a secant

5. Slope of a Tangent Calculus I -The study of ___________________________________ Slope : (in function notation) ax Move x closer to a Rates of change

6. A. THE DERIVATIVE (AT A POINT) WORDS: Layman’s description: The DERIVATIVE is a __________________ function. Built on the ________________ formula. The derivative (slope of a tangent line) is the limit of the slopes of the secants as the two points are brought infinitely close together Rate of change slope

7. Let T(t) be the temperature in Dallas( in o F) t hours after midnight on June 2, 2001. The graph and table shows values of this function recorded every two hours,. What is the meaning of the secant line (units)? Estimate the value of the rate of change at t = 10. t T 073 2 73 470 669 8 72 10 81 Estimate the rate of change tangent line As the points get closer together the ROC approach ROC at 10

8. EX: THE DERIVATIVE AT A POINT EX 1: at a = 4 Notation: Words: f(a) = 8

9. Equation of the Tangent To write the equation of a line you need: a) b) Point- Slope Form: point slope (4,8) m = 6

10. There is local linearity When the secant lines get very near a point it acts like a tangent line

11. Normal to a Curve The normal line to a curve at a point is the line perpendicular to the tangent at the point. Therefore, the slope of the normal line is the negative reciprocal of the slope of the tangent line. EX 1(cont): at a = 4 Find the equation of the NORMAL to the curve

12. EX: THE DERIVATIVE AT A POINT EX 2: at x = -2 Method: y = 1 Mantra:

13. At a Joint Point SKIP Piece Wise Defined Functions: The function must be CONTINUOUS Derivative from the LEFT and RIGHT must be equal. The existence of a derivative indicates a smooth curve; therefore,

14. Last Update: 08/12/10