Limits and Derivatives

1. Limits and Derivatives 2.7 Derivatives and Rates of Change 2.7 Derivatives and Rates of Change

2. Slope and Tangents • If a curve C has equation y = f(x) and we want to find the tangent line to C at the point P(a, f(a)), then we consider a nearby point Q(x, f(x)), where xa, and compute the slope of the secant line PQ: • Then we let Q approach P along the curve C by letting x approach a. 2.7 Derivatives and Rates of Change

3. Slope and Tangents • If mPQ approaches a number m, then we define the tangent t to be the line through P with slope m. (This amounts to saying that the tangent line is the limiting position of the secant line PQ as Q approaches P. ) 2.7 Derivatives and Rates of Change

4. Definition 1 The tangent line to the curve y = f (x) at the point P(a, f(a)) is the line through Pwith the slope provided the limit exists. 2.7 Derivatives and Rates of Change

5. Definition (Equation 2) Another expression for the tangent line that is easier to use 2.7 Derivatives and Rates of Change

6. Example • Find an equation for the tangent line to the curve y = 2/x at the point (2,1) on this curve. 2.7 Derivatives and Rates of Change

7. Tangent • We sometimes refer to the slope of the tangent line to a curve at a point as the slope of the curve at the point. • The idea is that if we zoom in far enough toward the point, the curve looks almost like a straight line. • Let’s zoom in on the point (1, 1) on the parabola y = x2 2.7 Derivatives and Rates of Change

8. Definition – Average Velocity • The average velocity over this time interval is which is the same as the slope of the secant line PQ. Note: This is just the difference quotient when x = a! 2.7 Derivatives and Rates of Change

9. Definition – Instantaneous Velocity • Now suppose we compute the average velocities over shorter and shorter time intervals [a, a + h]. • In other words, we let h approach 0. As in the example of the falling ball, we define the velocity (or instantaneous velocity) v(a) at time t = a to be the limit of these average velocities: • This means that the velocity at time t = a is equal to the slope of the tangent line at P. Note: This is just the slope of the tangent line!! 2.7 Derivatives and Rates of Change

10. Example: Free Fall I decided to jump out of a perfectly good plane that is 2.5 miles (13200 feet) above ground. Neglecting air resistance and assuming initial velocity is zero, the distance fallen is denoted by the function Verify that I haven’t hit the ground at 6 seconds. Find the instantaneous velocity at 6 seconds. 2.7 Derivatives and Rates of Change

11. Definition The derivative of a function f at a number a, denoted by f’(a) is if this limit exists. 2.7 Derivatives and Rates of Change

12. Note • The tangent line to y = f(x) at (a, f(a)) is the line through (a, f(a)) whose slope is equal to f ’(a), the derivative of f at a. 2.7 Derivatives and Rates of Change

13. Example • Find the derivative with respect to a of the function below 2.7 Derivatives and Rates of Change

14. Definition • Instantaneous rate of change 2.7 Derivatives and Rates of Change

15. Example • Find the IRC of the function below with respect to x when x1=-4 2.7 Derivatives and Rates of Change

16. Book Example – Page 150 #3 • Find the slope of the tangent line through the point (1, 3) of the parabola . • using Definition 1. • using Equation 2. • Find an equation of the tangent line in part (a). • Graph the parabola and the tangent line. As a check on your work, zoom in toward the point (1,3) until the parabola and the tangent line are indistinguishable. 2.7 Derivatives and Rates of Change

17. Book Example – Page 151 #12 • Shown are the graphs of the position functions of two runners, A and B, who run a 100-m race and finish in a tie. • Describe and compare how the runners run the race. • At what time is the distance between the runners the greatest? • At what time do they have the same velocity? 2.7 Derivatives and Rates of Change

18. Book Example – Page 151 #25 • If , find and use it to find an equation of the tangent line to the curve at the point (2,2). • Illustrate the first part by graphing the curve and the tangent line on the same screen. 2.7 Derivatives and Rates of Change

19. Book Example – Page 151 #13 • If a ball is thrown into the air with a velocity of 40 ft/s, its height (in feet) after t seconds is given by . Find the velocity when t=2. 2.7 Derivatives and Rates of Change

20. Book Example – Page 151 #34 • Each limit represents the derivative of some function f at some number a. State such an f and a in the following case: 2.7 Derivatives and Rates of Change