1. 2.2 Differentiation Techniques: The Power and Sum-Difference Rules 1.5

2. 2 OBJECTIVES Find the derivative using the Constant Rule. Find the derivative using the Power Rule. Find the derivative using the Constant Multiple Rule and the Sum and Difference Rules. Find the derivative of sine and cosine. Use derivatives to find rates of change

3. Leibniz’s Notation: When y is a function of x, we will also designate the derivative,, as which is read “the derivative of y with respect to x.”

4. THEOREM 1: The Power Rule For any real number k,

5. Example 1: Differentiate each of the following: a) b) c)

6. Example 2: Differentiate: a) b)

7. THEOREM 2: The derivative of a constant function is 0. That is,

8. THEOREM 3: The derivative of a constant times a function is the constant times the derivative of the function. That is,

9. Example 3: Find each of the following derivatives: a) b) c) a) b)

10. Example 3 (concluded): c)

11. THEOREM 4: The Sum-Difference Rule Sum: The derivative of a sum is the sum of the derivatives. Difference: The derivative of a difference is the difference of the derivatives.

12. Example 5: Find each of the following derivatives: a) b) a)

13. Example 5 (concluded): b)

14. 14 Finding the Equation of a Tangent Line Example: Find the equation of the tangent line at x = 1 and take a “peek” at the graph. Then verify using your calculator. “peek” at the graph

15. 15 Finding the Equation of a Tangent Line Example: Find the equation of the tangent line at x = 1 and take a “peek” at the graph. Then verify using your calculator. So, the slope of the tangent line, is -1, at x = 1. And if x = 1, then y = 1- 3(1)+ 4 = 2. So, using the point (1,2)

16. Example 6: Find the points on the graph of at which the tangent line is horizontal. Recall that the derivative is the slope of the tangent line, and the slope of a horizontal line is 0. Therefore, we wish to find all the points on the graph of f where the derivative of f equals 0.

17. Example 6 (continued): So, for Setting equal to 0:

18. Example 6 (continued): To find the corresponding y-values for these x-values, substitute back into Thus, the tangent line to the graph of is horizontal at the points (0, 0) and (4, 32).

19. Example 6 (concluded):

20. 20 Consider the function We could make a graph of the slope: slope Now we connect the dots! The resulting curve is a cosine curve.

21. 21 We can do the same thing for slope The resulting curve is a sine curve that has been reflected about the x-axis.

22. 22 Derivatives Involving sin x, cos x Try the following

23. 23 Consider a graph of displacement (distance traveled) vs. time. time (hours) distance (miles) Average velocity can be found by taking: A B The speedometer in your car does not measure average velocity, but instantaneous velocity. (The velocity at one moment in time.)

24. 24 Rates of Change Some Examples: population growth rates, production rates, velocity and acceleration *Common use for rate of change is to describe motion of an object along a straight line. Without Calculus WITH Calculus

25. 25 Motion Along a Straight Line s(t)= position function t s(t) (t+  t) s(t+  t) t (sec) s (ft) Ave. velocity during the time interval from t to (t+  t) = Instantaneous velocity at time t Slope of secant line

26. 26 Motion Along a Vertical Line v(t) > 0, v(t) < 0, v(t) = 0, stopped instantaneously s(t) = position (ft) v(t) = s´(t) = velocity (ft/sec) rate of change of position a(t) =v´(t) = s´´(t) = accel (ft/s 2 ) rate of change of velocity speed = |v(t)|Speed is never negative Note: Velocity is speed with a direction.

27. 27 Velocity is the first derivative of position.

28. 28 Example: Free Fall Equation Gravitational Constants: Speed is the absolute value of velocity.

29. 29 Acceleration is the derivative of velocity. example: If distance is in: Velocity would be in: Acceleration would be in:

30. 30 time distance acc pos vel pos & increasing acc zero vel pos & constant acc neg vel pos & decreasing velocity zero acc neg vel neg & decreasing acc zero vel neg & constant acc pos vel neg & increasing acc zero, velocity zero It is important to understand the relationship between a position graph, velocity and acceleration

31. 31 1Which of these position versus time curves best shows the motion of an object with constant positive acceleration? Determine the Concept The slope of an x (t) curve at any point in time represents the speed at that instant. The way the slope changes as time increases gives the sign of the acceleration. If the slope becomes less negative or more positive as time increases (as you move to the right on the time axis) then the acceleration is positive. ie: the graph is concave up!!!! If the slope becomes less positive or more negative then the acceleration is negative. ie: the graph is concave down!!! So the slope of an x (t) curve at any point in time represents the acceleration at that instant.

32. 32 a)The slope is both negative and increasing, therefore the velocity is negative and the acceleration is positve b)The slope is positive but decreasing, therefore, therefore, the velocity is positive and decreasing, and the acceleration is negative. c)The slope is positive and constant, therefore, the velocity is positive and the acceleration is 0 d)The slope is positive and increasing, therefore, the velocity is positive and increasing and the acceleration is positive. e)The slope of the curve is 0, therefore, the velocity and acceleration are both 0.

33. 33 Rates of Change: Average rate of change = Instantaneous rate of change = These definitions are true for any function. ( x does not have to represent time. )

34. Suppose that in June a chain of stores had combined daily sales of ice cream cones given by where s is the number (in hundreds) of cones sold and x is the day of the month.

35. (a) how many cones were sold on June 3rd? (b) at what rate were sales changing on June 10th? (c) at what rate were sales changing on June 28th? (d) on what day was the rate of change of sales equal to 10 cones per day?

36. 36 Example of Free Falling Object Example: A ball is thrown straight down from the top of a 220-foot building with an initial velocity of –22 feet/second. 1) What is its velocity after 3 seconds? 2) What is its velocity after falling 108 feet? 220 0

37. 37 Example: Find at x = 2. nDeriv ( x ^ 3, x,2) ENTER returns MATH: 8: (F(X),X,VALUE)

38. 38 Warning: The calculator may return an incorrect value if you evaluate a derivative at a point where the function is not differentiable. Examples: returns

39. 39 YOU SHOULD NOW BE ABLE TO: Find the derivative using the Constant Rule. Find the derivative using the Power Rule. Find the derivative using the Constant Multiple Rule and the Sum and Difference Rules. Find the derivative of sine and cosine. Use derivatives to find rates of change.

40. 40 Constant & Power Rules Ex: Power Rule Ex: Constant Rule

41. 41 Constant Multiple, Sum & Difference Rules Ex:

42. 42 Sine and Cosine Rules Example: