2.1 Tangents and Derivatives at a Point

1. 2.1 Tangents and Derivatives at a Point

2. Finding a Tangent to the Graph of a Function To find a tangent to an arbitrary curve y=f(x) at a point P(x0,f(x0)), we • Calculate the slope of the secant through P and a nearby point Q(x0+h, f(x0+h)). • Then investigate the limit of the slope as h0.

3. Slope of the Curve If the previous limit exists, we have the following definitions. Reminder: the equation of the tangent line to the curve at P is Y=f(x0)+m(x-x0) (point-slope equation)

4. Example • Find the slope of the curve y=x2 at the point (2, 4)? • Then find an equation for the line tangent to the curve there. Solution

5. Rates of Change: Derivative at a Point The expression is called the difference quotient of f at x0 with increment h. If the difference quotient has a limit as h approaches zero, that limit is named below.

6. Summary

7. 2.2 The Derivative as a Function We now investigate the derivative as a function derived from f by Considering the limit at each point x in the domain of f. If f’ exists at a particular x, we say that f is differentiable (has a derivative) at x. If f’ exists at every point in the domain of f, we call f is differentiable.

8. Alternative Formula for the Derivative An equivalent definition of the derivative is as follows. (let z = x+h)

9. Calculating Derivatives from the Definition The process of calculating a derivative is called differentiation. It can be denoted by Example. Differentiate Example. Differentiate for x>0.

10. Notations There are many ways to denote the derivative of a function y = f(x). Some common alternative notations for the derivative are To indicate the value of a derivative at a specified number x=a, we use the notation

11. Graphing the Derivative Given a graph y=f(x), we can plot the derivative of y=f(x) by estimating the slopes on the graph of f. That is, we plot the points (x, f’(x)) in the xy-plane and connect them with a smooth curve, which represents y=f’(x). Example: Graph the derivative of the function y=f(x) in the figure below.

12. What we can learn from the graph of y=f’(x)?

13. Differentiable on an Interval; One-Sided Derivatives If a function f is differentiable on an open interval (finite or infinite) if it has a derivative at each point of the interval. It is differentiable on a closed interval [a, b] if it is differentiable on the interior (a, b) and if the limits exist at the endpoints. Right-hand derivative at a Left-hand derivative at b A function has a derivative at a point if and only if the left-hand and right-hand derivatives there, and these one-sided derivatives are equal.

15. When Does A Function Not Have a Derivative at a Point A function can fail to have a derivative at a point for several reasons, such as at points where the graph has • a corner, where the one-sided derivatives differ. • a cusp, where the slope of PQ approaches  from one side and -  from the other. • a vertical tangent, where the slope of PQ approaches  from both sides or approaches -  from both sides. • a discontinuity.

17. Differentiable Functions Are Continuous Note: The converse of Theorem 1 is false. A function need not have a derivative at a point where it is continuous. For example, y=|x| is continuous at everywhere but is not differentiable at x=0.

18. 2.3 Differentiation Rules

19. The Power Rule is actually valid for all real numbers n.

20. Examples Example.

21. Constant Multiple Rule Note: Example.

22. Derivative Sum Rule Example.

23. Derivative Product Rule In function notation:

24. Example Example: Solution:

25. Derivative Quotient Rule In function notation:

26. Example Example: Solution:

27. Second- and Higher-Order Derivatives The derivative f’ of a function f is itself a function and hence may have a derivative of its own. If f’ is differentiable, then its derivative is denoted by f’’. So f’’=(f’)’ and is called the second derivative of f. Similarly, we have third, fourth, fifth, and even higher derivatives of f.

28. A general nth order derivative can be denoted by Example:

29. 2.4 The Derivative as a Rate of Change Thus, instantaneous rates are limits of average rates. When we say rate of change, we mean instantaneous rate of change.

30. Motion Along a Line: Displacement, Velocity, Speed, Acceleration, and Jerk Suppose that an object is moving along a s-axis so that we know its position s on that line as a function of time t: s=f(t). The displacement of the object over the time interval from t to t+∆t is ∆s = f(t+ ∆t)-f(t); The average velocity of the object over that time interval is

31. Velocity To find the body’s velocity at the exact instant t, we take the limit of the Average velocity over the interval from t to t+ ∆t as ∆t shrinks to zero. The limit is the derivative of f with respect to t.

32. Besides telling how fast an object is moving, its velocity tells the direction of Motion.

33. The speedometer always shows speed, which is the absolute value of velocity. Speed measures the rate of progress regardless of direction

34. The figure blow shows the velocity v=f’(t) of a particle moving on a coordinate line., what can you say about the movement ?

35. Acceleration The rate at which a body’s velocity changes is the body’s acceleration. The acceleration measures how quickly the body picks up or loses speed. A sudden change in acceleration is called a jerk.

36. Example Near the surface of the earth all bodies fall with the same constant acceleration. In fact, we have s=(1/2)gt2 , where s is the distance fallen and g is the acceleration due to Earth’s gravity. With t in seconds, the value of g at sea lever is 32 ft/ sec2 or 9.8m/sec2.

37. Example • Example: Figure left shows the free fall of a heavy ball bearing released from rest at time t=0. • How many meters does the ball fall in the first 2 sec? • What is its velocity, speed, and acceleration when t=2?

38. 2.5 Derivatives of Trigonometric Functions

39. Example Example: Solution:

40. Example Example: Solution:

41. Example: A body hanging from a spring is stretched down 5 units beyond Its rest position and released at time t=0 to bob up and down. Its position at any later time is s=5cos t. What are its velocity and acceleration at time t?

42. Since We have

43. Example Example: Solution:

44. 2.6 Exponential Functions In general, if a1 is a positive constant, the function f(x)=ax is the exponential function with base a.

45. If x=n is a positive integer, then an=a  a  …  a. If x=0, then a0=1, If x=-n for some positive integer n, then If x=1/n for some positive integer n, then If x=p/q is any rational number, then If x is an irrational number, then

46. Rules for Exponents

47. The Natural Exponential Function ex The most important exponential function used for modeling natural, physical, and economic phenomena is the natural exponential function, whose base is a special number e. The number e is irrational, and its value is 2.718281828 to nine decimal places.

48. The graph of y=ex has slope 1 when it crosses the y-axis.

49. Derivative of the Natural Exponential Function Example. Find the derivative of y=e-x. Solution: Example. Find the derivative of y=e-1/x.

50. 2.7 The Chain Rule