1. Chapter 3 Application of Derivatives
3.1 Extreme Values of Functions
Absolute maxima or minima are also referred to as global maxima or minima.

2. Examples

3. Extreme Value Theorem
The requirements in Theorems 1 that the interval be closed and finite, and
That the function be continuous, are key ingredients.

4. Local (Relative) Extreme Values

5. Finding Extrema

6. Critical Point
Theorem 2 says that a function’s first derivative is always zero at an interior point where the function has a local extreme value and the derivative is defined.
Hence the only places where a function f can possibly have an extreme value (local or global) are
Interior points where f’=0
Interior points where f’ is undefined,
Endpoints of the domain of f.

7. How to Find the Absolute Extrema
Thus the only domain points where a function can assume extreme values are critical points and endpoints.

8. Example
Example. Find the absolute maximum and minimum values of f(x)=10x(2-lnx) on the interval [1, ex].
Solution. We evaluate the function at the critical points and the endpoints and take the largest and the smallest of the resulting values.
The first derivative is
f’(x)=10(2-lnx)-10x(1/x)=10(1-lnx).
Let f’(x)=0, we have x=e. Then
Critical point value: f(e)=10e
Endpoint values: f(1)=20, and f(e2)=0.
So the function’s absolute maximum value
is10e at x=e. The absolute minimum value
is 0 and occurs at the right endpoint x=e2.

9. 3.2 The Mean Value Theorem

11. Mathematical Consequences

12. 3.3 Monotonic Functions and the First Derivative Test
A function that is increasing or decreasing on an interval is said to be
monotonic on the interval.

13. Graphs of functions Each tangent line Has positive slope.
Has negative slope.
Each tangent line
Has zero slope.

14. Theorem

15. Example
Example: Find the critical points of f(x)=x3 -12x-5 and identify the intervals on which f is increasing and on which f is decreasing.
Solution: The function f is everywhere continuous and differentiable. The
first derivative f ’(x)=3x2-12=3(x2-4)=3(x+2)(x-2) is zero at x=-2 and x=2.
These critical points subdivide the domain of f into intervals (-, -2), (-2, 2)
and (2, ) on which f ‘ is either positive or negative. We determine the sign
Of f ‘ by evaluating f at a convenient point in each subinterval.
Interval  <x< <x< <x<
f ‘ evaluated f ’(-3) = f ‘(0)= f ‘ (3) =15
Sign of f ‘
Behavior of f increasing decreasing increasing

17. First Derivative Test for Local Extrema

19. Example
Find the critical points of f(x)=(x2-3)ex. Identify the intervals on which f is increasing and decreasing. Find the function’s local and absolute extreme values.
Solution.
f ’(x)=(x2+2x-3)ex. Since ex is never zero, f ‘(x) is zero iff x2+2x-3=0. That is, x=-3 and x=1.
We can see that there is a local maximum about at x=-3 and a local minimum about at x=1. The local minimum value is also an absolute minimum, but there is no absolute maximum. The function increases on (-, -3) and (1, ) and deceases on (-3, 1).

21. 3.4 Concavity and Curve Sketching
Two ways to characterize the concavity of a differentiable function f on an open interval:
f is concave up on an open interval if its tangent lines have increasing slopes on that interval and is concave down if they have decreasing slopes.
f is concave up on an open interval if its graph lies above its tangent lines on that interval and is concave down if it lies below its tangent lines

22. Concavity and the Second Derivative Test for Concavity
If y=f(x) is twice-differentiable, we will use f’’ and y’’ interchangeable
When denoting the second derivative.

23. Example
Example: Find the intervals on which is concave up and the intervals on which it is concave down.
Solution:
Thus f(x) is concave up on the interval , and concave down on the interval

24. Inflection Points

25. Second Derivative Test for Local Extrema
This test requires us to know f’’ only at c itself and not in an interval about c.
This makes the test easy to apply.
However, this test is inconclusive if f’’=0 or if ‘’ does not exist at x=c.
When this happens, use the First derivative Test for local extreme values.

26. Strategy for Graphing y=f(x)

27. Graphical Behavior of Functions from Derivatives

28. 3.5 Parametrizations of Plane Curves

29. Parametric Formula for dy/dx
Example. Find dy/dx as a function of t if x = t - t2, y = t - t3.
Solution.

30. 3.7 Indeterminate Forms and L’Hopital’s Rule

31. How to Use L’Hopital’ Rule

32. Example
Example: Find
Solution: The limit is a indeterminate form of type 0/0.
Applying L’Hopital’s rule yields

33. Example
Example: Find
Solution: The limit is a indeterminate form of type 0/0.
Applying L’Hopital’s rule yields

34. Indeterminate Forms of Type /

35. Example
Example: Find
Solution: The limit is a indeterminate form of type
Applying L’Hopital’s rule yields
In fact, we can use LHopital’s rule to show that

36. Example
Example: Find
Solution: The limit is a indeterminate form of type
Applying L’Hopital’s rule yields
Similar methods can be used to find the limit of f(x)/g(x) is an
Indeterminate form of the types:

37. 3.8 Newton’s Method
Newton’s method is a technique to approximate the solution to an equation f(x)=0. Essentially it uses tangent lines in place of the graph of y=f(x) near the points where f is zero.
(A value of x where f is zero is a root of the function
f and a solution of the equation f(x)=0.)

39. To find a root r of the equation f(x)=0,
select an initial approximation x1. If f(x1)=0, then r=x1. Otherwise, use the root of the tangent line to the graph of f at x1 to approximate r. Call this intercept x2 .
We can now treat x2 in the same way we did x1. If f(x2 )=0, then r= x2 . Otherwise, we construct the tangent line to the graph of f at x2, and take x3 to be the x-intercept of the tangent line.
Continuing in this way, we can generate a succession of values x1,x2,, x3,,,x4…that will usually approach r.

40. Example
Use Newton’s Method to approximate the real solutions of x3-x-1=0
Solution: Let f(x)=x3-x-1. Since f(1)=-1 and f(2)=5, weh know by the Intermediate Value Theorem, there is a root in the interval (1, 2).
We apply Newton’s method to f with the starting values x0 =1. The result are displayed in the following table and figure.

41. At n=5, we come to the result x6=x5=1. 324717957
At n=5, we come to the result x6=x5= When xn+1=xn, Equation (1) shows that f(xn)=0. We have found a folution of f(x)=0 to nine decimals.

42. 3.9 Hyperbolic Functions

43. Identities and Derivatives for Hyperbolic Functions

44. Inverses Hyperbolic Sine, Cosine, and Secant of x

45. Inverse Hyperbolic Tangent, Cotangent, and Cosecant of x

46. Identities and Derivatives of Inverse Hyperbolic Functions