The total differential

Transcription

1 The total differential The total differential of the function of two variables The total differential gives the full information about rates of change of the function in the -direction and in the -direction. 1

2 Second order partial derivatives:, Note: If the two mixed second order partial derivatives are continuous then they will be equal. So, the order of taking partial derivatives of a function, can be interchanged 2

3 Examples:, 3 2, 2 6 2, 2 3 4, 3 4 3

4 At a local max or min, 0 and 0 Local maxima and minima Definition of a critical point: (, ) where 0 and 0 A critical point may be a local minimum, local maximum, or saddle. 4

5 Second derivative test Goal: determine type of a critical point, and find the local min/max. Note: local min/max occur at critical points General case: second derivative test. We look at second derivatives: ; ; The Hessian matrix (or simply the Hessian) is the square matrix of secondorder partial derivatives of a function 5

6 Given is and a critical point,. Define the second derivative test discriminant as,,,, Then If 0 and, 0 If 0 and, 0 If 0 If 0 local minimum local maximum saddle cannot be concluded 6

7 A saddle point is a point in the range of a function that is a critical point but not a local extremum. The name derives from the fact that the prototypical example in two dimensions is a surface that curves up in one direction, and curves down in a different direction, resembling a saddle or a mountain pass. 7

8 Example:, Critical points candidates: First derivative test applied We need to solve the following system of equations: The critical points are:, 3, 1 ;, 3, 1;, 0, 2 ;, 0,2 8

9 Maximum, minimum or saddle? Second derivative test applied: 2 2 6; 2,,,,,, ,,,,,, ,,,,,, ,,, 2 0,,,, , 6 0

10 The Integral of a Function. The Indefinite Integral Undoing a derivative: Antiderivative = Indefinite Integral Definition: A function is called an antiderivative of a function on same interval,, if differentiation undo 10

11 Note: Unlike derivatives, antiderivatives are note unique: Example: 1 3 is an antiderivative of on, because 1 3 But also for any constant because

12 Theorem: If is any antiderivative of on, then so is Every antiderivative of on has the form for some! Differentiation produces one derivative Antidifferentiation produces an infinite family of antiderivatives 12

13 differentiation 0 : symbol antidifferentiation A name for this family 13

14 The indefinite integral of the integral sign [elongated S ] - the integrand - indicates the independent variable - constant of integration - one of many antiderivative of The Indefinite Integral of represents the entire family of all antiderivatives of 14

15 Differentiation Antidifferentiation [indefinite Integration] Note: Sometimes we write:

16 Finding Antiderivatives (1) Use derivatives we know to build a table where 1 Derivative 1 1 Corresponding antiderivative 1 1 Add 1 to the power and divide by this new power 16

17

18 1 log log log 1 18

19 (2) Some Properties on Indefinite Integrals: a real number All applied earlier for limits + derivatives 19

20 2 2 2 Do not write: Note on constant of integration Do not forget constants of integrations Do not introduce them too soon Combine multiple constant into one What integration technique so far? (1) Use (create) a table (2) Rewrite an integrand (in order to use the table) 20

21 Examples:

22 The Indefinite Integration by Parts? Recall the product rule for derivatives, Integrate both sides 22

23 Shorthand notation: The integration by part formula Generally try to choose to be something that simplifies when you differentiate it. 23

24 Integration by parts formula: Example 1: 2 How to choose and? 2 and are easy to find: und But we cannot find the indefinite Integral of the product Then: 2 and 2, so

25 Integration by parts formula: Example 2:

26 Integration by parts formula: Example 3:

27 The Indefinite Integration by Substitution Idea: Suppose and exists Chain rule: So, Let, then: Substitution of for makes (when it works!) integration easier. 27

28 Straightforward Substitution Always consider Substitution first If on substitution fails, try another one! Always make a total change from to! Never mix variables! Substitution technique: Find something in the integrand to call to simplifies the appearance of the integral and whose is also present as a factor 28

29 Example:

30 Exercises: function substitution Integral

31 Summary A hard and fast set of rules for determining the method that should be used for integration does not exist. Some integrals can be done in more than one way. It is possible that you will need to use more than one method to compute an integral. There are integrals that cannot be computed in terms of functions that we know. 31

32 The Sigma shorthand for sums Greek S for sum Area Defined as a Limit Upper limit of summation term of the sum (common form) (integer): Index of summation:,,,, Lower limit of summation 32

33 Definition of Area under a Curve Continuous 0 Partition into equal subintervals Each width 1 33

34 Choose any point in each interval to calculate rectangle heights Definition: If is continuous on, 0 on, Then lim, 34

35 Net Area 0 Definition: Net Signed Area Net! 0 0 a b If is continuous on, 0 Then lim signed, 0 Approximating Area Numerically For large lim 35

36 The Definite Integral The Definite Integral Defined Extend our Net Area limit: lim Continuous function Equal length subinterval To compute the area under the graph of and above the interval, we proceed as follows: 36

37 1. Subdivide the interval, into unequal subintervals with endpoints: For each 1,2, 1, let, Note: The largest of the will be denoted 2. Inside each, select a point, evaluate,,,, and compute,,,, 37

38 3. Form the Riemann Sum. A Riemann sum is a summation of a large number of small partitions of a region. 4. Repeat Step 1-3 over and over with finer and finer subdivision of, (i.e. smaller and smaller and take a limit lim Partition in equal subinterval: means 0 guaranties each width shrinks Partition in unequal subinterval: 0 guaranties each width shrinks 38

39 Notice that if 0 on,, then the result of this procedure will be minus the area between the graph of and,. 0 0 If takes both positive and negative value on,, then the procedure yields the net signed area between the graph of and the interval,

40 1. is integrable on, if Definite Integral: Definition lim exists and does not depend on the choice of partition or the choice of point Riemann Sum 2. If is integrable, then the limit lim is called the Definite Integral of over, [or from to ] and is denoted 40

41 : lower limit of integration : upper limit of integration Be careful not to confuse and. They are entirely different types of things. The first is a number, the second is a collection of functions. Notation: 41

42 The definite Integral of a continuous Function = Net Area under a curve Theorem: If is continuous on, then is integrable on, And, Notation: 42

43 We will need methods for evaluating the number other than computing the limit that defines them. Some methods generally involve antidifferentiation, but some definite integrals can be evaluated by thinking of them as area. 43

44 Definite Integrals Using Geometry

45 Finding Definite Integrals: A new Definition and Properties 1. If is in Domain of, define 2. If is integrable on,, define 0 45

46 Theorem: If is integrable on any closed Interval containing,, Then No matter, how,, are ordered! Theorem: Suppose,, integrable on, a. If 0 for all in,, Then b. If for all in,, Then 0 46

47 There are two parts to this. Development: The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus, Part I Suppose: is continuous on, and [ differentiable means continuous] Partition 47

48 on each interval: The Mean Value Theorem for derivatives applied to on each interval 48

49 Taking a limit as 0 give us the definite Integral 49

50 FTC, Part I If is continuous on, and is any antiderivative for on,. then Notice. If is any antiderivative of, So, we can always omit writing here. Thus 50

51 The Fundamental Theorem of Calculus, Part II FTC, I provided suggests 51

52 The Fundamental Theorem of Calculus says: If is continuous on the Interval, then has an antiderivative on If is in then is one such antiderivative for meaning 52

53 Differentiation and Integration are Inverse Processes: FTC, PartI Integral of derivative recovers original function FTC, PartII Derivative of integral recovers original function. 53

54 Definite and Indefinite Integrals Related: is a function in is a number no involved! So, the variable of integration in a definite integral doesn t matter: The name of the variable is irrelevant. For this reason the variable in a definite integral is often referred to as dummy variable, place holder. 54

55 Some Examples:

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57 Definite Integration by Substitution. Extending the Substitution Method of Integration to definite Integrals to evaluate the number Substitution:,, Change - limits to -limits with the substitution: To get 57

58 Examples: 1. Find 1. substitution of : limits substitution: lower limit: 1 2 upper limit:

59 2. Find: 2 1. substitution: 2 2. limits substitution: lower limit: 1 1 upper limit:

60 The Definite Integral Applied Total Area Although We can find that " " 60

61 Example. Compute the area between 0,5 0,5 2 2, the axis and the lines 2,5 and 2,5: Nullpoints: 0,5 0, , f(x) = a x³ + b x² + c x + d x

62 Function: Antiderivative: Area 0,5 0, , , ,, 2 2, ,5 2 4,67 3,76 0,96 4,66 0,67 0,96 1,03 0,67 0,90 5,625 0,29 0,36 7,175 62

63 Area between Two Curves [one floor, one ceiling] 63

64 Examples: 1. Compute the area of the region between the graphs of and 6. To identify the top and the bottom and the interval, we need a sketch Intersections: , 2, 3,2 Area:

65 2. Compute the area of the region between two graphs: 4 6 and Intersections: , Area: