MA1032-Numerical Analysis-Analysis Part-14S2- 1 of 8

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1 MA1032-Numerical Analysis-Analysis Part-14S2- 1 of 8 RIEMANN INTEGRAL Example: Use equal partitions of [0,1] to estimate the area under the curve = using 1. left corner of the intervals 2. right corner of the intervals 3. midpoint of the interval 4. line joining the left and right corners of the interval Definitions: is a partition of [,] iff it is a ordered set of the form ={,,, } with =, = and > is a refinement of iff is a common refinement of, iff = [,] is the set of all partitions of [,] Definition: Upper and Lower Riemann Sums :[,] R is a bounded function, =,= where =sup{ [, ]},= where =inf{ [, ]} Definition: Upper and Lower Riemann Integrals =inf {, [,]} =sup {, [,]} Definition: is Riemann Integrable on [,] or R[,] iff = Riemann Integral of is the common value denoted by Theorem: is a refinement of 1.,, 2.,, Theorem: Theorem: R[,] iff >0 [,];,,< Theorem: If R[,] and [,] such that [, ] then <,, Theorem: [,] R[,] Theorems:, R[,] 1. + R[,] and += 2. R[,] 3. R[,] and () () 4. () 5. () () ( ) + 6. [,] R[,], R[,] and ()= Definition: is Uniformly continuous on R >0, >0,, ; < ( ) ( ) < Definition: is Lipschitz continuous on R >0,, ; ( ) ( ) ()+ Theorem: Lipschitz continuous Uniformly continuous Continuous ()

2 MA1032-Numerical Analysis-Analysis Part-14S2- 2 of 8 Example: Show that is not uniformly continuous on 0,1] but is. Theorem: Fundamental Theorem of Calculus If R[,] and there is a differentiable function such that = then = Theorem: Second Fundamental Theorem of Calculus If R[,] and [,] and = then 1. is continuous on [,]. 2. If is continuous at a point [,] then is differentiable at and =. Theorem: Integration by Parts, differentiable on [,], = R[,]and = R[,] h = Theorem: Change of Variable has continuous derivative [,]. is continous on [,]and let =, [,].Then for each [,], exixts and has value. Theorem: Mean Value Theorem for Integrals R[,]with.Then [,] such that =. If also [,] then, such that =. Definition: Improper Integrals of the first kind Suppose exixts for each. If lim exists and equal to R Otherwise we say that diverges we say that converges and has value Definition: Improper Integrals =lim, :[, R =lim, :,] R = +, :, R, R =lim,:,] R =lim,:[, R = +, :[,,] R,, Example: Find if it exists Example: Prove that if is bounded above and increasing, then lim is existing and finite Prove that () converges () converges Prove that if (), then the Laplace Transform of, = exists for all >. Theorem: Comparison Test Assume that the proper integral exists for each and suppose that 0 for all,then converges converges Theorem: Limit Comparison Test Assume both proper integrals and exist for each,where 0 and >0 If lim =c,then

3 MA1032-Numerical Analysis-Analysis Part-14S2- 3 of 8 1. c 0, converges converges 2. c=0 and converges converges 3. c= and diverges diverges Note: There are similar comparison tests for other improper integrals Example: Gamma Function is defined by Γ=. Show that 1. Γ exists for all >0 2. Γ= 1Γ 1 3. Γ= 1!for integer 1 4. we can use 2. to deine Γ for < 0 5. Γ does not exist for = 0, 1, 2, 3,. 6. Show that Γ = and = using ΓΓ1 = 7. Use the formula for the the dimesional ball = to ind volumes of 2,3,4,5 dimesional balls 8. Use the fact that Γ + 1~ 2 asmptotically as to ind 10! approximately 9. What is Γ 1?. It is called the Euler Constant and no one knows if it is rational or irrational! Prove that the Beta function, = 1 exists for all, > 0. It can be shown that, =.

4 MA1032-Numerical Analysis-Analysis Part-14S2- 4 of 8 MULTIVARIATE CALCULUS Definition: Function of two variables : R R Example: Draw the graphs of the following functions/surfaces 1., = + 2., = = 1 Definition: Limit lim,,, = > 0 > 0,0 <,,, <, < Note: Matric 0 <,,, < is a region around and excluding,. Some options for the matric are max, We will use the first matric. One can show that they are equivalent, what is needed is a region around,. Example: Use the definition to show that lim,, = 6 Example: Investigate the existence of the limit, lim,,, for the following functions,, 0,0 1., = 0,, = 0,0 2., =,, 0,0 0,, = 0,0 3., = sin,, 0,0 0,, = 0,0 Theorem: If lim,,, =, lim, and lim, exists then lim lim, = lim lim, =. Example: Use the above theorem to prove that lim,,, is not existing for, =,, 0,0. 0,, = 0,0 Prove by definition that if lim,,, along = and = 2 are different, then the limit is not existing. Definition: Continuity of at, lim,,, =, Definition: Partial derivatives, =, =, = lim,,, =, =, = lim Definition: and,, = lim,, = lim,, Theorem: Mean Value 1. and exists 2. =, + < } 3. + < Then 1. +,+,= +,+ +,+ 2. 0<,<1

5 MA1032-Numerical Analysis-Analysis Part-14S2- 5 of 8 Definition: Differentiability of at, 1. and exists at, 2. +, +, =, +, +, +, for all + < 3. lim,,, = lim, = 0,, Theorem: Example: Let, = +, = sin, 0 = 0. Show that but Definition: Higher order derivatives = = = = = = = = = = = = and so on Note: 1. We write to mean,,, 2. In a similar manner we write to mean that all the th order partial derivatives are continuous. There are 2 of them. 3. There are = = Example: Let!!!, =,, 0,0. 0,, = 0,0 Show that 0,0 0,0., th order partial derivatives that contains, times. Theorem: = Example: If =, then prove that the Laplace operator = = + + when = cos, = sin. Theorem: Chain rule 1. =,, =, = all in. Then = + 2. =,, =,, =, all in. Then = + and = + Note: The above may be written as = =, The determinant, det,, =, and =, is called the Jacobian or With = and =, the above may also be written as = and = = +,,, becomes We also see that = =, is acting as the true first derivative of =,. Therefore it is also called = grad or the Gradient of.

6 MA1032-Numerical Analysis-Analysis Part-14S2- 6 of 8 Example: Assume all functions are Show that if =,, =,, =,, =, then, =,,,,,. Show that if =,, =, then a functional relation of the form h, = 0 exists iff det,, 0. Definition: Directional Derivative of in the direction of the unit vector =, at,., = lim,, Theorem:,, 0 1., = 2. max, =, 3. min, =,, +, =,, =,, =, Theorem: Normal vector to a surface at,, =,,,, 1 =,, 1 Proof: Let = =,, be a curve on the surface of =, and =,, =,,,. Note that =,, is the tangent vector to the curve at,. Now, =, +, = = 0 Ie, =,,,, 1 =,, 1 is a vector perpendicular to the surface =, at,. Theorem: Equation of the tangent plane to the surface =, at,. =, +, =, =, Example: Let, = At the point 1,2 find 1. Direction in which the function increases most rapidly 2. Directional derivative in that direction 3. Equation of the tangent plane. Theorem: Taylor s expansion for one variable : R R If and, + h then + h = where is between and + h. h! +! Note: We can also write the above as If and + h for all [0,1] + Then + h = h! for some = + h with 0,1. h! h We agree to use the notation h h Note: The first two terms are the equation of the tangent line. Proof: Use generalized mean value theorem on =! and = Example: When = 1 + h = +! h +! h Example: Write the Taylor s expansion for = at = 0. Example: Derive the second derivative test to find the extrema of. What to do when = 0?

7 MA1032-Numerical Analysis-Analysis Part-14S2- 7 of 8 Theorem: Taylor s for two variables : R R and + h, + for all [0,1] Then + h, + = h +!, + for some = + h, + with 0,1. Proof: Use Taylor r expansion for = + h, + Example: When = 1 + h, + = +, +! h! h! h + + h, + =, + h +, + h +!! =, + h +, + h!! + 2h + h =, +, h +, +! h + 2 h + =, +,, h + h! h =, +, h +! h h =, +!, h +! h h + Note: The first two terms are the equation of the tangent plane. Definition: = = det= : determinant tr= + : trace : Hessian of Note: det>0 and >0<0 >0<0 tr>0<0 Example: Write the Taylor s expansion for, = and, = sin sin + at, = 0,0. Get the same answer by applying multiple one variable Taylor series expansions at 0. Definition:, is a critical point of, = or is not defined Definition: 1. has a relative maximum at,, + h, + in a neighbourhood of, 2. has a relative minimum at,, + h, + in a neighbourhood of, 3. has a saddle point at, is both above and below its tangent plane at,. Theorem: and, is a relative maximum/minimum/saddle point of, = Theorem: and, = then 1. det,>0 and tr,>0 then, is a relative mimimum 2. det,>0 and tr,<0 then, is a relative maximum 3. det,<0 then, is a saddle point 4. det,=0 inconclusive(why?) Example: Find the critical points and determine the nature of them ( relative maxima/minima/saddle points)., = , = + + 1, = + Example: Propose a method to determine the nature of critical points when det=0.

8 MA1032-Numerical Analysis-Analysis Part-14S2- 8 of 8 Theorem: Lagrange Multipliers If, and then the maxima/minima of, subjected to, = 0 are included in the set of solutions of, =, and, = 0. Example: Find the shortest distance from the point 1,0 to the parabola = 4. Find the directions of the axes of the ellipse = 0. Find the absolute maximum/minimum of, = on the closed disk

Topics Covered in Calculus BC

Topics Covered in Calculus BC Calculus BC Correlation 5 A Functions, Graphs, and Limits 1. Analysis of graphs 2. Limits or functions (including one sides limits) a. An intuitive understanding of the limiting