APSC 172: Calculus II Exam-AID

1. APSC 172: Calculus II Exam-AID Tutors: Will Cairncross, Murray Wong

2. Main Topics • Multivariate functions and partial derivatives • Integration • Power series

3. Multivariate Functions and Partial Derivatives Topics: • Theintuition behind multivariate calculus • Visualizing and sketching functions • Partial derivatives • Approximations • Chain rule • Gradient and the directional derivative • Extrema of functions

4. Intro to Multivariate Calculus • Examples • Equations of lines and planes • Surfaces • Functions of many variables

5. Example Find the equation for the plane through (1,5,2) with the normal <3,-2,1>

6. Example (Final 2009) Let a) Write down the equation of the level surface of f passing through (1,2,2). b) Find the equation of the tangent plane to the level surface of f at the point (1,2,2).

7. Partial Derivatives and Chain Rule • How does the function change when we change one variable? • Notation: “di-by-di-x” OR • Higher order: OR • Chain rule: It’s the same, but with more variables!

8. Example Find the total differential of

9. Example Find the second partial derivatives of

10. Gradient • Extension of the derivative to three dimensions • Tells the direction of maximum change of a function • Why?

11. Example Find the maximum rate of change of f(x,y,z) at (4,3,-1) and the direction in which it occurs.

12. Directional Derivatives • Gives the rate of change of a multivariate function in a particular direction. • Why the dot product? • The dot product acts like a “weighted average” between the components of u.

13. Example • Find the instantaneous rate of change of the value of f (x,y)at (5,1) in the direction given by v=<12,5>

14. Tangent Planes and Linear Approximations • In the same way we use a line to approximate a curve in a small region, the same can be done with a surface.

15. Example Find the equation to the tangent plane to at (3,-2,5). Use this tangent plane to approximate z at x = 3.007, y = -1.995.

16. Maxima and Minima • Extrema are points where the value of the function does not change with any infinitesimal change in one of the independent variables, ie. the function z = f (x,y) has an extremum (max or min) when:

17. Maxima and Minima continued 4 Cases: • M > 0 and fxx> 0 • Local minimum • M > 0 and fxx < 0 • Local maximum • M < 0 • Saddle point • M = 0 • Inconclusive

18. Example (Final 2010) Find the critical points of the function and decide in each case whether it is a maximum, minimum, or saddle point.

19. MULTIPLE INTEGRALS • Extension of integral to higher dimensions • The main concept remains the same • Add up infinitesimal bits of a function by finding an area/volume

20. Double Integrals over Rectangles • Used to find the volume of the region below a function of 2 variables z = f (x,y) • Simple case where the limits of integration are constants • Extension of the Riemann Sum to two variables:

21. Example Find the volume of over the region [0,1]×[0,1].

22. Double Integrals over General Regions • Concept is unchanged, but order of integration becomes more important. • Need to choose which variable to express as a function of the other • Eg. Triangle with vertices (-1,1), (0,0), (1,1)

23. Moments & Centres of Mass • The centre of mass with respect to a certain axis is the the line (parallel to that axis) of a knife-edge on which the shape would balance • We extend the idea of “adding up little bits of things” using mass • Just as small bits of areainfinitesimal area …small massesinfinitesimalmassesdensity

24. Double Integrals in Polar Coordinates • Just as we went from adding up infinitesimal strips of area to adding up infinitesimal boxes of area, we can add up any infinitesimal area we like! • If we choose to describe space in terms of (r,θ) coordinates we can express a bit of area as: • Just like in xyz, the challenge is finding the right boundaries for our integral.

25. Triple Integrals • Expand the idea of “adding up infinitesimal bits” to 3D regions by adding up cubes of sides dx, dy, dz • Like in double integrals, changing order will give the same numerical result BUT may make the integral more or less difficult to evaluate

26. Triple Integrals continued • As with double integrals, there are again three main types: • Type 1: Between two surfaces where z is a function of x and y • Integrate w.r.t z first

27. Triple Integrals continued • Type 2: Integration between two surfaces where x is a function if y and z • Integrate w.r.t. x first • Type 3: y is a function of x and z • Integrate w.r.t. y first

28. Steps for Triple Integrals • CAREFULLY sketch the region, taking note of the boundary points. • Decide in which direction to integrate first. • Remember that the last integral you do (the outer one) MUST be between two constants. • Carry out the integral (if they ask you to!)

29. Cylindrical Triple Integrals • These are nearly identical to our double integrals in polar coordinates, except another dimension is added • Now instead of “summing up” infinitesimal cubes, we are putting together regions that look like

30. Cylindrical Triple Integrals continued • Our formula to find the volume of one of these small blocks is • Like in polar coordinates, the r is included to turn our infinitesimal angle θ into a length rdθ

31. Introduction to Series • This is a major shift of gears in the course! But very important material, especially for engineers. • This is because all real engineering applications will have elements of approximation. • We start by looking at the basic concept of a series, before moving towards the ways series are used to approximate functions.

32. Introduction to Series continued • Questions for series: • What is an infinite sum? Does it have a definite value? • What does it means to equate a function f(x) to an infinite series? • How can we express a general function as the sum of an infinite series?

33. Geometric series • We have the formula: for geometric series, provided r < 1 • (would anyone to see the derivation again?) • This answers our questions for geometric series, but what about more complicated series where the constant changes with each term?

34. Power Series • Comparison test • Given two series • If an < bn for every nth term… • If Sb converges, then Sa must also converge. • If Sa does not converge, then Sbcannot converge.

35. Power Series continued • Absolute Convergence Theorem: • If the absolute value of a series converges, then so must the original series. • This makes sense, since absolute values |f(x)| can only make things bigger! (more positive) • Without the absolute value signs, there would definitely be a few negative terms to make things smaller.

36. Power Series continued • Ratio test: • 3 possible outcomes: • Limit diverges: The power series converges for all x • The limit is a finite number: This number defines the radius of convergence; the series converges on 0 < x < L • L = 0: The series converges only for x = 0 • For series not centered at the origin, the radius is defined as the “distance” from wherever the series is centered

37. Power Series continued • Differentiation and Integration: • Differentiation: the term-by-term derivative of a series… • has the same radius of convergence as the original series • is equal to the derivative of whatever function is represented by the original series • Integration: The term-by-term integral of a series… • has the same radius of convergence as the original series • is equal to the derivative of whatever function is represented by the original series

38. Taylor Series • A repeated application of integration by parts gives the formula: • This formula is an incredibly nice way to represent functions! • BELIEVE IT OR NOT… we already used this formula in our formulas for linear approximations and tangent planes • Note: A Taylor series centred at a=0 is a Maclaurin Series.

39. Taylor Series continued • By truncating (cutting off) the Taylor series of a function at a certain nth term, we are creating the nth-order Taylor polynomial Tn of the function. • We can use Taylor’s inequality to estimate the error in our approximation: • Good choice for M: Calculate the n+1st derivative and take its maximum value on your region [a-d,a+d]