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Tutors: Will Cairncross, Murray Wong

APSC 172: Calculus II Exam-AID

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Main Topics

1. Multivariate functions and partial derivatives2. Integration3. Power series

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Multivariate Functions and Partial Derivatives

Topics:• The intuition behind multivariate calculus• Visualizing and sketching functions

• Partial derivatives• Approximations• Chain rule• Gradient and the directional derivative• Extrema of functions

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Intro to Multivariate Calculus

• Examples• Equations of lines and

planes• Surfaces• Functions of many

variables

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Example

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

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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).

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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!

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Example

Find the total differential of

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Example

Find the second partial derivatives of

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Gradient

• Extension of the derivative to three dimensions• Tells the direction of maximum change of a

function– Why?

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Example

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

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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.

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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>

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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.

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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.

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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:

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Maxima and Minima continued

4 Cases:1. M > 0 and fxx > 0

– Local minimum

2. M > 0 and fxx < 0– Local maximum

3. M < 0– Saddle point

4. M = 0– Inconclusive

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Example (Final 2010)

Find the critical points of the function

and decide in each case whether it is a maximum, minimum, or saddle point.

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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

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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:

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Example

Find the volume of

over the region [0,1]×[0,1].

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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)

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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 areainfinitesimal area

…small massesinfinitesimal massesdensity

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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.

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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

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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

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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

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Steps for Triple Integrals

1. CAREFULLY sketch the region, taking note of the boundary points.

2. Decide in which direction to integrate first. – Remember that the last integral you do (the outer

one) MUST be between two constants.3. Carry out the integral (if they ask you to!)

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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

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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

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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.

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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?

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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?

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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 Sb cannot converge.

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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.

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Power Series continued

• Ratio test:

• 3 possible outcomes:1. Limit diverges: The power series converges for all x2. The limit is a finite number: This number defines the

radius of convergence; the series converges on 0 < x < L3. 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

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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

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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.

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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]