Post on 23-Feb-2016
description
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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]