Due Wednesday 12/5 at 9am

Section 11.6 Power Series as Functions
Section 11.7 Taylor Series

Reading Questions:

1. Give two good reasons for writing a known function ( such as cos(x) ) as a power series.

   SP says:
   

   First, a power series lets us find a very accurate approximation of values of trigonmetric functions (for instance) since they lack a finite algebraic formula.

   Also when a function is in a power series form, it is easier to antidifferentiate.

2. How does a Taylor series differ from a Taylor polynomial?

   SR says:
   

   Taylor series are extended versions of Taylor polynomials; Taylor polynomials are partial sums of Taylor series.

3. Why would you ever want to find the Taylor series of a function?

   EL, NGP, JK and WS combine to say:
   

   We want to find a Taylor series of a function because it is easy to evaluate polynomials at x_0. Taylor's Theorem guarantees that given appropriate conditions, the Taylor series of a function will converge to the value of the function.

   Taylor series allow a different form of interaction with the function, potentially simplifying tasks like finding the integral of that function.

   In finding integrals and solving differential equations ... sometimes solutions cannot be found (the usual way). ... The motions of a "simple" pendulum cannot be described with mere integrals and polynomials, but a Taylor series will describe the motion.

Reminder:

• Bring questions on PS 13 to class.

Section 11.5 Power Series

Reading Questions:

1. How do power series differ from the series we have looked at up to this point?

   MH says:
   

   A power series is in the form of a0 + a1(x-x0) + a2(x-x0) +... = sum(ak(x-x0)^k, k=0.. infinity). Here x is a variable, the constants ak are called the coefficients, and the constant x0 is called the base point. Unlike other series, a power series contains variables.

   It may converge for some values of x and diverge for others.

2. What is the interval of convergence of a power series? Explain in your own words.

   JH says:
   

   The interval convergence is the domain on which the power series will converge.

Due Monday 11/19 at 9am

Section 11.4 Absolute Convergence; Alternating Series

Reading Questions:

1. How closely does S₁₀₀ approximate the series sum((-1)^k (1/k), k=1 .. infinity) ? Why?

   WS says:
   

   Because this series fits the two conditions of the alternating series test
   (c₁ >c₂>c₃ > ... and the limit as k approaches infinity of c_k is 0)
   we can say that |S-S_n|< c_n+1.

   If we set n=100, we find |S-S₁₀₀|<c₁₀₁. So our approximation is at least within 1/101 of the series sum.

Due Friday 11/16 at 9am

Section 11.4 Absolute Convergence; Alternating Series

Reading Questions:

1. Give an example of a series that is conditionally convergent. Explain.

   KD says:
   

   A series that is conditionally convergent is sum((-1)^(k+1)/k,k=1..infinity) because the absolute value of the terms added up diverges but the sum as it is written converges.

2. Give an example of a series that is absolutely convergent. Explain.

   AJ says:
   

   sum(cos(k)/k^2,k=1..infinity) is an example of an absolutely converging series because the absolute value of each term adds up to a limit.

Due Wednesday 11/14 at 9am

Section 11.3 Testing for Convergence; Estimating Limits

Reading Questions:

1. Explain in a couple of sentences why the Ratio Test makes sense.

   PM says:
   

   The Ratio Test makes sense because the idea is that as k increases, (if the limit of a_k+1/a_k exists) the series becomes more and more comparable to a geometric series. Since we know how to determine convergence or divergence for a geometric series, we can use this comparison to determine the convergence or divergence of the series.

Due Wednesday 11/7 at 9am

Section 11.3: Testing for Convergence; Estimating Limits

Reading questions:

1. Explain in a couple of sentences why the Comparison Test makes sense.

   MC says:
   

   The comparison test makes sense because if a_k is smaller than b_k (for all k), then when the terms are added together in the series, the sum of the a_k will naturally be smaller (or at least not larger than) the sum of the b_k.

   For example, a_k=k+2 is smaller than b_k=k+3. If the value of k goes from 0..3, then the value of the sum of k+2 would be 2+3+4+5=14 and the value of the sum of k+3 would be 3+4+5+6=18. In this example (and others), b_k is larger than a_k and the sum of b_k is larger than the sum of a_k.

   Note: If a_k < b_k for all k, then a partial sum of the first n a_k terms will also be smaller than the partial sum of the first n b_k terms. However, the infinite sums may be equal. The important point is that the sum of the smaller terms can't shoot ahead and become larger than the sum of the larger terms.

2. Explain in a couple sentences why the Integral Test makes sense.

   JE says:
   

   Think of a (partial) sum as Riemann sum. Now, compare the sum -- now being represented as a Riemann sum -- to the corresponding integral. If the sum is less than a known, converging, integral, we know that the sum converges. if the sum is greater than a known, diverging integral, we know the sum diverges.

   By comparing the sum to an integral, we can figure out more information about the sum's convergence or divergence.

Due Friday 11/2 at 9am

Section 11.2 Infinite Series, Convergence, and Divergence

Reading questions:

1. There are two sequences associated with every series. What are they?

   EL says:
   

   Every series involes two sequences: 1.) the sequence of terms and 2.) the sequence of partial sums

   Note:
   The sequence of remainders (which are like tails) is a third sequence associated with every series.

2. Does the geometric series sum((1/4)^k,k=0..infinity) converge or diverge? Why?

   SP says:
   

   The geometric series converges because r=1/4 which is less than 1 and Theorem 4 states that if abs(r) < 1 the geometric series converges.

3. What does the nth Term Test tell you about each series? Explain.

   (a) sum(sin(k), k=0..infinity)
   

   LD says:
   

   the nth Term Test tells us that the sum diverges because the limit of sin(k) does not exist and so it can't be equal to zero.

   (b) sum(1/k , k=1..infinity)

   AS says:
   

   lim _k->inf 1/k=0, so the nth term test does not conclude anything.

Due Wednesday 10/31 at 9am

Section 4.2 More on Limits: Limits Involving Infinity and l'Hopital's Rule
Section 11.1 Sequences and Their Limits

Reading Questions:

1. Does l'Hopital's Rule apply to lim_{(x -> infty)} x² / e^x ? Why or why not?

   PM says:
   

   Yes it does apply because the limit as x approaches infinity of x^2/e^x is in indeterminate form.

   When you apply l'Hopital's rule twice, you see the limit is 0.

2. Does l'Hopital's Rule apply to lim_{(x -> infty)} x² / sin(x) ? Why or why not?

   WS says:
   

   No it does not.

   x^2 diverges towards infinity, but sin(x) occilates forever never approaching a certain value. Thus l'Hopital's Rule cannot be used.

3. Does the following sequence converge or diverge? Be sure to explain your answer.
   1, 3, 5, 7, 9, 11, 13, . . .

   JE says:
   

   This sequence diverges to infinity because it is not settling on a limit- all the numbers are odd and are going to continue to be increasingly odd to infinity.

4. Find a symbolic expression for the general term a_k of the sequence 1, 2, 4, 8, 16, 32, . . .

   AJ says:
   

   a_k = 2^k for a = 0, 1, 2, 3, ...

Due Friday 10/26 at 9am

Section 10.2: Detecting Convergence, Estimating Limits
Reading Questions:

Suppose that 0 < f(x) < g(x).

1. If int(f(x), x=1. .infty) diverges, what can you conclude about int( g(x), x=1. . infty)?

   EP says:
   

   If int(f(x), x=1..infty) diverges then int(g(x),x=1..infty) diverges as well.

2. If int(g(x), x=1. .infty) diverges, what can you conclude about int( f(x), x=1. . infty)?

   MC says:
   

   We cannot conclude anything about int(f(x), x=1..infty) because it is smaller than (or equal to) int(g(x), x=1..infty) and thus, COULD be finite, but may not be. We would have to do further testing.

3. If int(f(x), x=1. .infty) converges, what can you conclude about int( g(x), x=1. . infty)?

   KD says:
   

   If int(f(x), x=1. .infty) converges, then nothing conclusive has been said about int( g(x), x=1. . infty) because it may converge or diverge.

Due Monday 10/22 at 9am

Section 10.2: Detecting Convergence, Estimating Limits

Reading questions:

1. If 0 < f(x) < g(x) and int( g(x), x=1. . infty) converges, will int(f(x), x=1. .infty) converge or diverge? Why?

   AB says:
   

   int f(x) will converge because since int g(x) converges and is larger, then int f(x) must be a smaller finite number.

2. There are two types of errors that arise in Example 4 for approximating int( 1/(x⁵ +1), x=1..infty). What are the two types?

   NGP says:
   

   The two types of errors arise from the two components we use to find the approximation of
   int( 1/(x5 +1), x=1..infty).

   Because we would break this integral up into two integrals, we would have errors based on each one of those integrals.

   The first integral would be proper, meaning that we will be able to find a numerical approximation for it using left sum, right sum, or midpoint rule. We could then find the error of that approximation by bounding the error and using the methods we learned before.

   We would then have to find the error of the second integral (that comes from ignoring this second integral), which we could do by comparing it to a simpler improper integral ... to find an error for that integral.

   Then the numerical approximation we found using the first (proper) integral would have an error less than both error bounds added together.

Due Wednesday 10/17 at 9am

Section 10.1: Improper Integrals: Ideas and Definitions

Reading questions:

1. What are the two ways in which an integral may be improper?

   CN says:
   

   An integral can be improper if there is an infinite interval (meaning the interval of integration is infinite), or if there is an infinite integrand (meaning the integrand is unbounded somewhere on the interval of integration).

2. Explain why int( 1/x², x=1..infinity) is improper. Does the integral converge or diverge?

   MB says:
   

   int( 1/x^2, x=1..infinity) is improper because the right bound is infinite. This integral converges.

3. Explain why int( 1/x², x=0..1) is improper. Does the integral converge or diverge?

   JK says:
   

   This integral is improper because it is unbounded over the interval it is defined on. This integral is divergent: its value cannot be calculated. :(

Due Monday 10/15 at 9am

Section 9.2 Taylor's Theorem: Accuracy Guarantees for Taylor Polynomials

Reading Questions:

Let f(x)=sqrt(x).
1. Find P₃(x) for f at the base point x₀=64.

   SR says:
   

   P3	=	sqrt(64)+(x-64)/2*64^(1/2)-(x-64)^2/8*64^(3/2)+(x-64)^3/16*64^(5/2)
   	=	8+(x-64)/16-(x-64)^2/4096+(x-64)^3/524288

2. What can you say about the error committed by using P₃(x) as an approx for sqrt(x) on the interval [50,80]?

   MK says:
   

   | f''''(x)| = 15/(16(x^7/2))

   Graph of |f''''(x)|

   Therefore K4 = |f''''(50)|=15/[16(50)^7/2]= 1.06x10^-6.

   To make x-64 as large as possible, use x=80.

   Substituting in,
   

   | f(x) - Pn(x) |	<=	Kn+1|x-x0|n+1/(n+1)!
   	<=	(.00000106)(80-64)4/4!
   	<=	.0029

   The error is approximately 0.0029 which considerably good.

Due Friday 10/12 at 9am

Section 9.2 Taylor's Theorem: Accuracy Guarantees for Taylor Polynomials

Reading Questions:

What is the point of Theorem 2? Explain in your own words.

The purpose for Theorem 2, Taylor's Theorem, is to provide a maximum possible amount of error (upper-bound for error) between the approximation Pn(x) and the actual function f(x).

It uses the principle that the closer then (n+1)st derivative of a function is to 0 the less error there will be between Pn(x) and f(x).

MC adds:

...The factorial ensures that as as n increases, the error bound decreases.

And ED adds:

... The closer you get to that number (the base point), the more accurate your approximation will be.

Due Friday 10/5 at 9am (this is really the reading for next Wednesday's class)

Section 9.1 Taylor Polynomials

Reading questions:

Explain the basic idea of the Taylor polynomial for a function f(x) at x=x₀ in your own words.

JH says:


The idea of a Taylor polynomial is to develop a function that we can easily evaluate and work with that acts very similarly to our target function.

By matching up derivatives at a specific point we can bend our generic functions closer and closer to being identical to the target function.

Due Wednesday 10/3 at 9am

Section 8.1 Integration by Parts
Guide to Writing Mathematics

Reading questions:
Each integral can be evaluated using u-substitution or integration by parts. Which technique would you use in each case? You do not need to evaluate the integral, but explain your choice.

1. int( x*cos(x), x)

   EP says:
   

   I would use integration by parts in int( x*cos(x), x) because x and cos(x) are unrelated to each other.

   If u substitution was used with u=x, then du=dx, giving an equation that hasn't simplified.

2. int(x*cos(x²),x)

   WS says:
   

   u substitution shows potential in this case:
   the derivative of x^2 is 2x, which is pretty close to x.

   u substitution would likely switch all x's for u's (and du's) and do so in a useful fashion.

Due Monday 10/1 at 9am

Section 8.1 Integration by Parts

E-mail Subject Line: Math 104 Your Name 10/1

Reading questions:

1. Integration by parts attempts to undo one of the techniques of differentiation. Which one is it?

   MH says:
   
   

   Integration by parts attempts to undo the product rule.

2. Pick values for u and dv in the integral int( x * sin(x), x). Use parts to find an antiderivative for x * sin(x).

   EP says:
   
   

   For the integral int( x * sin(x), x), pick u= x, and dv=sin(x). Therefore, du=dx and v= - cos(x). Using integration by parts, int( x * sin(x), x)= x(-cos(x)) - int(-cos(x),x)= -xcos(x)+sin(x) (+C). This can be checked by differentiation.

Due Monday 9/24 at 9am

Section 7.2 Finding Volumes by Integration
Guide to Writing Mathematics

Reading questions:


1. Let R be the rectangle formed by the x-axis, the y-axis, and the lines y=1 and x=3. What is the shape of the solid formed when R is rotated about the x-axis?

   JK says:
   

   The solid is a cylinder with a radius of 1 and a length of three.

2. Let T be the triangle formed by the lines y=x, x=1 and the x-axis. What is the shape of the solid formed when T is rotated about the x-axis?

   PM says:
   

   When T is rotated around the x axis, a cone-shaped solid is formed

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Due Friday 9/21 at 9am

Section 7.1 Measurement and the Definite Integral; Arc Length

Reading questions:
Let f(x)=sin(Pi*x/2)+10 and g(x)=3x/10+10.

1. Set up the integral that determines the area of the region bounded by y=f(x) and y=g(x) between x=0 and x=5/3.

   ED says:
   

   

   Find the points of intersection on the graph. Conveniently they are at x=0 and x=5/3.

   Looking at the graph we can tell that f(x) is the upper bound and g(x) is the lower bound. Therefore we can write our integral as
   

   Area=int(f(x),x= 0 .. 5/3) ­ int(g(x), x= 0 .. 5/3) = int(f(x)-g(x), x=0 to 5/3).

2. Set up the integral that gives the length of the curve y=g(x) from x=-1 to x=3.

   AS says:
   

   g(x)=3x/10+10, and so g'(x)=3/10.

   Using the formula int(sqrt( (f'(x))^2+1 ), x=a..b), the length of the curve y=g(x) is

   int(sqrt((3/10)^2+1), x= -1..3)

Reminder:


• Begin PS 4. This problem set looks shorter than usual, but beware -- the problems take thought. Get started early, so you have time to ponder them and put various ideas together.

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Due Monday 9/17 at 9am

Section 6.2 Error Bounds for Approximating Sums
Reading questions:


How many subdivisions does the trapezoid method require to approximate int( cos(x³), x = 0. . 1) with error less than 0.0001?

EP says:


The trapezoid method requires at least 92 subdivisions to have an error bound less than 0.0001.

This is done using the equation:

|I-T_n| <= K₂(b-a)³/ (12n²).

• We know that (b-a)^3 is 1, because we know know the bounds of int(cos(x^3)).
• Next is to find K₂, which is an upper bound of the absolute value of the second derivative of cos(x³).

  Using Maple, with the command
  plot([abs(diff(cos(x^3),x,x)),10],x=0..1,color=[red,blue]);

  

  10 works as an upper bound.
• Next is to solve for n, which is the number of subdivisions. n is found as 91.29, so rounding up gives 92 subdivisions for n.

---

Due Friday 9/14 at 9am

Section 6.2 Error Bounds for Approximating Sums

Reading questions:


1. Explain in words what K₁ is in Theorem 3.

   CN says:
   

   K₁ is a constant whose graph is an upper bound for the graph of the absolute value of f'(x) on [a,b]. K₁ is used when working with left- and right-rule errors, similar to how in Theorem 1 the left- and right-rule sums correspond with f '(x) (since Thm 1 applies when f ' is always positive or always negative).

2. Explain in words what K₂ is in Theorem 3.

   MH says:
   

   K₂ is a constant which is used in a formula to determine a range of error for a trapezoid or midpoint sum approximation of the int(f(x)).

   K₂ is always going to be larger or equal to the largest value of |f ''(x)| on the interval [a,b].

3. Find values for K₁ and K₂ for int( x³, x= -3. . 1).

   KD says:
   

   To find K₁, we need to find the minimum and maximum slope values of x³ in the given interval. Having some prior knowledge of what the graph of x³ looks like, I know that the greatest slope (both positively and negatively) will be at x=-3. The derivative of x^3 is 3x^2 and plugging -3 in gives me 27. Therefore, K₁ can be any number higher than that.
   I choose K₁=30.

   To find K₂ we take the second derivative of x^3. This gives us 6x. Plug -3 in again (since this will give us the largest absolute value over [-3,1]). This gives us |-18|.
   I would choose K₂ to be 20.

---

Due Wednesday 9/12 at 9am

Problem Set Guidelines
Section 6.1 Approximating Integrals Numerically

Reading questions:


1. Why would we ever want to approximate an integral?

   PM says:
   

   We would want to approximate an integral because some integrals cannot be found using anti-differentiation and the Fundamental Theorem of Calculus.

2. Let f(x)=x² and I=int( f(x), x= -1. . 2). Does Theorem 1 apply to I? Explain.

   ED says:
   

   No, Theorem 1 does not apply because f(x) is not monotone. It decreases from [-1,0) then increases from (0,2).

3. Let f(x)=x² and I=int( f(x), x= -1. . 2). Does Theorem 2 apply to I? Explain.

   KD says:
   

   Theorem 2 applies to I because x^2 is concave up in the given interval so we know that Tn will overestimate I and Mn will underestimate I.

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Due Monday 9/10 at 9am

Section 5.6 Approximating Sums; The Integral as a Limit

Reading questions:


1. When approximating an integral, which would you expect to be more accurate, L₁₀ or L₁₀₀? Why?

   AJ says:
   
   

   L₁₀₀ would produce a more accurate estimate of the area under a curve because there would be less area outside the curve covered by the rectangles.

2. Give an example of a partition of the interval [0,3].

   TO says:
   

   If you wanted to use 6 subintervals on the interval [0,3] your partitions would be 0.5 units wide. Your partitions would be [0,0.5], [0.5,1], [1,1.5]... [2.5,3].

3. Explain the idea of a Riemann sum in your own words.

   MB says:
   

   A Riemann sum is the addition of multiple rectangles, each of which is chosen to approximate the area under part of a curve. As a result, a good Riemann sum is as close as you need to the actual value of the integral.

---

Due Friday 9/7 at 9am

Section 5.4 Finding Antiderivatives; The Method of Substitution

> Reading questions:


1. Explain the fundamental difference between a definite integral and an indefinite integral. Please go deeper than saying one has limits of integration and one doesn't. The first is a real number -- why? what does it represent? Then think similarly about indefinite integrals.

   LD says:
   
   

   A definite integral results in a real number, because you are ... solving for the area underneath the curve. ...An indefinite integral results in a "family" of values--indefinite integrals show ALL of the antiderivatives.

2. Substitution attempts to undo one of the techniques of differentiation. Which one is it?

   EL says:
   

   Substitution undoes the differentiation technique of the chain rule.

3. What are the three steps in the process of substitution?

   NGP says:
   

   1. Substitute: if you have int(f(x) dx), you need to choose a function u=u(x), find the derivative of u, and then plug both back into the original integral so int(f(x) dx) is in a new form, int( g(u) du).
   2. Antidifferentiate in u: solve int(g(u)du), meaning find the antiderivative of g(u).
   3. Resubstitute: once you have found the antiderivative of g(u), plug u(x) back into the new integral.

      To double check your answer you can then take the derivative of that and it should match up with your original f(x).

Reminder:


• Begin working on PS 2. This is a group assignment -- that means you must work with one other person (or in some cases, two other people). Start introducing yourself to people in the class, and try to find someone you think you can work well with.

  These groups are not permanent -- you're welcome to work with different people different weeks.

---

Due Wednesday 9/5 at 9am

Problem Set Guidelines
Section 3.4 Inverse Functions and Their Derivatives (Appendix S in your book, I believe)


Reading questions:


1. What is the domain of the function arccos(x)? Why is this the domain?

   MC says:
   

   The domain of arccos(x) is [-1, 1].

   This is because the range of cos(x) is [-1,1]. The x-values of a function are the equivalent of the y-values of its inverse.

2. Explain how we can tell lines which are neither horizontal nor vertical have inverses.

   JE says:
   

   We can tell lines which are neither horizontal nor vertical have inverses because they all pass the horizontal line test. A line that passes this test is a one-to-one function and therefore has an inverse.

3. Why do you think we are studying the inverse trig functions now?

   BS says:

   I think that the inverse trig functions will allow us to find the integrals/antiderivatives of complex equations/fractions like Problem 4.

4. Find one antiderivative of 1 / (1+x²).

   SP says:
   

   One antiderivative of 1/(1+x^2) is arctan(x).

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Due Friday 8/31 at 9am

guidelines for submitting reading assignments
suggestions for reading a math text
course policies
syllabus

Section 5.1: Areas and Integrals
Section 5.2: The Area Function
Section 5.3: The Fundamental Theorem of Calculus

E-mail Subject Line: Math 104 Your Name 8/31

Reading questions:

1. Why do you think it makes sense to call
   int(f(x),x=a..b)/(b-a) the average value?

   Notes:

   • The above is written in Maple notation -- it's as good a way as any to write mathematical ideas without symbols, with the added benefit that it gets you used to some Maple notation. The above says the integral of f(x), from x=a to x=b, all divided by b-a.
   • The text doesn't specifically address this question; the reason I'm asking this is because this is exactly the sort of question you should be learning to ask yourself (and attempting to answer) when you read.
   JH says:
   

   The average value of a function indicates the average output of that function, which I will call 'average'.

   The idea is that if you graphed the line

   y = 'average', and then found the area of the rectangle formed between y = 'average', the x-axis, x = a, and x = b you would have the exact same area as int(f(x),x=a..b) denotes.

   SO the rectangle with a height defined by 'average' and a width defined by the interval length (b-a) has the same area as int(f(x),x=a..b).

   So to find the average value if you have area you must divide the area by the rectangle,s width (b-a).

   int(f(x),x=a..b)/(b-a) does exactly that.

2. Find the signed area between x^5 and the x-axis from x=1 to x=2. DG says:
   

   In order to find the area of the region we must take the antiderivative (FTC, version 2).

   In this case it is (1/6)*x^6.

   Then we subtract the outputs: F(b)-F(a).

   The answer is 64/6 - 1/6 = 63/6.

3. If f(x) is continuous, must it have an antiderivative? If your answer is yes, does that mean there must be a nice formula (or any formula at all) for the antiderivative?

   Combining a few people's responses: AB & AS says:
   
   

   if f(x) is continuous then it has ... measurable area between the function and the x axis. According to the FTC, the area function is an antiderivative, and so an antiderivative exists.

   This does not guarantee that a formula for the antiderivative can be found.

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Janice Sklensky
Wheaton College
Department of Mathematics and Computer Science
Science Center, Room 109
Norton, Massachusetts 02766-0930
TEL (508) 286-3973
FAX (508) 285-8278
jsklensk@wheatonma.edu

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