Here is material about the administration of the course:

- Canvas page (where you must log in for full access, available while the course is in session).
- Help with DjVu (if you have trouble reading the files below).
- Course policies (DjVu).
- Class hours: Mondays through Fridays from 11:00 to 11:50 in ESQ 100D.

- Name: Toby Bartels, PhD;
- Canvas messages.
- Email: TBartels@Southeast.edu.
- Voice mail: 1-402-323-3452.
- Text messages: 1-402-805-3021.
- Office hours:
- Mondays and Wednesdays from 9:30 to 10:30,
- Tuesdays and Thursdays from 2:30 PM to 4:00, and
- by appointment,

- The first set, on single-variable Calculus (including material from Calculus 1);
- The second set (TBA), on multivariable Calculus (including material for Calculus 3).

- General review:
- Date due: January 14 Tuesday.
- Reading from my notes (first set): Review at least Chapter 4 through Section 4.3 (pages 39–42).
- Reading from the textbook: Review at least Sections 5.4 and 5.5.
- Exercises due:
- If
*F*′(*x*) =*f*(*x*) for all*x*, then what is ∫*f*(*x*) d*x*, that is the indefinite integral of*f*(*x*) with respect to*x*? - If ∫
*f*(*x*) d*x*=*F*(*x*) +*C*, then what is ∫^{b}_{x=a}*f*(*x*) d*x*, that is the definite integral of*f*(*x*) with respect to*x*as*x*runs from*a*to*b*?

- If

- Work:
- Date due: January 15 Wednesday.
- Reading from the textbook: Section 6.5 (pages 386–389).
- Exercises due:
- Suppose that a variable force is applied
on object moving from
*a*to*b*along the*x*-axis, such that the amount of the force in the direction of the positive*x*-axis is a function*F*of the object's position along the*x*-axis. Write down an integral for the work done on the object by that force. - Suppose that a spring with a (positive) spring constant
*k*is stretched or compressed a (signed) distance*x*from its equilibrium length. Write down a formula for the force needed to hold the spring in that position.

- Suppose that a variable force is applied
on object moving from

- Moments:
- Date due: January 16 Thursday.
- Reading from the textbook: Section 6.6 (pages 392–400).
- Exercises due:
- If you wish to find
the total mass of the plate
described in Exercise 6.6.24 on page 400 of the textbook,
will you do an integral with respect to
*x*or an integral with respect to*y*? - Give the formula for the centre of mass
(
*x̄*,*ȳ*) in terms of the total mass*M*and the moments*M*_{x}and*M*_{y}.

- If you wish to find
the total mass of the plate
described in Exercise 6.6.24 on page 400 of the textbook,
will you do an integral with respect to

- Differential equations:
- Date due: January 17 Friday.
- Reading from my notes (first set):
- Section 4.4 (page 42);
- Chapter 5 through Section 5.2 (pages 45–47);
- Optional: The rest of Chapter 5 (pages 47&48).

- Reading from the textbook: Section 7.2 "Separable Differential Equations" (pages 416–418).
- Exercises due: Fill in the blanks with vocabulary words:
- An equation with differentials or derivatives in it is a(n) _____ equation.
- A differential equation of the form
d
*y*/d*x*=*g*(*x*)*h*(*y*) is called _____.

- Exponential growth:
- Date due: January 21 Tuesday.
- Reading from the textbook:
- Section 7.2 through "Exponential Change" (pages 415&416);
- The rest of Section 7.2 (pages 418–422).

- Exercises due:
- Fill in the blanks with a vocabulary word: A quantity is growing _____ if it is growing in such a way that its rate of growth is proportional to its size.
- Suppose that a quantity
*y*undergoes exponential growth with a relative growth rate constant of*k*and an initial value of*y*_{0}at time*t*= 0. Write down a formula for the value of*y*as a function of the time*t*.

- Integration by parts:
- Date due: January 22 Wednesday.
- Reading from my notes (first set): Section 4.5 (pages 42&43).
- Reading from the textbook: Chapter 8 through Section 8.1 (pages 436–442).
- Exercises due:
Use the formula
∫
*u*d*v*=*u**v*− ∫*v*d*u*for integration by parts.- To find
∫
*x*e^{x}d*x*, what should be*u*and what should be d*v*? - To find ∫
*x*ln*x*d*x*, what should be*u*and what should be d*v*?

- To find
∫

- Partial fractions:
- Date due: January 23 Thursday.
- Reading from the textbook: Section 8.4 (pages 456–461).
- Exercises due:
Suppose that a rational expression has the denominator
(
*x*^{2}+ 3)(*x*+ 3)^{2}.- What are the denominators of its partial fractions?
- For each of these, indicate the maximum degree of the numerator.
(If you wish,
you may do this by writing a general form for each numerator,
such as
*A*or*A**x*+*B*.)

- Trigonometric integration:
- Date due: January 24 Friday.
- Reading from the textbook: Section 8.2 through "Eliminating Square Roots" (pages 445–447).
- Exercises due:
- To find ∫ sin
^{2}*x*d*x*, what trigonometric identity would you use? That is, sin^{2}*x*= _____? - For which (if any) of the following other integrals
would you also need to use the identity from the previous exercise?
(Say Yes or No for each one.)
- ∫ sin
^{2}*x*cos^{2}*x*d*x*, - ∫ sin
^{2}*x*cos^{3}*x*d*x*, - ∫ sin
^{3}*x*cos^{2}*x*d*x*, - ∫ sin
^{3}*x*cos^{3}*x*d*x*.

- ∫ sin

- To find ∫ sin

- Tricky trigonometric integration:
- Date due: January 27 Monday.
- Reading from the textbook: The rest of Section 8.2 (pages 447–449).
- Exercises due:
Rewrite the following expressions
using only powers of sin
*x*and/or cos*x*(and possibly multiplication):- sec
^{2}*x*, - sec
*x*tan*x*, - csc
^{2}*x*, - csc
*x*cot*x*.

- sec

- Trigonometric substitution:
- Date due: January 28 Tuesday.
- Reading from the textbook: Section 8.3 (pages 451–454).
- Exercise due:
For each of the following expressions,
give a trigonometric substitution
that would tend to help when finding integrals containing that expression:
- √(
*x*^{2}+ 25), the square root of*x*^{2}+ 25; - √(
*x*^{2}− 25), the square root of*x*^{2}− 25; - √(25 −
*x*^{2}), the square root of 25 −*x*^{2}.

- √(

- Integration using computers and tables:
- Date due: January 29 Wednesday.
- Reading from the textbook:
- Section 8.5 (pages 463–467);
- Skim "A Brief Table of Integrals" (pages T1–T6).

- Optional:
Look at these some or all of
these computer algebra systems that will do integrals (as well as much more):
- Sage (free to download, free for most uses online, $149 per year to make intensive computations online);
- Maxima (free to download);
- Maple ($99 to download);
- Wolfram Mathematica ($161 to download, $81 per year online);
- Wolfram Alpha (free for some uses online, $57 per year for advanced features, $3 for a smartphone app with intermediate features);
- David Scherfgen's integral calculator (free to use online, not a full CAS, but shows steps).

- Exercise due:
Which entry
in the table of integrals in the back of the textbook (pages T1–T6)
tells you how to integrate
∫
*x*^{2}(*x*^{2}+ 3)^{−1/2}d*x*? (Hints: That table doesn't use negative exponents; it uses fractions instead. And it doesn't use fractional exponents; it uses roots instead.)

- Numerical integration:
- Date due: January 30 Thursday.
- Reading from the textbook: Section 8.6 through "Simpson's Rule" (pages 469–472).
- Exercises due:
- Which numerical method of integration approximates a function with a piecewise-constant function? (Hint: This is not from Section 8.6; you already know it from Chapter 5.)
- Which numerical method of integration approximates a function with a continuous piecewise-linear function?
- Which numerical method of integration approximates a function with a continuous piecewise-quadratic function?

- Error estimation:
- Date due: February 4 Tuesday.
- The rest of Section 8.6 (pages 473–476).
- Exercises due:
Suppose that you are attempting to approximate the integral of a function
*f*on an interval [*a*,*b*] and*f*is at least three times differentiable on [*a*,*b*]. Also suppose that*M*_{2}≥ |*f*″(*x*)| and*M*_{3}≥*f*″′(*x*) whenever*x*∈ [*a*,*b*]. Answer the following questions using only*a*,*b*,*M*_{2}, and*M*_{3}(and*n*or*ε*when one of these appears in the question):- If you approximate the integral with the Trapezoid Rule
using
*n*subintervals, then what is the maximum possible absolute error in the approximation? - If you wish to approximate the integral with the Trapezoid Rule
and have an absolute error no more than
*ε*, then what is the largest possible number of subintervals that you might need? - If you approximate the integral with Simpson's Rule
using
*n*subintervals, then what is the maximum possible absolute error in the approximation? - If you wish to approximate the integral with Simpson's Rule
and have an absolute error no more than
*ε*, then what is the largest possible number of subintervals that you might need?

- If you approximate the integral with the Trapezoid Rule
using

- Improper integrals:
- Date due: February 5 Wednesday.
- Reading from the textbook: Section 8.7 through "Improper Integrals with a CAS" (pages 478–484).
- Exercise due:
Consider the integral
∫
^{∞}_{x=−∞}(*x*^{2}+ |*x*|^{1/2})^{−1}d*x*, that is the integral of (*x*^{2}+ |*x*|^{1/2})^{−1}d*x*as*x*runs from −∞ to ∞. List all of the reasons why this integral is improper.

- Comparison tests:
- Date due: February 6 Thursday.
- The rest of Section 8.7 (pages 484–486).
- Exercises due:
Suppose that you want to know whether
∫
^{∞}_{x=1}(*x*+ sin*x*)/*x*^{2}d*x*, the infinite integral of (*x*+ sin*x*)/*x*^{2}, converges.- Knowing that the infinite integral of 1/
*x*^{2}converges, can you use the Direct Comparison Test to decide? - Knowing that the infinite integral of 1/
*x*^{2}converges, can you use the Limit Comparison Test to decide? - Knowing that the infinite integral of 1/
*x*diverges, can you use the Direct Comparison Test to decide? - Knowing that the infinite integral of 1/
*x*diverges, can you use the Limit Comparison Test to decide?

- Knowing that the infinite integral of 1/

- Sequences:
- Date due: February 7 Friday.
- Reading from the textbook:
- Chapter 9 through Section 9.1 "Representing Sequences" (pages 495–497);
- Section 9.1 "Recursive Definitions" (pags 502&503).

- Exercises due:
For each of the following sequences (each called
*a*), write down the values of*a*_{0}through*a*_{4}(so 5 values in all for each sequence).*a*_{n}=*n*! (the factorial of*n*);*a*_{n}= (−1)^{n};*a*_{0}= 0,*a*_{1}= 1,*a*_{n+2}=*a*_{n}+*a*_{n+1}.

- Limits of infinite sequences:
- Date due: February 10 Monday.
- Reading from my notes (first set): Chapter 6 through Section 6.1 (pages 49&50).
- Reading from the textbook: The rest of Section 9.1 (pages 497–502, 503&504).
- Exercises due:
Suppose that
*f*is a function defined everywhere, and*a*is a sequence given by*a*_{n}=*f*(*n*).- If lim
_{x→∞}*f*(*x*) (the limit of*f*at infinity) exists, then must lim_{n→∞}*a*_{n}(the limit of*a*at infinity) be the same? - If lim
_{n→∞}*a*_{n}(the limit of*a*at infinity) exists, then must lim_{x→∞}*f*(*x*) (the limit of*f*at infinity) be the same? - If
*c*is a constant, what is lim_{n→∞}*c*^{n}/*n*! (the limit as*n*→ ∞ of*c*^{n}divided by*n*factorial)?

- If lim

- Finite series:
- Date due: February 11 Tuesday.
- Reading from my notes (first set): Section 6.2 (pages 50&51).
- Exercises due:
- If
*a*_{n}=*n*^{2}for all*n*, write Σ^{4}_{n=0}*a*_{n}, the sum of*a*from 0 to 4, explicitly as a sum of 5 constants. - If
*a*is any sequence, write Σ^{j}_{n=i}*a*_{n}, the sum of*a*from*i*to*j*, as an integral of the function*f*(*x*) =*a*_{⌊x⌋}.

- If

- Infinite series:
- Date due: February 13 Thursday.
- Reading from the textbook:
- Section 9.2 introduction (pages 508&509);
- Section 9.2 "The
*n*th-Term Test for a Divergent Series" and the rest (pages 512–515).

- Reading from my notes (first set): Section 6.3 (pages 51&52).
- Exercises due:
Fill in each blank with either ‘sequence’ or ‘series’:
- lim
_{n→∞}*a*_{n}, the limit of*a*_{n}as*n*goes to infinity, is the limit of an infinite _____. - Σ
^{∞}_{n=0}*a*_{n}, the sum of*a*_{n}as*n*runs from zero to infinity, is the sum of an infinite _____.

- lim

- Evaluating special series:
- Date due: February 14 Friday.
- Reading from my notes (first set): Section 6.4 (pages 52&53).
- Reading from the textbook: Section 9.2 "Geometric Series" (pages 510–512).
- Exercises due:
Finish these formulas and attach any conditions necessary for them to be true:
- The sum of
*b*_{n+1}−*b*_{n}as*n*runs from*i*to infinity (where*i*is a natural number and*b*is an infinite sequence of real numbers): Σ^{∞}_{n=i}(*b*_{n+1}−*b*_{n}) = _____. - The sum of
*r*^{n}as*n*runs from*i*to infinity (where*i*is a natural number and*r*is a real number): Σ^{∞}_{n=i}*r*^{n}= _____.

- The sum of

- The Integral Test:
- Date due: February 17 Monday.
- Reading from the textbook: Section 9.3 through "The Integral Test" (pages 518–521).
- Exercises due:
- Does the Integral Test apply
to the function
*f*(*x*) = sin^{2}(π*x*)? Why or why not? - For which values of
*p*does Σ^{∞}_{n=1}(1/*n*^{p}), the sum of 1/*n*^{p}as*n*runs from 1 to infinity, converge, and for which values does it diverge?

- Does the Integral Test apply
to the function

- Integral estimates for series:
- Date due: February 18 Tuesday.
- Reading from the textbook: the rest of Section 9.3 (pages 521&522).
- Exercises due:
Suppose that
*f*is a function meeting the conditions of the Integral Test and*a*is the corresponding sequence, so that*a*_{n}=*f*(*n*). Also suppose that the infinite integral of*f*converges. Write down an upper and lower bound of the infinite series of*a*from*i*using a*finite*series of*m*terms and some infinite integrals. That is, write down a compound inequality _____ ≤ Σ^{∞}_{n=i}*a*_{n}≤ _____, where the blanks are expressions with finite series and infinite integrals (but no infinite series).

- Comparison tests for series:
- Date due: February 19 Wednesday.
- Reading from the textbook: Section 9.4 (pages 524–528).
- Exercises due:
Suppose that you want to know whether
Σ
^{∞}_{n=1}(*n*+ (−1)^{n})/*n*^{2}, the infinite series of (*n*+ (−1)^{n})/*n*^{2}, converges.- Knowing that the infinite series of 1/
*n*^{2}converges, can you use the Direct Comparison Test to decide? - Knowing that the infinite series of 1/
*n*^{2}converges, can you use the Limit Comparison Test to decide? - Knowing that the infinite series of 1/
*n*diverges, can you use the Direct Comparison Test to decide? - Knowing that the infinite series of 1/
*n*diverges, can you use the Limit Comparison Test to decide?

- Knowing that the infinite series of 1/

- Alternating series:
- Date due: February 20 Thursday.
- Section 9.6 introduction (pages TBA).
- Exercises due:
Identify which of these series are alternating.
(Say Yes or No for each.)
- The sum of (−1)
^{n}/(2 +*n*); - The sum of (2 + (−1)
^{n})/*n*; - The sum of (cos
*n*)/*n*.

- The sum of (−1)

- Absolute convergence:
- Date due: February 25 Tuesday.
- Reading from the textbook:
- Section 9.5 introduction (pages TBA);
- Section 9.6 "Conditional Convergence" & "Rearranging Series" (pages TBA).

- Exercises due:
- If a series converges, is it necessarily true that its series of absolute values also converges?
- If the series of absolute values converges, is it necessarily true that the original series converges?

- The Ratio and Root Tests:
- Date due: February 26 Wednesday.
- Reading from the textbook: Section 9.5 "The Ratio Test" & "The Root Test" (pages TBA).
- Exercises due:
- Under what circumstances
does the Ratio Test
*not*tell you whether a series converges or diverges? - Under what circumstances
does the Root Test
*not*tell you whether a series converges or diverges? - If the Ratio Test doesn't tell you, is it possible that the Root Test will?

- Under what circumstances
does the Ratio Test

- Convergence tests:
- Date due: February 27 Thursday.
- Reading from the textbook:
- Section 9.6 "Summary of Tests" (pages TBA).
- Reading from my notes (first set): Section 6.5 (pages TBA).
- Exercises due: TBA.

- Power series:
- Date due: February 28 Friday.
- Section 9.7 (pages TBA).
- Exercises due:
Which of the following are (or are equivalent to)
power series (in the variable
*x*)? (Say Yes or No for each.)- Σ
^{∞}_{n=0}*n*^{2}(*x*− 3)^{n} - Σ
^{∞}_{n=5}(2*x*− 3)^{n} - Σ
^{∞}_{n=0}(√*x*− 3)^{n} - 5 + 7
*x*− 3*x*^{3}

- Σ

- Taylor polynomials:
- Date due: March 3 Tuesday.
- Reading from my notes (first set): Section 6.6 (pages TBA).
- Reading from the textbook: Section 9.8 "Taylor Polynomials" (pages TBA), but don't worry yet about when they talk about infinite series and convergence.
- Exercises due:
- Suppose that
*f*is a function,*a*is a number, and*f*is infinitely differentiable at*a*. Given a natural number*k*, let*P*_{k}be the Taylor polynomial of order*k*generated by*f*at*a*(so*P*_{k}is also a function). Then the derivatives of*f*at*a*and the derivatives of*P*_{k}at*a*must be equal up to what order? That is, they must agree through their first derivatives (up to order 1), through their second derivatives (up to order 2), through their*k*th derivatives (up to order*k*), or what? - More TBA.

- Suppose that

- Taylor remainders:
- Date due: March 4 Wednesday.
- Reading from the textbook:
- Section 9.8 "Taylor Polynomials", the discussion about infinite series and convergence;
- Section 9.9 through the very beginning of "Estimating the Remainder" (pages TBA).

- Exercises due:
- If a function
*f*is to have a good approximation on an interval by a polynomial of degree at most*k*, then it's best if its derivative of what order is close to zero on that interval? That is, the derivative near zero should be its first derivative (order 1), its second derivative (order 2), its*k*th derivative (order*k*), or what? - More TBA.

- If a function

- Taylor series:
- Date due: March 5 Thursday.
- Reading from my notes (first set): The beginning of Section 6.7 (pages TBA).
- Reading from the textbook: the rest of Section 9.8 (pages TBA).
- Exercises due:
- Fill in the blank:
If
*f*is a function that is infinitely differentiable everywhere, then the Maclaurin series generated by*f*is the Taylor series generated by*f*at ___. - True or false:
Whenever
*a*is a constant and*f*is a function whose derivatives of all orders exist at*a*, if the Taylor series generated by*f*at*a*converges anywhere, then it must converge to*f*there.

- Fill in the blank:
If

- The Binomial Theorem:
- Date due: March 6 Friday.
- Reading from the textbook: Section 9.10 through "The Binomial Series for Powers and Roots" (pages TBA.
- Reading from my notes (first set): The parts of Section 6.7 about the Binomial Theorem (pages TBA).
- Exercises due:
- Using the Binomial Theorem,
expand (
*x*+ 1)^{6}. - Using the Binomial Theorem,
write (1 +
*x*^{2})^{−1}as an infinite series (assuming that*x*^{2}< 1 so that the series converges), and simplify the expression for the terms.

- Using the Binomial Theorem,
expand (

- More common Taylor series:
- Date due: March 9 Monday.
- Reading from the textbook:
- The rest of Section 9.9 (pages TBA);
- The rest of Section 9.10 (pages TBA).

- Reading from my notes (first set): The rest of Section 6.7 (pages TBA).
- Exercises due:
- Assuming that
*a*and*b*are real numbers, fill in the blank using trigonometric operations applied to real numbers (and other operations as appropriate): exp (*a*+ i*b*) = e^{a+ib}= ___. - Can any limit using L'Hôpital's Rule be done using Taylor polynomials or series instead? Explain why or why not.

- Assuming that

- More to come!

This web page and the files linked from it were written by Toby Bartels, last edited on 2020 February 16. Toby reserves no legal rights to them.

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