Calculus I Assignments

MTH 111: Calculus with Analytic Geometry I
Dr. Elmer K. Hayashi
Fall 2002
Assignments

Aug 28-30	Sep 2-6	Sep 9-13	Sep 16-20	Sep 23-27
Sep 30-Oct 4	Oct 7-11	Oct 14-18	Oct 21-25	Oct 28-Nov 1
Nov 4-8	Nov 11-15	Nov 18-22	Nov 25-26	Dec 2-6

Textbook: Stewart, Calculus, Fourth Edition
Help Sessions each week: Mon. and Thu. at 7 p.m., Tue. and Wed. at 4 p.m. in Calloway 3.

Wed, 08/28/2002. Review of Differentiation.: The last operation that would be performed to evaluate an expression tells you what differentiation rule to use first to differentiate the expression. Review all differentiation rules, see pages 147-152, 173, and 177.
Do on page 183, problems 11, 19, 33, 36, 37, 43, 44, 47 for practice. On Friday, ask about any problems that you do not understand, or come by my office before Friday.

Fri, 08/30/2002. Review of Graph Analysis.: Review how to deal with absolute values as discussed in class. Understanding the unit circle definition of sine and cosine is essential for working with all trig functions, see page A26 and following. The first derivative of a function tells us where the graph of the function is increasing and where it is decreasing. The second derivative tells us about the concavity of the graph of the function, review Chapter 4. Maple can make your work a lot easier, but you have to really understand the theory to take full advantage of the power of Maple. Sign up for an Maple orientation session. Write up problem 48 on page 183 and 32 on page 248 to turn in next Tuesday. For an extra challenge, determine the concavity information for problem 16 on page 248. Also do problems 15 and 37 for practice with trig functions.

Mon, 09/02/2002. Review of Graphing.: Review section 4.3 and note the summary of curve sketching given in section 4.5
On page 270, do problems 11, 15, 17, 23, and 37 for practice.

Tue, 09/03/2002. Horizontal and Vertical Asymptotes.: Review sections 4.4 on limits at infinity and horizontal asymptotes, and section 2.2 on one-sided infinite limits and vertical asymptotes.
Use difference of squares (a-b)(a+b) = a^2 - b^2 to rewrite an expression that is indeterminate. Remember that sqrt(x^2) = abs(x), and that abs(x) = x if x > 0, and abs(x) = -x if x < 0.
On pages 260-261, do problems 3, 11, 13, 17, 19, 27, 33, 41, 42 for practice.
Write up problem 42 on page 261 and problem 16 on page 308 to turn in on Friday.

Wed, 09/04/2002. Review and More about Maple.: Install wfucalc library on your computer by following the directions on the handout given in class. Complete your review of chapters 1-4. Take advantage of the help sessions on Mon and Thu from 7-8 p.m. and Tu and Wed from 4-5 p.m. in Calloway 3 if you need extra help with your review, or stop by and see me to get the real scoop.
For more practice on limits, asymptotes, and graphing, see PDF file with practice problems

Fri, 09/06/2002. Area Approximation using Rectangles.: To understand applications of integration, it is essential that you understand the summation notation for right endpoint approximation of areas. Extra effort here will pay off in greater understanding later. The area under a curve can be approximated using rectangles, and is defined as the limit of such approximations. The left endpoint method, the right endpoint method, and the midpoint method are the commonly used methods of approximationg area with rectangles. Practice using the Maple commands illustrated in Area Approximation using Rectangles
Study example 1 on pages 314-315, and examples 3 and 4 on pages 319-322.
Look at problems 1, 3, 11, 15, 17, 19 on pages 322-323 to check your understanding. Practice using Maple on problems 9 and 10.

Mon, 09/09/2002. Properties of the Definite Integral.: The definition of the definite integral as the limit of Riemann Sums gives us the interpretation that the definite integral of a function is the area between the graph of the function and the x-axis adding the area above the x-axis and subtracting the area below the x-axis. The properties of the definite integral are important in helping to evaluate or estimate the value of the integral.
Read pages 324-327. Study examples 4, 5, 6, 7 on pages 329-333.
On pages 334-336, do problems 5, 7, 9, 11, 13, 29, 35, 39, 43, 45, 49, 63 to check your understanding.

Tue, 09/10/2002. Fundamental Theorem, Part 1.: The derivative of a function defined as a definite integral of a continuous function where the independent variable is the upper limit of integration is just the integrand expressed as a function of the independent variable.
Learn the Fundamental Theorem of Calculus, and study examples 1, 2, and 3 on pages 340-341.
On pages 344-345, do problems 1, 5, 7, 9, 11, 13, 15 to check your understanding.
Write up the following problems to turn in on Friday: problem 16 on page 323, problems 50, 64 on page 336, problem 16 on page 344.
See Instructions on using Maple for area approximation and with piecewise functions

Wed, 09/11/2002. Fundamental Theorem, Part 2.: The Fundamental Theorem of Calculus basically says that differentiation and integration are essentially inverse operations in that one undoes the other. Thus the Fundamental Theorem, Part 2, asserts that integration of a function is the same as antidifferentiation of the function.
Study example 7 on page 343, the table of integrals on page 347, and example 5 on page 349.
On page 344-345, do problems 17, 19, 27, 29, 31, 33, 49, and on page 353, do problems 21, 23, 25, 31, 37 to check your understanding.

Fri, 09/13/2002. Integration by Substitution.: If you wish you could integrate with respect to u (instead of x), then you can have your wish if make sure you have du (instead of dx).
Study examples 1-7 on pages 357-360.
Practice on problems 1-50 on page 361-362 to check understanding.

Mon, 09/16/2002. Area between two curves.: To find the area between two curves, integrate top - bottom from left to right. Set up the integral by considering the right endpoint approximation of the area.
Study examples 1, 2, 4, 5, 6 on pages 372-376.
On pages 376-377, do problems 1, 3, 11, 15, 21, 29, 31, 40, 41 to check understanding.

Tue, 09/17/2002. Volumes of revolution using disks or washers.: The volume of a solid of revolution can be found by using rectangular approximation of the region, and then rotating the rectangles to obtain disks or washers. Use the disk or washer method when vertical rectangles are rotated about the x-axis, or when horizontal rectangles are rotated about the y-axis.
Study examples 1-4 on pages 380-383.
Do problems 1, 3, 5, 7, 9, 31, and 43 on page 387 to check understanding.
Write up problem 32 on page 377 and problems 4, 6, 34 on page 387 to turn in on Friday.
For the exam next week, practice on Exam 1 from Spring 2002 in PDF format

Wed, 09/18/2002. Volumes of revolution using cylindrical shells.: Volumes of revolution can also be computed using cylindrical shells instead of washers if vertical rectangles must be used when rotating about the y-axis, or horizontal rectangles must be used when rotating about the x-axis.
Study examples 5 and 6 on pages 383-384, and examples 1 and 2 on page 391.
Do problems 11, 13, 15, 17, 33, 35 on page 387, and 3, 5, 7 on page 392-393 to check understanding.

Fri, 09/20/2002. Average Value of a function.: The average value of a function on an interval is just the value of a constant function that would yield the same definite integral over the given interval. Finding volumes of solids obtained by rotating about different axes just requires a well-drawn graph, and careful analysis of radii.
On page 393, do problems 6, 7, 13, 18, 23, 25 for more practice on volumes.
On page 400, do problems 5, 7, 9, 12 to check understanding of average value.

Mon, 09/23/2002. Review.: Don't forget to take advantage of help sessions on Mon. and Thu. at 7 p.m., and Tue. and Wed. at 4 p.m. in Calloway 3.
Problems 9 13, 14, 15 on page 401 will give more practice on volumes.

Tue, 09/24/2002.: First Hour Exam on differentiation, limits at infinity, asymptotes and graphing analysis, Fundamental Theorem of Calculus, area approximation using rectangles, limit definition of definite integral, integration by substitution, area between two curves, volumes of solids of revolution.

Wed, 09/25/2002. Plotting Parametrically in Maple.: Take Home Exam due on Friday.
See Plotting Parametrically in Maple

Fri, 09/27/2002. Inverse Functions,: If y is a function of x, we sometimes want to consider x as a function of y. We can only do this if y is a 1-1 function of x, and the resulting function is called the inverse function.
Read about inverse functions on pages 407-411
Study examples 1-5 on pages 408-411.
Do problems 1, 3, 5, 7, 15, 23, 27, 29, 31, 33 on pages 414-415 to check understanding.

Mon, 09/30/2002. Derivative of an inverse function.: The chain rule can be used to remember the relationship between the derivative of a one-to-one function and the derivative of its inverse, see Theorem 7 on page 412. Study examples 6-8 on page 413-414.
On page 415-416, do problems 35, 37, 39, 41, 43, 45 to check understanding.

Tue, 10/01/2002. Natural Logarithm Function.: We define ln(x) as a definite integral, and use the Fundamental Theorem to derive the derivative formula for ln(x). Know the properties of the logarithm function because they are important.
Study examples 1-3, 5-8 on pages 447-448.
Do problems 19, 21, 25, 27, 33, 35 on page 452 to check your understanding.
Write up problem 42 on page 415 and problems 26 and 42 on page 452 to turn in on Friday.

Wed, 10/02/2002. Logarithmic Differentiation, Integration involving ln x.: Applying the logarithm rules before differentiating can simplify differentiation. Similarly, logarithmic differentiation can be used to simplify differentiation. The new derivative formula for ln x gives a new formula for integration as well.
Study examples 9-14 on page 449-452.
Do problems 39, 41, 43, 51, 56, 57, 65, 69, 71, 73 on pages 452-453 to check understanding.

Fri, 10/04/2002. The Exponential Function: The exponential function is the inverse of the logarithm function, and its properties follow from the properties of the natural logarithm function and the relationship between a function and it inverse.
Study examples 4-9 on page 456-458.
Do problems 29, 33, 35, 41, 43, 49, 61, 67, 69, 71, 73, 77 on pages 459-460 to check your understanding.

Mon, 10/07/2002. Laws of Exponents.: Limits at infinity involving exponentials is treated no differently than limits at infinity involving powers. Use the fact that exp(x) is the inverse ln(x) to help solve equations involving exponentials or logarithms.
Study the laws of exponents on page 456 and 461. Study examples 1-3 on pages 454-456, and learn equation 1 on page 461 and equation 6 on page 466.
Do the odd problems 1-27 on pages 458-459 to check understanding.

Tue, 10/08/2002. Logarithms and exponentials to different bases.: Equation 1 on page 461 and equation 6 on page 466 should be used to convert exponentials to the exponential function and any logarithm to the natural logarithm. Working with ln(x) and exp(x)=e^x is much simpler.
Read about the Power Rule versus Exponential Rule on page 464. Study examples 4 and 5 on pages 465-466.
Do problems 3-10, 13, 21, 22, 23, 25, 27, 29, 37, 39, 41 on pages 467-468 to check understanding.
Write up problems 24, 26, 80 on pages 459-460 and problmes 387, 40 on page 468 to turn in on Friday.

Wed, 10/09/2002. Inverse Trig Functions.: The trig functions are not 1-1, so we have to restrict their domains in order to define the inverse trig functions.
Study examples 1-3 on ages 470-471, and the table of derivatives on page 474.
Do problems 1, 3, 5, 6, 12, 13, 14, 23, 25, 27, 29, and 31 on pages 476-477 to check understanding.

Fri, 10/11/2002. More on Inverse Trig Functions.: The derivative formulas for the inverse trig functions provide us with a new set of integration formulas that we can use with the method of substitution.
Study examples 5, 6, 8, 9, 10 on pages 473-476.
Do problems 9, 10, 14, 37, 39, 47, 59, 61, 63, 65, 67, 69, 73 on pages 476-477 to check understanding.

Mon,10/14/2002.: L'Hospital's Rule is a method for evaluating the behavior of indeterminates of the form 0/0 or infinity/infinity. Be sure to check that a limit is of the proper form for evaluating using L'Hospital's Rule; in particular, the limit may not actually be indeterminate.
Study examples 1-5 on pages 487-488.
On page 493, do 5-33 (at least odd problems) to check understanding.

Tue, 10/15/2002.: If an indeterminate is not of the form necessary to apply L'Hospital's Rule, then it must be rewritten to get into one of the proper forms. Remember also that factoring the largest term from the numerator and denominator may actually work better than L'Hospital's Rule for certain limits.
Study examples 6-8 on pages 489-490.
On page 494, do problems 39, 41, 43, 45, 47, 49, 51, 52, 53, 73, 75 to check understanding.

Wed, 10/16/2002.: To evaluate indeterminate powers, first find the limit of the logarithm of the given function. Remember to exponentiate to obtain the final answer.
On page 494, do problems 56, 57, 59, 62, 67 to check understanding.

Fri, 10/18/2002. Fall Break.: No Class.

Mon, 10/21/2002. Review.: on pages 497-498, do problems 11, 13, 17, 19, 25, 33, 35, 37, 49, 61, 65, 67, 73, 75 to review for next exam.

Tue, 10/22/2002. Review: Review for the exam tomorrow.
Here is the Second Hour Exam, Spring 2002 in PDF Format.

Wed, 10/23/2002.: Second Hour Exam covering sections 7.1, 7.2*-7.4*, 7.5, 7.6 on exponentials, logarithms, inverse trig functions, and L'Hospital's Rule. Take home part of exam is due on Friday.

Fri, 10/25/2002. Integration by Parts.: Case 1. If the integrand is a power of x times either cos(x), sin(x), or exp(x), then integrate by parts with u = the power of x. If the integrand contains cos(w), sin(w), or exp(w) where w is more complicated than just x, then you might do a change of variable to w first. Case 2. If the integrand contains ln(x), arctan(x), arcsin(x), or any function we know how to differentiate, but not integrate, then integrate by parts with u = ln(x), u = arctan(x), u = arcsin(x), or u = the function you can differentiate but not integrate. Do problems 1-10, 17, 19, 21 on page 508 to check understanding.
Maple 6 Worksheet on Integration by Parts in Maple.

Mon, 10/28/2002. Integration by Parts.: Sometimes a simple substitution like u=ln(x) or u=ax can simplify a problem if done before doing integration by parts; look for opportunities to do this. Products fo two of cos(ax), sin(ax), e^(ax) can be done by integrating by parts twice, and then solving for the integral that reappears on the second time around. Study examples 4-6 on pages 506-508. Do problems 11, 15, 23, 25, 27, 29, 31, 33, 35, 41 on pages 508-509 to check understanding.

Tue, 10/29/2002. Trig Integrals.: In an integral that has a power of sin(x) times a power of cos(x), let u=sin(x) if the power of cos(x) is odd; let u=cos(x) if the power of sin(x) is odd. If both powers are even, use the half angle formulas to reduce to a problem with lower powers. Repeat the above process until all integrals are evaluated.
Study examples 1-4 on pages 510-512.
Do problems 1, 7, 11, 13, 15, 17 on page 516 to check understanding.
Write up problems 12, 18, 22, 24 on page 508 to turn in on Friday.

Wed, 10/30/2002. Trig Integrals.: In an integral that has a power of tan(x) times a power of sec(x), let u=tan(x) if the power of sec(x) is even; let u=sec(x) if the power of tan(x) is odd. If the power of sec(x) is odd and the power of tan(x) is even, then pray for Divine guidance.
Study examples 5-8 on pages 513-515.
Do problems 21, 23, 25, 27, 29, 31, 39, 51, 53, 57, 59 on pages 516-517 to check understanding.

Fri, 11/01/2002. Trig Substitution.: Integration by trig substitution involves three steps. First, make the appropriate trig substitution to simplify terms of the form a^2-x^2, a^2+x^2, x^2-a^2. Second, evaluate the resulting trig integral using techniques learned in section 8.2. Third, use inverse trig functions as studied in section 7.5 to express your answer in terms of the original variable.
See using Maple to Change Variables
Study examples 1, 3, 4, 5 on page 518-520.
Do problems 1, 2, 3, 5, 7, 9, 11, 13 on pages 522-523 to check understanding.

Mon, 11/04/2002. Completing the Square.

Mon, 11/11/2002. Error estimation.: To estimate the maximum possible error in a numerical approximation of a definite integral, one must obtain an estimate for the size of the second derivative (trapezoid and midpoint approximations) or the fourth derivative (Simpson's method). Graphing the derivatives in Maple is one way to obtain such estimates.
Study examples 2-3 on pages 549-550, and examples 6-7 on pages 553-554.
Do problems 21, 23, 25, 39 on pages 555-557 to check understanding.
Write up problems 26, 32 on page 555-556 to turn in on Wednesday.

Tue, 11/12/2002. Improper Integrals with infinite limits.: If an integral has an infinite limit, then it is an improper integral. One must evaluate the integral over a finite interval, and then let the appropriate endpoint go to infinity or -infinity to see if the integral is convergent or divergent. If both limits are infinite, then break the integral up into two integrals, each with only one infinite limit.
Study examples 1-3 on pages 549-550.
Do problems 13, 15, 19, 25 on page 565, and problem 33 on page 569 to check understanding.

Wed, 11/13/2002. Improper Integrals with discontinuous integrands.: If the integrand is discontinuous at a point in the interval of integration, then one must look at the integral on an interval not containing that bad point, and then let the endpoint approach the bad point to see if the integral is convergent.
Study examples 5-7 on page 562.
Do problems 27, 32, 36, 39 on page 565, and problems 35, 37, 41, 45, 47 on page 569 to check understanding.

Fri, 11/15/2002. Arc Length and Surface Area.: Finding arc length and surface area are applications of integration where using Simpson's method to approximate the value of the integral is particularly valuable because the integral obtained is often quite difficult, if not impossible, to evaluate exactly.
Study examples 1-3 on pages 577-578 and examples 1-2 on page 585-586.
Do problems 7, 11, 19, 35 on pages 580-581 and problems 13, 15, 21 on page 587 to check understanding.

Mon, 11/18/2002.: Complete your review for the exam. Here is a copy of the Third Exam from Spring 2002. Note that problems 2 and 9 on this exam require knowledge of parametric equations, which we have not covered yet. You can substitute something like problem 25 on page 581 or problem 21 on page 587 for those two problems.
You may also want to try practice problems for exam 3.

Tue, 11/19/2002.

Wed, 11/20/2002.: Third Hour Exam covering Integration by substitution, integration by parts, trig integrals, trig substitution, completing the square, inverse trig function evaluation, trig identities, partial fractions, approximate integration, improper integrals, arc length, and surface area. This covers chapter 8 and 9.1 and 9.2. Take home part will be due on Friday.

Fri, 11/22/2002.

Parametric equations are useful for plotting and analyzing graphs that are not graphs of functions. Polar coordinates are especially convenient for working with circular curves that are so messy in rectangular coordinates. Slope is still change in y divided by change in x.
Study example 2 on pages 684-685, example 6 on page 697, and example 9 on page 701.
Do problems 3, 5, 7, 9, 11 on page 687, and problems 57, 61, 63, 65 on page 703 to check understanding.

Mon, 11/25/2002.: The equations x = r cos(theta), y = r sin(theta), and r^2 = x^2 + y^2 can be used to convert rectangular coordinates to and from polar coordinates. The object is to choose the coordinate system that makes our work easier for a particular given curve.
Study examples 4-5 on pages 696-697, examples 7-8 on page 698, and example 10 on pages 701-702.
Do problems 15, 17, 19, 21, 23, 25, 27, 29, 31, 37, 49, 56 on pages 702-703 to check understanding.

Tue, 11/26/2002.

Wed, 11/27/2002. Thanksgiving Holiday: No Class.

Mon, 12/02/2002.: Study examples 1 and 2 on pages 705-706. Do problems \7, 8, 11, 25, 27, 31, 33 on pages 708-709 to check understanding.

Tue, 12/03/2002.: Study examples 1 and 2 on page 691 and example 4 on page 707.
Do problems 3, 5, 9, 11, 13 on page 693 and 45, 47, 49, 51 on page 708 to check understanding.

Wed, 12/04/2002.: Review Chapters 5, 6, 7, 8, and 11 for the final exam.

Fri, 12/06/2002.

Tue, 12/10/2002. Final Examination.: 9:00 a.m.-12:00 p.m.
Calloway 10.

Created 06/25/2002. Last modified 12/03/2002. Email to ekh@wfu.edu