Calculus I
Dr.
Elmer K. Hayashi
Spring 2002
Assignments
Text: Calculus by James Stewart, fourth edition.
- Tuesday, Jan 15, 2002.
- Relearn all the differentiation formulas in Chapter 3.
Remember that the last operation that would be performed to
evaluate expression tells you what differentiation rule to
use first to differentiate the expression. On page 183 (sec 3.6),
work problems 11, 19, 33, 36, 37, 43, 44, 47 and practice justifying
each step of your work.
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- Wednesday, Jan 16, 2002.
- Review one way of dealing with absolute values by studying
problem 74 on page 185. Review definition of trig functions and identities beginning
on page A26 in the back of the text. Review ideas of graphing functions
in Chapter 4. Apply the method of reducing the solution of a difficult
problem to the solution of simpler problems by using a sign chart to
determine where a function is positive or negative.
Write up the following problems to turn in next Tuesday:
problem 48 on page 183, problem 74c on page 185, and problem 32 on page 248.
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- Fri, Jan 18, 2002.
- The absolute value function can be dealt with by separating its domain
into parts or by using the fact that sqrt(x^2) = absolute value of x.
sin(x) is an odd function, and cos(x) is an even function.
Review sections 4.3-4.5. In particular, relearn the guidelines
for sketching a curve given on pages 264-265.
Study carefully example 7 on page 246, example 4 on page 254-255, and example 1 on page 266.
in each of these three sections. Finish up the three problems to
be turned in on Tuesday.
Monday, Jan 21, 2002.
Holiday
Tuesday, Jan 22, 2002.
Review guidelines for graphing. f is increasing if f' is positive.
f is concave upward if f' is increasing and f'' is positive. Review
asymptotes and finding limits at infinity. To find limits at infinity,
factor out the largest term in the numerator and the largest term in the
denominator. This will make all terms except the main ones go to zero.
For practice look at problems 11, 15, 17, 23, 37 on page 270, and
problems 11, 13, 17, 19, 27 on page 261.
Write up problem 4 on page 261, problem 14 on page 270, and problem 16 on page 308
to turn in on Friday.
Wednesday, Jan 23, 2002.
Use difference of squares (a-b)(a+b) = a^2 - b^2 to eliminate square roots
in a limit problems. Remember the sqrt(x^2) = abs(x) and abs(x) = x if x is greater than 0,
and abs(x) = -x if x is less than 0.
Do problems 9 and 12 at bottom of page 307 for practice.
Will and Phil will be in Calloway 3 from 1-4 on Thursday with practice problems,
and they will be available to answer questions as needed.
Friday, Jan 25, 2002.
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 approximating 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.
- Monday, Jan 28, 2002.
- 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 about the definition of the definite integral on pages 324-327.
Skip examples 2 and 3. Study examples 4, 5, 6, 7, and read the accompanying
text on pages 329-333. On pages 334-336, do problems 5, 7, 9, 11, 13, 29,
35, 39, 43, 45, 49, and 63 to check your understanding of what you read.
Calculus I study and help sessions will be held every Monday night
during the semester from 4-7 p.m. in Calloway 3. Will and/or Phil will be
available to help you with any questions you may have.
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- Tuesday, Jan 29, 2002.
- 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.
Do problems 1, 5, 7, 9, 11, 13, 15 to check your understanding.
Write up problem 16 on page 323, and problems 36, 54, and 64 on page 335 to turn in on Friday.
More Maple commands for
Area Approximation and Piecewise Functions.
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- Wednesday, Jan 30, 2002.
- The Fundamental Theorem of Calculus basically says that differentiation and integration
are essentially inverse operations in that one undoes the other. Roughly speaking, (part 1)
the derivative of an integral is the integrand, and
(part 2) the integral of a function is the antiderivative of the function.
Study example 7 on page 343, 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.
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- Friday, Feb 1, 2002.
- If u = f(x), then du = f'(x) dx. Changing variables in integration requires
that substitution to replace (f'(x) dx) by du is done carefully in addition to replacing
f(x) by u.
Study examples 1-7 on pages 357-360.
Do as many as you can of problems 1-50 on pages 361-362 to check your understanding.
See how to do Change of Variables in Maple.
- Monday, Feb 4, 2002.
- The area between two curves can be approximated by summing the areas of n rectangles by
any of the methods we have discussed. If we then take the limit as n goes to infinity, we
obtain a definite integral that can be evaluated using the Fundamental Theorem.
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 your understanding.
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- Tuesday, Feb 5, 2002.
- The volume of a solid of revolution can be approximated using n disks or washers.
If we then take the limit as n goes to infinity, we obtain a definite integral that can be evaluated
using the Fundamental Theorem.
Study examples 1-4 on pages 380-383.
Do problems 1, 3, 5, 7, 9, 31, and 43 on page 387 to check your understanding.
Write up problem 32 on page 377 and problems 4, 6, and 34 to turn in next Friday.
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- Wednesday, Feb 6, 2002.
- Rotation about axes other than the x-axis or y-axis requires careful analysis of the graphs
to determine the proper radii. 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.
Do problems 11, 13, 15, 17, 33, and 35 on page 387 to check your understanding.
Study examples 1 and 2 on page 391.
Do problems 3, 5, and 7 on pages 392-393 to check your understanding.
This Thursday, the graduate assistants will help you with an old exam.
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- Friday, Feb 8, 2002.
- Do problems 6, 7, 13, 18, 23, 25 on page 393 for more practice on volumes.
- Monday, Feb 11, 2002.
- The average value of a function over an interval is the constant value
that would give the same area as the definite integral of the function over that
interval.
Study example 1 on page 399.
Do problems 5, 7, 9, 12 on page 400 and problems 9, 13, 14, 15 on page 401
to check your understanding.
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- Tuesday, Feb 12, 2002.
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- Wednesday, Feb 13, 2002.
- Hour Exam and Take Home Exam
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- Friday, Feb 15, 2002.
- I 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 1-1 and 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.
- Monday, Feb 18, 2002.
- 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.
On page 35, 37, 39, 41, 43, 45 on page 415-416 to check understanding.
Write problems 2, 36, 42 on pages 414-415 to turn in on Wednesday.
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- Tuesday, Feb 19, 2002.
- The natural logarithm function is defined as a definite integral.
The fundamental theorem is used to find its derivative formula and
to prove the properties of logarithms.
Study examples 1-3 on pages 445-447 and examples 5-8 on page 448.
Do problems 1-8, 9, 11, 19, 21, 25, 27, 33 on page 452 to check understanding.
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- Wednesday, Feb 20, 2002.
- Applying the logarithm rules before differentiating can simplify the differentiation.
Similarly, logarithmic differentiation can be used to simplify differentiation.
Study examples 9-14 on pages 449-452.
Do problems 39, 41, 43, 51, 56, 57, 65, 71, 73 on page 452-453 to check understanding.
Write up problems 52, 68, 70, 72 on pages 452-453 to turn in next Monday.
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- Friday, Feb 22, 2002.
- The exponential function is the inverse of the logarithm function, and its
properties follow from the relationship between a function and its inverse.
Study examples 4-9 on pages 456-458.
Do problems 29, 33, 35, 41, 43, 49, 61, 67, 69, 71, 73, 77 on pages 459-460
to check understanding.
- Monday, Feb 25, 2002.
- The Laws of Exponents follow from the Laws of Logarithms (since logarithms are exponents),
and are important for simplifying and solving expressions involving exponentials. Problems
involving different bases should be converted immediately to base e (or natural logarithms)
because working with ln(x) and e^x is much simpler than working in other bases.
Study examples 1-3 on pages 454-456, and learn equation 1 on page 461 and equation 6
on page 466.
Do odd problems 1-27 on pages 458-459 to check understanding.
Write up problems 24, 26, 58, 80 on pages 459-460 to turn in next Wednesday.
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- Tuesday, Feb 26, 2002.
- Problems involving logarithms or exponentials to different bases
are done most easily by converting to natural logarithms and the exponential function.
Study Laws of Exponeents on page 456 and 461.
Read about 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.
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- Wednesday, Feb 27, 2002.
- The domain of the trig functions must be restricted in order to obtain a one-to-one
function. By restricting the domain, we can define the inverse trig functions unambiguously.
Study examples 1-3 on pages 470-471, and the table of derivatives on page 474.
Do problems 1, 3, 5, 6, 12, 13, 14, 23, 25, 27, 29, 31 on pages 476-477 to check
understanding.
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- Friday, Mar 1, 2002.
- 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 59, 61, 63, 65, 67, 69 on page 477 to check understanding.
Write up problems 8, 10, 14, 34, 72 on pages 476-477 to turn in on Tuesday.
Monday, Mar 4, 2002.
L'Hospital's Rule is a method for evaluating the behavior of indeterminates
of the form 0/0 or infinity/infinity.
Study examples 1-5 on pages 487-488.
On page 493, do 5-33 (at least odd problems) to check understanding.
Tuesday, Mar 5, 2002.
If a limit is not of the form necessary to apply L'Hospital's Rule,
then it must be rewritten to get it into one of the proper form.
Study examples 6, 7, and 8 on pages 489-490.
Do problems 39, 41, 43, 45, 47, 49, 51, 52, 53, 73, 75 on page494 to check understanding.
Wednesday, Mar 6, 2002.
To evaluate indeterminate powers, first find the limit of the logarithm.
Study examples 9 and 10 on page 491
Do problems 56, 57, 59, 62, 67 on page 494 to check understanding.
Do problem 48 on page 477.
Go to the problem session on Thursday between 1 and 4 for more practice
problems, and to get help with homework and an old exam
Friday, Mar 8, 2002.
No class.
- Monday, Mar 18, 2002.
- Integrals that evaluate to arcsin(u), arctan(u), or arcsec(u) can
be done using trig substitution (x=a sin(t), x=a tan(t), or x=a sec(t)).
Do problems 66, 67, 70 on page 477 using trig substitution.
Review 7.1, 7.2*-7.4*, 7.5, and 7.7 for the exam on Wednesday.
Do problems 11, 13, 17, 19, 25, 33, 35, 37, 49, 61, 65, 67, 73, 75
on pages 497-498 for more practice.
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- Tuesday, Mar 19, 2002.
- Review 7.1, 7.2*-7.4*, 7.5, and 7.7 for the exam on Wednesday.
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- Wednesday, Mar 20, 2002.
- Hour Exam and Take Home Exam due on Friday.
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- Friday, Mar 22, 2002.
- The integration by parts formula is given on page 504 of our text.
The integration by parts formula is a product rule that allows us to
convert an integral that is the product of two functions to a second
integral that is hopefully simpler to evaluate. If an integral contains
ln(x), arcsin(x), arctan(x), or any function that we know how to differentiate
but not integrate, then using u = ln(x), u = arcsin(x), u = arctan(x), or
u = the function will generally result in an integral that we have a better
chance of evaluating. Secondly if an integral contains a power of x, (x^n),
times either cos(ax), sin(ax), or e^(ax), then u = x^n will lead to a integral
that is simpler and repeated application of integration by parts will lead
to integral that we can finally evaluate.
Study examples 1, 2 and 3 on pages 504-506,
and do problems 1-10 to check your understanding.
See Using Maple to do integration by parts
for examples showing how to use Maple to help with integration problems.
- Monday, Mar 25, 2002.
- Products of 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 and 5 on pages 506-507.
Write up problems 12, 18, 22, 24 on page 508 to turn in on Wednesday.
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- Tuesday, Mar 26, 2002.
- Sometimes a simple substitution like u=ln(x) or u=ax can simplify
a problem if done before doing integration by parts.
Do problems 23, 25, 27-32, 41, 45 to check understanding.
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- Wednesday, Mar 27, 2002.
- In an integral that has a power of sin(x) times a power of cos(x),
if the power of cos(x) is odd, put u= sin(x), and if the power of sin(x)
is odd, put u=cos(x). If both powers are even, use the half angle formulas.
In and integral that has a power of tan(x) times a power of sec(x),
if the power of sec(x) is even, put u=tan(x), and if the power of tan(x) is
odd, then put u=sec(x).
Study examples 1-5 on pages 510-513.
Do problems 1, 7, 9, 11, 13, 15, 17, 23, 25 on page 516 to check your
understanding.
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- Friday, Mar 29, 2002.
- Good Friday Holiday; class will not meet.
Monday, Apr 1, 2002.
The integral of a power of tan(x) can be done by replacing tan^2(x)
by sec^2(x)-1, and effectively reducing the power of the tan(x). Continue
this same substitution until you have evaluated the integral completely
or you have the integral of tan(x), which is ln|sec(x)| = -ln|cos(x)|.
Study exaqmples 5, 6, 7 on page 513-515.
Do problems 21, 27, 29, 31, 39, 51, 53, 57, 59 on page 516 to check
understanding.
Tuesday, Apr 2, 2002.
Integration by trig substitution involves at least three steps.
First, make the appropriate trig substitution to eliminate square roots.
Second, evaluate the resulting trig integral. Third, use inverse trig
functions to express your answer in terms of the original variable.
Study examples 1, 3, 4, 5 on pages 518-520.
Do problems 1, 2, 3, 5, 7, 9, 11, 13 on pages 522-523 to check understanding.
Wednesday, Apr 3, 2002.
Sometimes one must complete the square before applying a trig substitution.
See using Maple to complete the square.
Rational functions can be dealt with by expanding in partial fractions.
See using Maple to expand a rational function into partial fractions.
Study example 7 on page 522 and examples 1 and 2 on pages 524-526.
Write up problem 26 on page 523 and problem 20 on page 532 to turn in on Friday.
Friday, Apr 5, 2002.
Study examples 1-4 on pages 536-537
Do problems 13, 15, 19, 31 on page 556 and
problems 1, 5, 7, 9, 11, 13, 15, 17, 19, 23, 27, 29 on pages 568-569
to check understanding.
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- Monday, Apr 8,Study example 1 on pages 547-548 and examples 4 and 5 on pages 552-553.
On pages 555-556 do problems 11, 13, 15, 19, 31, and 33 to check understanding. 2002.
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- Tuesday, Apr 9, 2002.
- Study examples 2 and 3 on pages 549-550 and examples 6 and 7 on pages 553-554.
Do problems 21, 23, 39 on pages 555-557 to check understanding.
Problem 25 will be done in problem session on Thursday.
Write up problems 26 and 32 to turn in on Friday.
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- Wednesday, Apr 10, 2002.
- Study examples 1-3 on pages 559-560 and examples 5-7 on page 562.
Do problems 13, 15, 19, 25, 27, 32, 36, 39 on page 565 to check understanding.
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- Friday, Apr 12, 2002.
- Review sections 8.7 and 8.8.
Do problems 33, 35, 37, 41, 45, 47, 63 and 65 on page 569 to check understanding.
Monday, Apr 15, 2002.
Sections 9.1 and 9.2, Arc Length and Surface Area.
Study examples 1-3 on pages 577-578 and examples 1 and 2 on pages 585-586.
Do problems 7, 11, 19, 35 on pages 580-581 (On 19 and 35, you can use
Simpson's method to approximate the value of the integral). Also do
problems 13, 15, 21 on page 587 to check understanding.
Tuesday, Apr 16, 2002.
Review of Homework Problems.
Review Chapters 8 and 9.
No new problems assigned.
Wednesday, Apr 17, 2002.
Sections 11.1-11.2 Parametric Equations, Tangents and Areas
Study examples 1-3 on pages 675-676.
Do problems 5, 7, 9, 11, 17 on page 679 and problems 3, 5, 9
on page 687 to check understanding.
Review for exam next Wednesday by working Practice Exam on Chapters 8 and 9
Get help with the Practice Exam at Thursday's problem session, 1-4 p.m.
Friday, Apr 19, 2002.
Section 11.3 Arc Length and Surface Area
Study examples 1 and 2 on page 691.
Do problems 3, 9, 11, 13 on page 693, and problem 22 on page 680
to check understanding.
- Monday, Apr 22, 2002.
- Section 11.4 Polar Coordinates
Study examples 1-6 on pages 695-697 and example 9 on page 700.
Do problems 15, 17, 19, 21, 23, 25, 61, and 65 to check understanding.
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- Tuesday, Apr 23, 2002.
- Review
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- Wednesday, Apr 24, 2002.
- Hour Exam and Take Home Exam
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- Friday, Apr 26, 2002.
- Section 11.3 Equations and Graphs in Polar Coordinates
Study examples 7, 8, 9 on pages 698-700
Write up problems 16, 24, 28, 64, 76 on pages 702-703 to turn in on Tuesday.
- Monday, Apr 29, 2002.
- Finding areas in Polar Coordinates
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.
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- Tuesday, Apr 30, 2002.
- Arc Length of polar curves.
Study example 4 on page 707.
Do problems 37, 39, 41, 45, 47, 49 on pages 708-709 to check understanding.
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- Wednesday, May 1, 2002.
- Review
- Wednesday, May 8, 2002.
- Final Exam, 9:00 a.m. - 12:00 p.m.
Carswell 208
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