Marywood University Department of Mathematics Course Information for Math 203 Calculus III by Dr. J.

The Ring Nebula (Almost a Conic Section)

The third course in the four-semester Calculus sequence. It covers techniques of integration, an introduction to differential equations, sequences and infinite series, Taylor series, polar coordinates and conic sections. The textbook is "Calculus" by Howard Anton published by Wiley. The chapters from this text covered in class are Chapters 9, 10, 11, and 12. The course grade is determined by 3 tests, 4 homework sets, 2 Maple computer labs, and the final examination. Useful websites for the course are listed at the end of this page. Prerequisite: Math 202.

Test #1 Test #2 Test #3

Test #1 Sample Problems (Evaluation of integrals, Differential equations)

• Find dy/dx for each of the following functions.
  • y = cosh(x⁵)
  • y = (sinh 2x)(tanh 3x)
  • y = cosh²x - sinh²x
• Evaluate these integrals by an appropriate substitution.
  • Int [x (sec²(x²)] dx
  • Int [1/(x ln x)] dx
  • Int [cosh(sqrt(x))/sqrt(x)] dx
  • Int[sqrt(x² - 25)/x] dx (for x >= 5) Answer: sqrt(x² - 25) - 5 sec^-1(x/5) + C
• Write out the form of the partial fraction decomposition: [3x - 7]/[(x+4)³(x²+8x+3)²]
• What trigonometric substitution is needed to evaluate this integral? (Do not actually do the integration.)
  • Int [x³ sqrt(5 - x²)] dx
• Find the following integrals using either integration by parts or partial fractions.
  • Int [Arcsin x] dx
  • Int [1/(x² + 8x + 7)] dx
  • DefInt (0, 2) [ x e^2x] dx Answer: (3e⁴ + 1)/4
• Evaluate the improper integeral: Defint(e, +infinity) [1/(x ln³x)] dx Answer: 1/2
• Solve the following differential equations:
  • e^-y sin x - y' cos²x = 0
  • (1 + y²) y' = (e^x) y
  • y' + 2xy = x
• Polonium-210 is a radioactive element with a half-life of 140 days. If 10 milligrams are placed in a lead container, how many milligrams will be present after 50 days? (Ans: 7.81 days) How long will it take for 70% of the original sample to decay? (Ans: 243.2 days)
• The half-life of a certain radioactive substance used in radiology is 8 days. How long will it take 10 gm of the substance to decay to 3.8 gm?

Test #2 Sample Problems (Infinite series)

• True or False.
  • If a monotone decreasing sequence is bounded from below, then it converges.
  • If a monotone increasing sequence is bounded from below, then it converges.
  • If lim uk = 0, then the series sigma(uk) converges.
  • If S ak and S bk are both convergent, then S (ak + bk) must also be convergent.
  • If S ak and S bk are both divergent, then S (ak + bk ) must also be divergent.
• Find the limits of the following sequences { an }. Give reasons.
  • i) an = (2n+1)(n+2)/(3n²)
  • ii) an = cos(¼n/2)
  • iii) an = (n)^1/n
• Show that the sequence { n e^-2n } is monotone. Does it converge? By what theorem?
• Express the repeating decimal 0.181818 . . . as a fraction.
• Find the sum of the following convergent series.
  • i) S [(-3/4)^k-1]
  • ii) S [1/(k²+5k+6)] [Hint: Find the nth partial sum using partial fractions.]
• Determine whether each of the following series converges or diverges. You must give a reason for your answer.
  • i) S [ke^(-k²)]
  • ii) S [k² / (4+k²)]
  • iii) S (2^k / k³)
• Find the Taylor series about xo = 1 for the function f(x) = sqrt(x) specifically writing out the first four terms. Use this series to approximate sqrt(1.01) to four decimal places.
• Classify S [(-1)^k+1 / k^4/3] as absolutely convergent, conditionally convergent, or divergent.

Test #3 Sample Problems (Conic sections, Polar coordinates, Parametric equations)

• Classify each of the following equations as representing circle, ellipse, parabola, or hyperbola.
  • i) x² + y² + xy + x - y = 3
  • ii) 2x² - y² + 4xy - 2x + 3y = 6
  • iii) x² + 4xy + 4y² - 3x = 6
  • iv) x² + y² + 3x - 2y = 10
• Find the equation of the circle with center at (2,2) which passes through the point (4,5).
• Find an ellipse with one vertex at (3,1), the nearer focus at (1,1), and eccentricity 2/3.
• Find the equation of the hyperbola with foci at (0,0) and (0,4) if it passes through the point (12,9).
• A point P moves so that the ratio of its distances from two fixed points is a constant k (not = 1). What conic section does P trace out and why?
• The earth moves in an elliptical orbit with the sun at one of the foci. If the length of half the major axis is 93 million miles and the eccentricity is 0.017, find the least and greatest distances of the earth from the sun. (The points on the orbit where these occur are known as perihelion and aphelion respectively.)
• Determine the Cartesian equation of r = 5sec(pi/3 - q).
• One focus of an ellipse of eccentricity 1/4 is at the origin. The corresponding directrix is the line r cosq = 8. Find the polar equation of the ellipse. (Hint: Draw a picture and use PF = ePD.)
• Find the area inside the lemniscate r² = 4 cos 2q and outside the circle r = sqrt(2).
• Find dy/dx and d²y/dx² at the point t = 2 for the curve described parametrically by x = exp(t) , y = exp(-t) without eliminating the parameter.
• Find the length of the cardioid r = 2(1 - cosq).
• A comet has a parabolic orbit with the sun at its focus. When the comet is 100 sqrt(2) million miles from the sun, the line from the sun to the comet makes an angle of 45º with the axis of the parabola. What will be the minimum distance between the comet and the sun?

Useful Websites

http://www.math.psu.edu/dna/graphics.html (Great graphics for learning Calculus from Penn State)

http://www.ping.be/math (Tutorials about most of the main topics in Calculus and Linear Algebra)

http://www.calculus.net (Very interesting practice problems and projects in Calculus and Diff. Equations)

http://www.math.duke.edu/modules/materials/index.html (Modules and projects created by Duke University for enhancing the learning of Calculus, Diff. Equations, Linear Algebra, and Engineering Mathematics)

http://www.maplesoft.com/apps/powertools/ (Maple 6 Resource Center)

http://www.maa.org (Mathematical Association of America)

http://archives.math.utk.edu (Math Archives)

http://forum.swarthmore.edu (Math Forum)

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Comments to Dr. Craig M. Johnson, Chair, Dept. of Mathematics: johnsonc@ac.marywood.edu

Last update: March 10, 2001

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