Marywood University Department of Mathematics Course Information for Math 201 by Dr. J.

Isaac Newton (1642-1727)

Math 201 Calculus I

This is the first course in the 12-credit calculus sequence. Topics include functions, graphing, limits and continuity, differentiation and its applications, relative and absolute extrema, and optimization. The textbook is "Calculus", Alternate Edition, written by Thomas and Finney and published by Addison Wesley. The course grade is based upon three tests, weekly quizes, two Maple labs, and a final examination. Prerequisite: Mastery of algebra, trigonometry and analytic geometry. Useful websites for the course are listed at the end of this page.

 Test #1 Sample Problems

• Find the domain and the range of the function f(x) = 2 + sqrt(16 - 9x²).
• For what value of a does the function f(x) = (x² - 4)/(x - a) have a removable discontinuity at x = a ?
• Sketch the following graphs. Be certain of your x and y-intercepts.
  • h(x) = (x + 1)² - 4
  • y = 3 sin 2x + 1
  • f(x) = sqrt(x - 5) + 2
  • f(x) = cos(x + p/2)
• Here is the graph of some function y = f(x). On the same set of axes, draw the graph of y = f (x - 2)+ 3. Label the corners of this graph.

• For the function y graphed below, find the following limits. If a limit which does not exist can be expressed as -infinity or +infinity, please do so. Otherwise, write DNE (does not exist).

• lim y(x) = ________ as x approaches 2+  
• lim y(x) = ________ as x approaches 2-
• lim y(x) = ________ as x approaches 2
• lim y(x) = ________ as x approaches -1
• lim y(x) = ________ as x approaches +infinity
• lim y(x) = ________ as x approaches - infinity
• On what intervals is y continuous?

• Find the points of discontinuity, if any.
  • f(x) = tan x over [0, 2p]
  • f(x) = [x] over the real numbers (The greatest integer function.)
  • f(x) = (x² + 3x - 4)/(x + 4) over the real numbers
  • f(x) = sin (1/ x) over the positive real numbers
• True/False
  • f o g = g o f for any two functions f and g.
  • A curve in the xy-plane is the graph of a function if and only if no vertical line intersects the curve more than once.
  • If f is continuous at c, then lim f(x) = f(c) as x approaches c
  • If lim f(x) = L as x approaches c for any function f, then f(c) = L.
  • All polynomials are continuous everywhere.
  • All trigonometric functions are continuous everywhere.
  • If f is continuous on the interval [a,b] and f(a) > 0 and f(b) < 0, then there is at least one solution to the equation f(x) = 0 in the interval (a, b).
• Find the following limits. If a limit which does not exist can be expressed as -infinity or +infinity, please do so. Otherwise, write DNE (does not exist).
  • lim (2x² - 6x + 5) as x approaches 3
  • lim [(x² + 6x + 8)/(x + 2)] as x approaches -2
  • lim sin(1/x) as x approaches 0^-
  • lim [1/(x² - 1)] as x approaches 1⁺
• Find g(x) if f(x) = 3x - 8 and (f o g)(x) = x² + x .
• If f(x) = 2x³ - 9x + 8, explain why there exists at least one number a such that f(a) = pi. (Hint: Use IVP.)
• Use the definition of the derivative to find f '(x) for f(x) = 1/x .

Test #2 Sample Problems

• Find dy/dx for each of the following.
  • y = -10x^4 + sec 2x
  • y = 6x⁴ - 5x³ + 1/x²
  • y = (4x⁵ + 6x³ - 1)⁵⁰
  • y = 12 tan³x
  • y = sqrt(x)cos⁴(3x)
  • y = (x²-1)/(x²+1)
  • y = sin(x²y²)
  • y = sqrt(1 + sin(x²)
• Find the equation of the line tangent to the graph of g(x) = sqrt(x) at x = 9.
• Find the values of A and B for the function f(x) = A sin x + B cos x if f(0) = 2 and the tangent line to the graph at (0, 2) has slope m = 4.
• On Earth, in the absence of air, assume that a rock thrown vertically upward reaches a height of s = 24t - 4.9t^2 meters in t seconds. How long would it take the rock to reach its maximum height? What is that maximum height?
• Sketch the graph of the derivative of the function whose graph is shown.

• Compute: i) d³y/dx³ for y = (x + 8)³ and ii) dⁿy/dxⁿ for y = (x + a)ⁿ 
• Use the definition of the derivative to find f '(x) for f(x) = sin x.
• If y = cos 4x, then find the value of d³y/dx³ at x = pi/8.
• Find all critical points of h(x) = x sqrt(8 - x^2).
• Find all relative extrema for f(x) = 2x² - x⁴.
• Find the maximum and minimum values of f(x) = cuberoot(3x - 2) over the interval [0, 4].
• Find the function f(x) whose derivative is f '(x) = x^2 + 1 - sin x and such that f(0) = 3.
• Draw a graph of a single function f that satisfies all four of these criteria:
  • f continuous everywhere except at x = 3.
  • f(-1) = 0, f(0) = 1.5, f(1) = 2, f(3) = 5
  • f '(x) > 0 for x < 1, f '(1) = 0, f '(x) < 0 for x > 1.
  • lim f(x) = 1 as x --> 3, lim f(x) = 0 as x --> infinity
• Assume that you only know that the formula d[xⁿ]/dx = nx^n-1 works for n any integer. Use implicit differentiation to prove that d[x^r]/dx= rx^r-1 also holds for r any rational number.
• You are sitting 3000 ft from the launch site of a rocket. After launch, the rocket rises vertically at the rate of 500 ft/sec. At what rate is the elevation angle changing between you and the rocket at the instant the rocket is 3000 ft high?
• Water runs into a conical tank at the rate of 9 cubic ft/min. The tank stands point down and has a height of 10 ft and a radius of 5 ft. How fast is the water level rising when the water is 6 ft deep? (Ex. 4 p. 175)
• A 13-ft ladder is leaning against a house when its base starts to slide away. By the time the base is 12 feet from the house, the base is moving at the rate of 5 ft/sec. At that time, how fast is the top of the ladder sliding down the wall and how fast is the angle between the ladder and the ground changing?
• State Rolle's Theorem precisely.
• Let f(x) = 1/x . Show that there is no c in the interval (-1, 2) such that f '(c) = [f(2) - f(-1)]/[2 - (-1)] . Explain why this does not violate the Mean Value Theorem.

Test #3 Sample Problems

• Use a linear approximation to estimate the value of cuberoot(26.9).
• Determinte the percentage error in computing the area of a circle if the radius measurement has a percentage error of ±3%.
• Find the intervals on which f is: (i) increasing (ii) decreasing (iii) concave up (iv) concave down and (v) list the x-coordinates of any inflection points.
  • f(x) = 4+ 15x - x³
  • f(x) = x^4/3 - x^1/3
  • f(x) = sin x cos x on [0, 2p]
• Use derivative tests to graph the function f(x) = x²/(1+ x²). Be sure to draw any asymptotes and to include both coordinates of any intercepts, extrema, or inflection points.
• The position function of a particle moving along a coordinate line is given by s = t + 9/(t+1) for t >= 0. Analyze the motion of the particle and draw a schematic picture of the motion. Be sure to label the key points.
• The area A of a circular sector ("piece of pie") of radius r and arclength s is given by A = 0.5 r s. If the perimeter is 100 meters, what value of r will produce a maximum area?
• A cylindrical tin can has to hold 50 cm³ of tomato juice. Find the radius of the can which requires a minimum amount of material for its construction.
• Use Newton's method to determine an algorithm (i.e. a formula for x_n+1 in terms of x_n ) for approximating 4^throot(30) by solving x⁴ - 30 = 0. Also, if x₀ = 2, find x₁.

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: January 5, 2002

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