• 2 Pages 13_2
    13_2

    School: Morningside

    13.2 Lecture Notes Math 206 11/17/08 Mammenga PARTIAL DERIVATIVES A. We want to do calculus on functions of several variables. What does the derivative of such a function really mean? B. Let f (x, y) be a function of two variables. The partial deri

  • 3 Pages 11_5HW
    11_5HW

    School: Morningside

    11.5 Homework Solutions Math 206 Mammenga POWER SERIES ASSIGNMENT: Find the radius of convergence and the interval of convergence of the following power series. 1. (1)n xn n2n n=1 Solution: We must use the Ratio Test to determine the interval of con

  • 2 Pages 12_9
    12_9

    School: Morningside

    12.9 Lecture Notes Math 206 11/6/08 Mammenga THE CROSS PRODUCT A. The cross product of two vectors in R3 , say u = (u1 , u2 , u3 ) and v = (v1 , v2 , v3 ), is given by the following determinant: uv= i j k u1 u2 u3 v1 v2 v3 = (u2 v3 u3 v2 )i (u1 v3

  • 7 Pages 8_3HW
    8_3HW

    School: Morningside

    8.3 Homework Solutions Math 206 Mammenga TRIGONOMETRIC ANTIDERIVATIVES ASSIGNMENT: Calculate each of the following integrals. You must show your work for full credit (but I encourage you to check your solution using Derive)! 1. sin3 x dx Solution: S

  • 2 Pages 13_7
    13_7

    School: Morningside

    13.7 Lecture Notes Math 206 11/25/08 Mammenga THE CHAIN RULE A. How do we use the Chain Rule when differentiating the composition of multi-variable functions? Let's remember the Chain Rule for differentiating the composition of a single-variable func

  • 3 Pages 13_7HW
    13_7HW

    School: Morningside

    13.7 Homework Solutions Math 206 Mammenga THE CHAIN RULE ASSIGNMENT: 1 1. Let z = u2 +v2 , where u = cos 2t and v = sin 2t. Find dz both by using dt the chain rule and by expressing z explicitly as a function of t before differentiating. Solution: W

  • 2 Pages 12_7
    12_7

    School: Morningside

    12.7 Lecture Notes Math 206 10/31/08 Mammenga THE DOT PRODUCT A. The dot product of two vectors, say v = (v1 , v2 ) and w = (w1 , w2 ), is v w = v1 w 1 + v2 w 2 . What if v and w had been 3-tuples? n-tuples? What type of value is the dot product sc

  • 1 Page Exam1
    Exam1

    School: Morningside

    Calculus II Exam 1 Mammenga Sept. 22, 2008 There are a total of 50 points on this exam. Read each question/problem carefully. Write complete answers in your blue book and make sure your writing is legible. Dont forget to check your work. Good luck!

  • 1 Page PracticeExam1
    PracticeExam1

    School: Morningside

    Practice Exam 1 Math 206 Mammenga 1. Find the area of the region bounded by the parabol y = -x2 + 1, the line 2 y = - 3 x - 2 , and the portion of the unit circle in the second quadrant. 3 1 1 2. Let R be the region that is bounded by y = cos x and

  • 2 Pages 11_1
    11_1

    School: Morningside

    11.1 Lecture Notes Math 206 09/23/08 Mammenga SEQUENCES AND THEIR LIMITS A. A sequence is an infinite list of numbers, often denoted by {ak } . k=1 Some examples are listed below: 1, 2, 3, 4, 5, . . . 1 1 1 1 1, , , , , . . . 2 3 4 5 1, -3, 9, -27, 8

  • 2 Pages 11_3
    11_3

    School: Morningside

    11.3 Lecture Notes Math 206 09/26/08 Mammenga TESTING FOR CONVERGENCE; ESTIMATING LIMITS A. In this section, we discuss some tests for convergence that do not tell us what the series actually converges to. We only use the tests to determine whether i

  • 2 Pages 13_3HW
    13_3HW

    School: Morningside

    13.3 Homework Solutions Math 206 Mammenga LINEAR APPROXIMATION IN SEVERAL VARIABLES ASSIGNMENT: 1. Suppose that f (x, y, z) = 3xy + y 2 yz. (a) Find a linear approximation for this function when x = 1, y = 1, and z = 1. Solution: Lets compute some

  • 2 Pages 10_2
    10_2

    School: Morningside

    10.2 Lecture Notes Math 206 09/15/08 Mammenga DETECTING CONVERGENCE, ESTIMATING LIMITS A. Sometimes it is difficult to determine whether an improper integral converges using the limit technique described in the previous section. The following two tes

  • 2 Pages 11_6
    11_6

    School: Morningside

    11.6 Lecture Notes Math 206 10/03/08 Mammenga POWER SERIES AS FUNCTIONS A. Recall that a power series is just a function of x, and the domain of a power series (i.e. the set of values you can "plug in" for x) is the interval of convergence for that p

  • 4 Pages 13_4HW
    13_4HW

    School: Morningside

    13.4 Homework Solutions Math 206 Mammenga THE GRADIENT AND DIRECTIONAL DERIVATIVES ASSIGNMENT: 1. Draw by hand the gradient at each point with integer coordinates in 1 the rectangle [0, 2] [0, 2] for the function f (x, y) = x2 + 2 y. Then draw the

  • 2 Pages 13_1
    13_1

    School: Morningside

    13.1 Lecture Notes Math 206 11/14/08 Mammenga FUNCTIONS OF SEVERAL VARIABLES A. A function of two variables, f (x, y), produces a single output value for each pair of input values, (x, y). A function of n variables produces a single output value for

  • 3 Pages 12_6HW
    12_6HW

    School: Morningside

    12.6 Homework Solutions Math 206 Mammenga MODELING MOTION ASSIGNMENT: 1. Suppose that a(t) = 2, v(0) = 1, and p(0) = 0. (a) Find equations for p(t) and v(t). Solution: Since the indefinite integral of acceleration is velocity, we have v(t) = 2t + c

  • 2 Pages 12_5HW
    12_5HW

    School: Morningside

    12.5 Homework Solutions Math 206 Mammenga VECTOR-VALUED FUNCTIONS, DERIVATIVES, AND INTEGRALS ASSIGNMENT: 1. Let P = (3, -7) and Q = (1, 1). (a) Find a vector-valued function that describes a line through P and Q. Solution: We can do this by adding

  • 1 Page PracticeFinal
    PracticeFinal

    School: Morningside

    Practice Final Exam Math 206 Mammenga 1. Evaluate the integral 1 2x - 3 dx. (x2 - 3x + 12)7/3 2. Determine if the infinite series k=0 k (e + 1)-k converges or diverges. If it converges, give the sum. If it diverges, explain why. 3. Consider

  • 2 Pages 12_4HW
    12_4HW

    School: Morningside

    12.4 Homework Solutions Math 206 Mammenga VECTORS ASSIGNMENT: 1. Suppose that - = (3, 5), - = (-2, 2), and - = (0, -4). u v w - + 3- - 2- . (a) Find the vector u v w Solution: Using the values listed above, we have - + 3- - 2- = (3, 5) + 3 (-2

  • 1 Page PracticeExam3
    PracticeExam3

    School: Morningside

    Practice Exam 3 Math 206 Mammenga 1. Find a parametrization for the specified regions. (a) The part of the line y = x contained within the circle x2 + y 2 = 1. (b) The trace of x2 + y 2 + z 2 + 6z = 16 in the plane z = 0. 2. Let r2 = cot . (a) Find

  • 3 Pages 10_1HW
    10_1HW

    School: Morningside

    10.1 Homework Solutions Math 206 Mammenga IMPROPER INTEGRALS: IDEAS AND DEFINITIONS ASSIGNMENT: dx . 0 2x 6 (a) Explain why the integral is improper. 1 Solution: The integrand, , is not dened for x = 3 which is a point 2x 6 in our interval of integ

  • 2 Pages 13_4
    13_4

    School: Morningside

    13.4 Lecture Notes Math 206 11/20/08 Mammenga THE GRADIENT AND DIRECTIONAL DERIVATIVES A. Let f (x, y) be a function of two variables and (x0 , y0 ) a point of its domain; assume that both partial derivatives exist at (x0 , y0 ). The gradient of f at

  • 2 Pages 12_2
    12_2

    School: Morningside

    12.2 Lecture Notes Math 206 10/23/08 Mammenga CURVES AND PARAMETRIC EQUATIONS A. A parametric equation has a rule or formula for the x-coordinate and another rule or formula for the y-coordinate. This means that both x and y depend upon the same inde

  • 5 Pages 7_2HW
    7_2HW

    School: Morningside

    7.2 Homework Solutions Math 206 Mammenga FINDING VOLUMES BY INTEGRATION ASSIGNMENT: 1. The base of a solid object is the region bounded by the parabola y = 1 x2 and 2 the line y = 2. Cross sections of the object perpendicular to the y-axis are equi

  • 3 Pages 12_1HW
    12_1HW

    School: Morningside

    12.1 Homework Solutions Math 206 Mammenga THREE-DIMENSIONAL SPACE ASSIGNMENT: 1. Consider the equation x2 + y 2 = 3. (a) Plot the equation x2 + y 2 = 3 in xy-space. Solution: (b) Plot the equation in xyz-space. Solution: (c) What is the relations

  • 3 Pages 7_4HW
    7_4HW

    School: Morningside

    7.4 Homework Solutions Math 206 Mammenga SEPARATING VARIABLES: SOLVING DIFFERENTIAL EQUATIONS SYMBOLICALLY ASSIGNMENT: 1. Solve the following DEs. (a) y = xy 4 Solution: First, re-write the y as y = xy 4 dy x = 4 dx y 4 y dy = x dx y 4 dy = x dx dy

  • 4 Pages 13_6HW
    13_6HW

    School: Morningside

    13.6 Homework Solutions Math 206 Mammenga MAXIMA AND MINIMA ASSIGNMENT: 1. Does the function f (x, y) = 2y - sin x have any stationary points? If so, find one. If not, explain why not. Solution: To find a stationary point, we find the partials of f

  • 3 Pages 11_1HW
    11_1HW

    School: Morningside

    11.1 Homework Solutions Math 206 Mammenga SEQUENCES AND THEIR LIMITS ASSIGNMENT: 1. Find the limits of the following sequences or explain why the limit does not exist. 47 (a) ak = k2 Solution: lim ak = 47 k k 2 lim k k = 47 lim = 47 0 = 0. 1 k2

  • 3 Pages 11_4HW
    11_4HW

    School: Morningside

    11.4 Homework Solutions Math 206 Mammenga ABSOLUTE CONVERGENCE; ALTERNATING SERIES ASSIGNMENT: Determine whether each of the given innite series converges absolutely, converges conditionally, or diverges. You must justify your answers. (2)n 1. n! n=

  • 4 Pages Syllabus
    Syllabus

    School: Morningside

    Calculus II, Math 206 4 credits, Fall 2008 MTRF 11:45 12:35 pm, SC 163 Instructor: Dr. Brenda Mammenga Office: SC 148 Phone: x5466 (office), (712) 5600525 (home) Email: mammenga@morningside.edu Webpage: http:/gandalf.morningside.edu/~mammenga Offic

  • 5 Pages 7_3HW
    7_3HW

    School: Morningside

    7.3 Homework Solutions Math 206 Mammenga WORK ASSIGNMENT: 1. Suppose that a spring has a natural length of 4 inches and that it requires 14 lb of force to stretch it 1 inch. (a) Find the amount of work done by stretching the spring three inches beyo

  • 2 Pages 12_8
    12_8

    School: Morningside

    12.8 Lecture Notes Math 206 11/3/08 Mammenga LINES AND PLANES IN THREE DIMENSIONS A. To describe a line in three dimensions, we need to know a point it passes through, denoted by P0 = (x0 , y0 , z0 ), and its direction vector, denoted by v = (a, b, c

  • 3 Pages 12_9HW
    12_9HW

    School: Morningside

    12.9 Homework Solutions Math 206 Mammenga THE CROSS PRODUCT ASSIGNMENT: 1. Compute the cross product of the following vectors. (a) (3, 1, -4) and (0, 3, 5) Solution: Using the diagonals method, i j k 3 1 -4 0 3 5 i j 3 1 0 3 = 5i + 0j + 9k - 0k - (

  • 2 Pages 9_1
    9_1

    School: Morningside

    9.1 Lecture Notes Math 206 10/07/08 Mammenga TAYLOR POLYNOMIALS A. We want to approximate functions that are not nice with polynomials (which are nice). What constitutes a nice function? B. Denition of a Taylor polynomial: Let f be any function whose

  • 2 Pages 7_3
    7_3

    School: Morningside

    7.3 Lecture Notes Math 206 09/09/08 Mammenga WORK A. Work is defined as the product of force and distance. However, in "real-life" situations, the amount of force applied is rarely constant. This then requires the use of an integral: b Work done by

  • 2 Pages 11_6HW
    11_6HW

    School: Morningside

    11.6 Homework Solutions Math 206 Mammenga POWER SERIES AS FUNCTIONS ASSIGNMENT: 1. Find the power series representation of f (x) = cos(3x). Solution: We just need to replace x with 3x in the standard power se (-1)n (3x)2n (-1)n 9n x2n ries of cos

  • 3 Pages 11_2
    11_2

    School: Morningside

    11.2 Lecture Notes Math 206 09/25/08 Mammenga INFINITE SERIES, CONVERGENCE, AND DIVERGENCE A. A series is a sum of a sequence and is written in the form an = a0 + a1 + a2 + a3 + . n=0 There are MANY sequences that converge themselves, but whose

  • 2 Pages 12_1
    12_1

    School: Morningside

    12.1 Lecture Notes Math 206 10/17/08 Mammenga THREE-DIMENSIONAL SPACE A. Cartesian coordinates in 3-space how does this work? The axes should be plotted consistent with the "right-hand rule." B. The distance between P = (x1 , y1 , z1 ) and Q = (x2 ,

  • 2 Pages 13_3
    13_3

    School: Morningside

    13.3 Lecture Notes Math 206 11/18/08 Mammenga LINEAR APPROXIMATION IN SEVERAL VARIABLES A. Suppose we want to approximate values of a two-variable function f (x, y) near a known value and that we know the derivative at that known value. We can do so

  • 3 Pages 12_7HW
    12_7HW

    School: Morningside

    12.7 Homework Solutions Math 206 10/31/08 Mammenga THE DOT PRODUCT ASSIGNMENT: 1. Let v = (2, 6) and w = (1, 1). (a) Find the vector that is the projection of v in the w-direction. Call this vector a. Solution: Using the formula given above, a= (2,

  • 3 Pages 8_3
    8_3

    School: Morningside

    8.3 Lecture Notes Math 206 09/02/08 Mammenga TRIGONOMETRIC ANTIDERIVATIVES A. When trying to integrate an expression involving the product of trigonometric functions, it is usually best to rst convert the expression into the form (sinm x) (cosn x).

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