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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

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

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

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

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 multivariable functions? Let's remember the Chain Rule for differentiating the composition of a singlevariable func

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

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 3tuples? ntuples? What type of value is the dot product sc

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!

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

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

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

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

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

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

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

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

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

12_5HW
School: Morningside
12.5 Homework Solutions Math 206 Mammenga VECTORVALUED FUNCTIONS, DERIVATIVES, AND INTEGRALS ASSIGNMENT: 1. Let P = (3, 7) and Q = (1, 1). (a) Find a vectorvalued function that describes a line through P and Q. Solution: We can do this by adding

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

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

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

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

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

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 xcoordinate and another rule or formula for the ycoordinate. This means that both x and y depend upon the same inde

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 yaxis are equi

12_1HW
School: Morningside
12.1 Homework Solutions Math 206 Mammenga THREEDIMENSIONAL SPACE ASSIGNMENT: 1. Consider the equation x2 + y 2 = 3. (a) Plot the equation x2 + y 2 = 3 in xyspace. Solution: (b) Plot the equation in xyzspace. Solution: (c) What is the relations

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, rewrite the y as y = xy 4 dy x = 4 dx y 4 y dy = x dx y 4 dy = x dx dy

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

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

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=

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

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

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

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  (

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

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 "reallife" situations, the amount of force applied is rarely constant. This then requires the use of an integral: b Work done by

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

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

12_1
School: Morningside
12.1 Lecture Notes Math 206 10/17/08 Mammenga THREEDIMENSIONAL SPACE A. Cartesian coordinates in 3space how does this work? The axes should be plotted consistent with the "righthand rule." B. The distance between P = (x1 , y1 , z1 ) and Q = (x2 ,

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 twovariable function f (x, y) near a known value and that we know the derivative at that known value. We can do so

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 wdirection. Call this vector a. Solution: Using the formula given above, a= (2,

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).