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E-21a Math Fall 2009 HW #5 problems Problems to turn in on Thurs, Oct 8: Section 11.2: In problems 10 and 11, find the limit, if it exists, or show that the limit does not exist. 6 x3 y xy 10. lim 11. lim 4 4 ( x , y ) (0,0) 2 x y ( x , y ) (0,0) x2 y 2 Section 11.3:
37. Find the indicated partial derivative: f ( x, y, z )
x ; yz
f z (3, 2,1)
68. Show that the Cobb-Douglas production function P ( L, K ) bL K satisfies the equation
L P K P ( ) P L K
Section 11.4: 4. Find an equation of the tangent plane to the given surface at the specified point: z y ln x , (1, 4, 0)
28. Use differentials to estimate the amount of metal in a closed cylindrical can that is 10 cm high and 4 cm in diameter if the metal in the top and bottom is 0.1 cm thick and the metal in the sides is 0.05 cm thick. 31. If R is the total resistance of three resistors, connected in parallel, with resistances R1, R2, and R3, then 11 1 1 R R1 R2 R3 If the resistances are measured in ohms as R1 = 25, R2 = 40, and R3 = 50, with a possible error of 0.5% in each case, estimate the maximum error in the calculated value of R.
38. Suppose you need to know an equation of the tangent plane to a surface S at the point P(2,1,3). You dont have an equation for S but you know that the curves r1 (t ) 2 3t ,1 t 2 ,3 4t t 2 and r2 (u ) 1 u 2 , 2u 3 1, 2u 1
both lie on S. Find an equation of the tangent plane at P. Section 11.5: 2. Use the Chain Rule to find dz : z x ln( x 2 y ), x sin t , y cos t dt 32. The radius of a right circular cone is increasing at a rate of 1.8 in/sec while its height is decreasing at a rate of 2.5 in/sec. At what rate is the volume of the cone changing when the radius is 120 inches and the height is 140 inches?
34. The voltage V in a simple electrical circuit is slowly decreasing as the battery wears out. The resistance R is slowly increasing as the resistor heats up.. Use Ohms Law, V = IR, to find how the current I is changing at the moment when R = 400, I = 0.08A, dV 0.01 V/sec , and dR 0.03 /sec . dt dt Section 11.6: In problems 8 and 10, (a) find the gradient of f ; (b) Evaluate the gradient at the given point P ; (c) Find the rate of change of f at P in the direction of the vector u. 236 43 8. f ( x, y ) y ln x , P (1, 3) , u 5 , 5 10. f ( x, y, z ) x yz , P(1,3,1), u 7 , 7 , 7
In problems 12 and 14, find the directional derivative of the function at the given point in the direction of the vector v. x 12. f ( x, y ) ln( x 2 y 2 ), (2,1), v 1, 2 14. f ( x, y, z ) , (4,1,1) , v 1, 2,3 yz
1
30. Suppose that you are climbing a hill whose shape is given by the equation z 1000 0.005 x 2 0.01y 2 and you are standing at point a with coordinates (60, 40, 966). a) If you walk due south, will you start to ascend or descend? At what rate? b) If you walk northwest, will you start to ascend or descend? At what rate? c) In which direction is the slope largest? What is the rate of ascent in that direction? At what angle above the horizontal does the path in that direction begin?
Additional practice problems: Section 11.2:
13. Find the limit, if it exists, or show that the limit does not exist:
2x2 y ( x , y ) (0,0) x 4 y 2 lim
Section 11.4: 1. Find an equation of the tangent plane to the given surface at the specified point: z 4 x 2 y 2 2 y , (1, 2, 4) .
13. Find the linear approximation of the function f ( x, y ) 20 x 2 7 y 2 at (2, 1) and use it to approximate f (1.95,1.08) . 15. Find the linear approximation of the function f ( x, y, z ) x 2 y 2 z 2 at (3, 2, 6) and use it to approximate the number (3.02) 2 (1.97) 2 (5.99) 2 . 25. The length and width of a rectangle are measured as 30cm and 24cm, respectively, with an error in measurement of at most 0.1cm in each. Use differentials to estimate the maximum error in the calculated area of the rectangle.
Section 11.5:
dz : z sin x cos y, x t , y t dt 29. The temperature at a point (x, y) is T ( x, y ) , measured in degrees Celsius. A bug crawls so that its position
1. Use the Chain Rule to find
after t seconds is given by x 1 t , y 2 1 t , where x and y are measured in centimeters. The 3 temperature function satisfies Tx (2,3) 4 and Ty (2,3) 3 . How fast is the temperature rising on the bugs path after 3 seconds?
33. The length l, width w, and height h of a box change with time. At a certain instant the dimensions are l = 1 meter, and w = h = 2 meters, and l and w are increasing at a rate of 2 meters/sec while h is decreasing at a rate of 3 meters/sec. At that instant find the rates at which the following quantities are changing: (a) The volume (b) The surface area (c) The length of the diagonal.
Section 11.6: In problem 7, (a) find the gradient of f ; (b) Evaluate the gradient at the given point P ; (c) Find the rate of change of f at P in the direction of the vector u. 5 7. f ( x, y ) 5 xy 2 4 x 3 y , P (1, 2) , u 13 , 12 13
13. Find the directional derivative of the function at the given point in the direction of the vector v: g ( s, t ) s 2 et , (2, 0) , v i j 29. Suppose that over a certain region of space the electrical potential V is given by V ( x, y, z ) 5 x 2 3 xy xyz a. Find the rate of change of the potential at P(3, 4,5) in the direction of the vector v i j k . b. In which direction does V change most rapidly at P? c. What is the maximum rate of change at P?
2

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Harvard - MATH - E-21a

Math E-21a Fall 2009 HW #6 Solutions Section 11.3:1Section 11.5:23Section 11.6:4

Harvard - MATH - E-21a

Math E-21a Fall 2009 HW #6 problems Problems to turn in on Thurs, Oct 15: Section 11.3: z z and : yz ln( x z ) 42. Use implicit differentiation to find x y 48. Find all the second partial derivatives: f ( x, y ) ln(3 x 5 y ) 62. Level curves are shown for

Harvard - MATH - E-21a

Math E-21a Fall 2009 HW #7 Solutions Section 11.7:123450. Since the plane passing through (1, 2, 3) must cut out a tetrahedron in the 1st octant, its intercepts must all be positive call them a, b, and c. The volume of this tetrahedron will be V 1 ab

Harvard - MATH - E-21a

Math E-21a Fall 2009 HW #7 problems Problems to turn in on Thurs, Oct 22: Section 11.7: In problems 6-14, find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function wi

Harvard - MATH - E-21a

HW #8 Solutions Math E-21a Fall 2009 Problem 1:Problem 2:Problem 3:Problem 4:1Problem 5:Problem 6: [Note: The figure shown is a rather poor one. Yours is probably a lot better.]Heres an alternative: Use the parameterization from part (a) which give

Harvard - MATH - E-21a

Math E-21a Fall 2009 HW #8 problems Read sections 11.8 and 12.1 (and maybe 12.2-12.3) and do the following problems: Problems to turn in on Thurs, Oct 29: 1. (Prob. 11.8/16) Use the Method of Lagrange Multipliers to find the maximum and minimum values of

Harvard - MATH - E-21a

Math E-21a Fall 2009 HW #9 Solutions Section 12.2:Section 12.3:12Section 12.4:34Section 12.5:56

Harvard - MATH - E-21a

Math E-21a Fall 2009 HW #9 problems Problems to turn in on Thurs, Nov 5: Section 12.2: 12. Calculate the iterated integral: 18. Calculate the double integral: Section 12.3:011 0xy x 2 y 2 dx dyRx dA, R [0,1] [0,1] ( x, y ) : 0 x 1, 0 y 1 1 xy4y dA

Harvard - MATH - E-21a

Math E-21a Fall 2009 HW #10 Solutions Section 12.7:123Section 12.8:456

Harvard - MATH - E-21a

Math E-21a Fall 2009 HW #10 problems Problems to turn in on Thurs, Nov 12: Section 12.7: 4. Evaluate the iterated integral: 0 xE12xy 02 xyz dz dy dx10. Evaluate the triple integral2x 2 y z 4 . 12. Evaluate the triple integral (1,1,0), and (0,1,1)

Harvard - MATH - E-21a

Math E-21a Fall 2009 HW #11 Solutions Section 12.9:Section 13.1:12Section 13.2:34Section 13.3:5

Harvard - MATH - E-21a

Math E-21a Fall 2009 HW #11 problemsProblems due Thurs, Nov 19:Section 12.9: (these were inadvertently omitted from last weeks PDF of the assignment) 12. Evaluate the integral (4 x 8 y ) dA , where R is the parallelogram with vertices (1,3) , (1, 3) , (

Harvard - MATH - E-21a

Math E-21a Fall 2009 HW #12 Solutions Section 13.4:1Section 13.5:2345

Harvard - MATH - E-21a

Math E-21a Fall 2009 HW #12 problemsProblems due Thurs, Dec 3:Section 13.4: 4. Evaluate the line integral by two methods: (a) directly and (b) using Greens Theorem. x dx y dy , C consists of the line segments from (0,1) to (0,0) and from (0,0) to (1,0)

Harvard - MATH - E-21a

Math E-21a Fall 2009 HW #13 Solutions Section 12.6:12Section 13.6:3456

Harvard - MATH - E-21a

Math E-21a Fall 2009 HW #13 problems To be turned in Thurs, Dec 10: Section 12.6: 2. Find the area of the part of the plane 2 x 5 y z 10 that lies inside the cylinder x 2 y 2 9 .4. Find the area of the part of the plane with vector equation r (u , v) 1 v

Harvard - MATH - E-21a

Math E-21a Fall 2009 HW #14 Solutions Section 13.7:1Section 13.8:234

Harvard - MATH - E-21a

Math E-21a Mega-List of Things You May Want to Know Fall 2009 1. Write equations for surfaces in R3 that are characterized geometrically by distances (spheres, cylinders, etc.), and determine their intersection with specified planes. 2. Sketch or identify

Harvard - MATH - E-21a

Math E-21a Practice Exam #1 Solutions Fall 20091) True/False questions (circle one) you do not need to show your reasoning: a) The vector v 1, 2, 4 is perpendicular to the plane with equation 4 x 2 y 2 z 100 . False: In fact, the vector v 1, 2, 4 is para

Harvard - MATH - E-21a

Math E-21a Midterm Exam 1 Topics Fall 20091. Given vectors in R2 or R3, do addition, scalar multiplication, dot and cross products, scalar and vector projection. 2. Manipulate vector expressions symbolically (distributive law, triple product, A (B C), et

Harvard - MATH - E-21a

Math E-21a Fall 2009 Practice Exam #2 solutions1) The level sets of a function f ( x, y ) are shown. Determine the signs of the following derivatives at the point P. Indicate either +, , or 0 in the spaces provided.fx =+ fy = f xx = f yy = f xy = +2)

Harvard - MATH - E-21a

Math E-21a Things to know for the 2nd midterm exam Higher order derivatives, interpretation of 2nd derivatives, equality of mixed partials (Clairauts Theorem), quadratic approximation. Extrema of functions and the 2nd Derivative test; local maxima, local

Harvard - MATH - E-21a

Math E-21a Practice Final Exam Fall 2009The following problems appeared on previous Final Exams. Neither the number of problems nor the type of problems are necessarily indicative of problems that will appear on this semesters exam, but they should provi

Harvard - MATH - E-21a

Math E-21a Practice Final Exam solutions Fall 20091) True or False (circle one you need not show your work): a) The dot product of two nonzero vectors gives the cosine of the angle between these two vectors. FALSE - u v u v cos , so in general the dot pr

Harvard - MATH - E-21a

Math E-21a Some useful facts f dx f dy + = f v for a path in R2; Basic Chain Rule: d [ f ( x(t ), y (t ) ] = dt x dt y dt d f ( x(t ), y (t ), z (t ) = f dx + f dy + f dz = f v for a path in R3. [ ] dt x dt y dt z dt Directional Derivative of a function f

Harvard - HISTORY OF - harce183

Fall 2009 Tuesday, 7:35-9:35 PM Emerson Hall, Room 108Alexander von Hoffman 1033 Mass. Ave., Rm. 502/ 617-495-9915 alexander_von_hoffman@harvard.eduHARC E-183 THE ARCHITECTURE OF BOSTON (11489) Course Requirements 1) Attendance at classes and tours 2) R

Harvard - HISTORY OF - harce183

Classical Revival Boston Public Library; 1888-95; Copley Square; Charles F. McKim Boston Symphony Hall; 1900; Huntington Avenue; McKim Massachusetts Horticultural Society; 1901; Huntington Avenue; Wheelwright & Haven Museum of Fine Arts; 1907-09; Huntingt

Harvard - HISTORY OF - harce183

"Of Greater Lasting Consequence": Frederick Law Olmsted and the Fate of Franklin Park, Boston Author(s): Alexander von Hoffman Source: The Journal of the Society of Architectural Historians, Vol. 47, No. 4 (Dec., 1988), pp. 339-350 Published by: Society o

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GEOPHYSICS, VOL. 62, NO. 3 (MAY-JUNE 1997); P. 713722, 7 FIGS., 1 TABLE.P -wave reection coefcients for transversely isotropic models with vertical and horizontal axis of symmetryAndreas Ruger (AVO). The variation of P -wave amplitudes with incidence a

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Answers to Mastery ChecksModule 1: C+ Fundamentals1. C+ is at the center of modern programming because it was derived from C and is the parent of Java and C#. These are the four most important programming languages. 2. True, a C+ compiler produces code

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GEOL 4330Introduction to Geophysics Assignment No. 3Fall, 2010Answer BOTH questions Please read each question CAREFULLY 1. Table A lists the position (latitude, , and longitude, ) of the paleomagnetic pole for North America for various times over the p

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Pure appl. geophys. 164 (2007) 21412156 00334553/07/11214116 DOI 10.1007/s00024-007-0266-8 Birkhauser Verlag, Basel, 2007Pure and Applied GeophysicsAnisotropy Effects on Microseismic Event LocationANDREW KING1 and SHAHRIAR TALEBI2AbstractSeismic anis

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Houston Downtown - GEOL - 7320

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Houston Downtown - GEOL - 7320

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Houston Downtown - GEOL - 7320

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Houston Downtown - GEOL - 7320

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Houston Downtown - GEOL - 7320

Basic Earth ImagingJon F. Claerbout c November 17, 2010Contents1 Field recording geometry 1.1 RECORDING GEOMETRY . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 1.2 Fast ship versus slow ship . . . . . . . . . . . . . . . . . . . . . . . . 1

Houston Downtown - GEOL - 7320

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Houston Downtown - GEOL - 7320

IMAGE ESTIMATION BY EXAMPLE: Geophysical Soundings Image Construction Multidimensional autoregressionJon F. Claerbout Cecil and Ida Green Professor of Geophysics Stanford University with Sergey Fomel Stanford University c February 2, 2011Contents1 Basi

Houston Downtown - GEOL - 7320

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Houston Downtown - GEOL - 7320

1 .1 Exploding ReflectorsT he basic equipment for reflection seismic prospecting is a source for impulsive sound waves, a geophone (something like a microphone), and a multichannel waveform display system. A s urvey line is defined along the earth's surf

Houston Downtown - GEOL - 7320

Why S pace and Time?I n the previous chapter we learned how t o extrapolate wavefields down into the earth. The process proceeded simply, since it is just a multiplication in the frequency domain by exp[ik,(w, k , ) ~ ] . Finite-difference techniques wil

Houston Downtown - GEOL - 7320

3.0 Offset, Another DimensionE arlier chapters have assumed t h a t the shot and the geophone are located in the same place. The reality is t hat there is often as m uch as a 3k m horizontal separation between them. The 3-km offset is comparable t o the

Houston Downtown - GEOL - 7320

T h e Craft of Wavefield ExtrapolationT his chapter attends t o those details t h a t enable us t o do a high-quality job of downward continuing wavefields. There will be few new seismic imaging concepts here. There will, however, be interesting examples

Houston Downtown - GEOL - 7320

5.0 Some F rontiersI n this final chapter of the book are gathered together imaging concepts t h a t have been published, but have not yet come into routine industrial use. The first part of this chapter develops the mathematical concept of linear m oveo

Houston Downtown - GEOL - 7320

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Adrian College - STAT - MC101

They will have less than five unfilled orders 94.7347% of the time.P(X = 0) = 0.13533528 P(X = 1) = 0.27067057 P(X = 2) = 0.27067057 P(X = 3) = 0.18044704 P(X = 4) = 0.09022352Hypergeometric distribution with parameters N = 9, M = 4, n = 3.Accordin

Wyoming - CHEM - 2420

CHEM 234, Spring 2008Final ExamIan R. GouldPRINTED FIRST NAMEPRINTED LAST NAMEASU ID or Posting IDPerson on your LEFT (or Aisle)!PRINT YOUR NAME ON EACH PAGE! READ THE DIRECTIONS CAREFULLY! USE BLANK PAGES AS SCRATCH PAPER work on blank pages will

Wyoming - CHEM - 2420

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Wyoming - CHEM - 2420

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Wyoming - CHEM - 2420

Wyoming - CHEM - 2420

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Wyoming - CHEM - 2420

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Wyoming - CHEM - 2420

Massachusetts Institute of Technology Dr. Kimberly L. Berkowski5.13, Fall 2006 Organic Chemistry IIEXAM #3 EXTRA PROBLEMSKEYWhat to expect on Exam #3: 1. ~1 Labeling experiment 2. ~2 Mechanisms 3. ~2 Syntheses 4. ~5 transformations supply missing prod

Wyoming - CHEM - 2420

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Wyoming - CHEM - 2420

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Wyoming - CHEM - 2420

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