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inside lume, of a positively charged conducting sphere. 10) Which of the following statements about Gauss's law are correct? (There may be more than one correct choice.) A) The electric flux passing through a Gaussian surface depends only on the amount of charge inside that surface, not on its size or shape. B) Gauss's law is valid only for symmetric charge distributions, such as spheres and C) cylinders. If there is no charge inside of a Gaussian surface, the electric field must be zero at points of that surface. D) If a Gaussian surface is completely inside an electrostatic conductor, the electric field must always be zero at all points on that surface. E) Only charge enclosed within a Gaussian surface can produce an electric field at points on that surface.
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Middle East Technical University - PHYSICS - PHYS 110

01234 894 7 7 7 7 7 7 7 5267 868 7 7 7 7 7 7 7 !"$&( #%')/ :$72 #;)# )$0 =-$ $ $ ')$ $?#- <,/ ) > % ) @ ( +,-$0 "#2 (45%- ) 83%49(- $"$</8$11, '- '&(2-#.")$ "# * . ) 13 6$7$.) 1,.$-#., $, ;,- B$ $ . $0 $ "$, #F $ #.")$ "#$<11$ H "-; "1) $</"-@ A$ . . ;,

Middle East Technical University - PHYSICS - PHYS 110

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Middle East Technical University - PHYSICS - PHYS 110

01234 894 7 7 7 7 7 7 7 7 5267 868 7 7 7 7 7 7 !"$&( #%')$ #-"!$, 0*$'&($24#"./$/ * -# 2".$ #+" * 5$9 $2+"1* 2 $ 2$88#$* *+ $- + / $ $ '1 3 "25$#+" -* #+7$-/.+ $ 82./$4+#:$ 7$* 5 +"2 7 , . * % 2 6. ,# 2 2 7# 2 2$/+ +"2#2.$/ * -# $ $ #9 !"1$ #+"2$89 9, $

Middle East Technical University - PHYSICS - PHYS 110

Middle East Technical University - PHYSICS - PHYS 110

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Alpha particles (charge = +2e, mass = 6.68 10-27 kg) are accelerated in a cyclotron to a final orbit radius of 0.60 m. The magnetic field in the cycl

Middle East Technical University - PHYSICS - PHYS 110

Middle East Technical University - MATHEMATIC - MATH 119

MATH 120RECITATION QUESTIONS (WEEK 1)1. Determine whether the sequence converges or diverges.If it converges,find the limit. a) an = (-1)n n3 n3 + 2n2 + 1 ln n b) an = ln 2n n! c) an = n 22. Use a graph of the sequence to decide whether the sequence is

Middle East Technical University - MATHEMATIC - MATH 119

MATH 120RECITATION QUESTIONS WEEK 21. Use the integral test to show thatn=0 e-n converges.22. Show thatn=21 converges if and only if p > 1. n(ln n)p3. Does the seriesn=2ln n n23converge? 1 1 - sin ) converges or diverges. n n4. Determine whe

Middle East Technical University - MATHEMATIC - MATH 119

MATH 120RECITATION QUESTIONS WEEK 31. Find the radius of convergence and interval of convergence of the series.a)n=1 xn 5n n5 xn (ln n)nb)n=22. A function f is defined by f (x) = 1 + 2x + x2 + 2x3 + x4 + that is, its coefficients are c2n = 1 and c

Middle East Technical University - MATHEMATIC - MATH 119

MATH 120RECITATION QUESTIONS WEEK 41. Find the Maclaurin series for f (x) = (1 + x)-3 , and the radius and interval of convergence, also. 2. Find the Maclaurin series of f (x) = cos x2 and its radius of convergence. Compare the graph of f and those of i

Middle East Technical University - MATHEMATIC - MATH 119

MATH 120WEEK 5 (RECITATION QUESTIONS)1. Find an equation of the sphere that passes through the point (4, 3, -1) and has center (3, 8, 1). 2. Describe in words the region of3represented by the equation x2 + y 2 = 1. 33. If v lies in the first quadran

Middle East Technical University - MATHEMATIC - MATH 119

MATH 120WEEK 6 (RECITATION QUESTIONS)1. Reduce the equation 4y 2 + z 2 - x - 16y - 4z + 20 = 0 to one of the standard forms, classify the surface, and sketch it. 2. Sketch the region bounded by the surfaces z = x2 + y 2 and x2 + y 2 = 1 for 1 z 2.3. Fi

Middle East Technical University - MATHEMATIC - MATH 119

MATH 120RECITATION QUESTIONS (WEEK 7)1. Using different directions to approach the origin, show that the following limits do not exist. a) b) lim(x,y)(0,0)xy x2 + y 2xy 2 (x,y)(0,0) x2 + y 4 lim2. Using the formal definition of the limit prove a) b)

Middle East Technical University - MATHEMATIC - MATH 119

MATH 120RECITATION QUESTIONS (WEEK 8)1. Find an equation of the tangent plane to the given surface at the specified point. a) z = b) z = e 4 - x2 - 2y 2 ,x2 -y 2(1, -1, 1),(1, -1, 1)2. Explain why the function f (x, y) =x is differentiable at the

Middle East Technical University - MATHEMATIC - MATH 119

MATH 120RECITATION QUESTIONS (WEEK 9)1. Let f (x, y) =2xy x2 +y 2if (x, y) = (0, 0) 0 if (x, y) = (0, 0)Show that Du f (0, 0) exists in every direction u = cos, sin , 0 2, but the formula Du f (0, 0) = f (0, 0) u is true only for u = i or u = j . 2.

Middle East Technical University - MATHEMATIC - MATH 119

INTEGRATION PROBLEMS, MATH 120, WEEK 10DAVID PIERCE1. FindRf when R = [0, 2] [1, 2] and f (x, y) = 3xy 2 - 2x2 y. x 1 and 0 y 1.2. Find the average value of y 2 /(x2 + 1) when 03. Express as an iterated integral the volume enclosed by: the two sheet

Middle East Technical University - MATHEMATIC - MATH 119

MATH 120WEEK 11 (RECITATION QUESTIONS)1 + cos . 21. Find the area inside the larger loop and outside the smaller loop of the limacons r =2. Find the volume of the solid bounded by the cylinders x2 + y 2 = r2 and y 2 + z 2 = r2 . 3. Use polar coordinat

Middle East Technical University - MATHEMATIC - MATH 119

MATH 120WEEK 12 (RECITATION QUESTIONS)1. Find the area of the surface : the part of the plane z = 2 + 3x + 4y that lies above the rectangle [0, 5] [1, 4]. 2. Find the area of the surface : the part of the sphere x2 + y 2 + z 2 = 4z that lies inside the

Middle East Technical University - MATHEMATIC - MATH 119

MATH 120 SPRING 2008 WEEK 13 RECITATION (1) Plot the gradient vector fields and level curves of the functions given below: (a) f (x, y) = x2 - y 2 (b) f (x, y) = + x2 + y 2 (2) Evaluate xy 2 dsCwhere C is given by r(t) = (cos t, t, sin t). (3) Evaluate

Middle East Technical University - MATHEMATIC - MATH 119

MATH 120WEEK 14 (RECITATION QUESTIONS)1. Determine whether or not F(x, y) = (x3 + 4xy)i + (4xy - y 3 )j is a conservative vector field.If it is, find a function f such that F = f 2. (a) Find a function f such that F = (b) Use part (a) to evaluate where

Middle East Technical University - MATHEMATIC - MATH 119

Math119 Recitation 2September 25, 20061. Evaluate the following limits (a) limx8 (b) (c)x2/3 -4 x1/3 -2 3 limx0 1+x-1 x 3+x2 - 3-x2 limx0 x21 x -1 x-1(d) limx11 (e) limx0 ( x -(f) 2. Let1 x2 ) limx0 2+x- 2-x 3 2+x- 3 2-xf (x) = Find limx0 f (x).

Middle East Technical University - MATHEMATIC - MATH 119

Math120 Recitation 3October 4, 20061. Let f (x) = 4 - x2 x-1 if x 2, if x > 2.(a) Find limx2- f (x) and limx2+ f (x). (b) Does limx2 f (x) exists? (c) Sketch the graph of f . 2. Show by means of an example that limxa f (x)g(x) may exists even though ne

Middle East Technical University - MATHEMATIC - MATH 119

Math119 Recitation 5October 18, 20061. Find a formula for f n (x) for f (x) = (1 + x)-1 . 2. Find the given derivative by observing the pattern f 35 (x); where f (x) = x sin x 3. Find constants A, B, C such that the function y = Ax2 + Bx + C satisfies t

Middle East Technical University - MATHEMATIC - MATH 119

Math 119 Recitation 6October 21, 20061. Evaluate the following limits or explain why they do not exist.|x-2| (a) limx2 |x|-2 . (b) Let 5 - x2 f (x) 5 - x3 for -1 x 1. Find limx0 f (x).(c) limx11-x3 . 1- x(d) limx (e) (f)x+sin x 2x+7-5 sin x . x+x l

Middle East Technical University - MATHEMATIC - MATH 119

Math 119 Recitation 7November 1, 20061. Verify that the function f (x) = sin(2x) satisfies (a) f is continuous on [-1, 1], (b) f is differentiable on (-1, 1), (c) f (-1) = f (1). and find all numbers c (-1, 1) such that f (c) = 0. 2. Verify that the fun

Middle East Technical University - MATHEMATIC - MATH 119

Math 119 Recitation 8November 6, 20061. Find the dimensions of the isoceles triangle of largest area that can be inscribed in a circle of radius r. 2. A right circular cylinder is inscribed in a sphere of radius r. Find the largest possible surface area

Middle East Technical University - MATHEMATIC - MATH 119

Math 119 Recitation 9November 16, 20061. Use mid-point rule to approximate 2. Express the limit limnn i=1 5 x-1 dx 1 x+1with n = 4.xi sin xi x on [0, ] as a definite integral.5 (1 -13. Use formal definition of integral to evaluate 4. Prove thatb a

Middle East Technical University - MATHEMATIC - MATH 119

Math 119 Recitation 10November 21, 20061. Find the area of the region bounded by y 2 = 2x and x - y = 4. 2. Let R be the region between the parabolas y = x2 and y = 8x. (a) Find the area of R. (b) Find the volume of the solid of revolution that we obtai

Middle East Technical University - MATHEMATIC - MATH 119

Math 119 Recitation 12December 5, 20061. Differentiate (a) y = arcsin 1 - x2 (b) y = arctan 1-x 1+x (c) y = arctan (sin x) (d) y = tanh (ln x) 2. Find the tangent line to the curve arctan ( 2x ) - x = 0 at the point (1, 2). y y2 3. Evaluate the integral

Middle East Technical University - MATHEMATIC - MATH 119

Math 119 Recitation 13December 20, 20061. Investigate the convergence using the comparison and limit comparison tests (a) (b)1 dx 0 (1-x3 ) x2 +x4 5 dx 2 (x-3)2 (x-2)3/22. Use the limit and the comparison tests to deterimine the convergence of the fol

Middle East Technical University - MATHEMATIC - MATH 119

MATH-119WEEK 6 RECITATION QUESTIONS1. Given that the graph of f passes through the point (1, 6) and that the slope of its tangent line at (x, f (x) is 2x + 1,find f (2) 2. The graph of the velocity function of a car is shown in the figure.Sketch the gra

Middle East Technical University - MATHEMATIC - MATH 119

MATH-119 WEEK 11 RECITATION QUESTIONS Evaluate the integrals 1. x arctan xdx./42.0x sec2 xdx. x4 (ln x)2 dx23. 4.1x2 dx . 9 - x25.dx . 6x - x2 - 8 ex e2x - 1 dx.ln 36.ln 27.sin2 x cot3 xdx./68.0cos x sec(2x)dx. x2 ex dx39.andx3 ex

Middle East Technical University - MATHEMATIC - MATH 119

Middle East Technical University - MATHEMATIC - MATH 119

Middle East Technical University - MATHEMATIC - MATH 119

Middle East Technical University - MATHEMATIC - MATH 119

Middle East Technical University - MATHEMATIC - MATH 119

Middle East Technical University - MATHEMATIC - MATH 119

Calculus With Analytic GeometryHakan TOR February 26,20101. Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows. (a) Shift 3 units upward. (b) Shift 3 units downwards. (c) Shift 3 units to the

Middle East Technical University - MATHEMATIC - MATH 119

Calculus With Analytic GeometryHakan TOR March 4,20101. The graph of g is given. (a) At what numbers is g discontinuous? Why? (b) At what numbers is g not differentiable? Why?Solution: (a) Although limx-2- g(x) = limx-2+ g(x), the function g(x) is disc

Middle East Technical University - MATHEMATIC - MATH 119

Middle East Technical University - MATHEMATIC - MATH 119

Calculus With Analytic GeometryHakan TOR March 19,20101. For each of the numbers a, b, c, d, e, r, s, and t, state whether the function whose graph is below has an absolute maximum or minimum, a local maximum or minimum, or neither a maximum nor a minim

Middle East Technical University - MATHEMATIC - MATH 119

Middle East Technical University - MATHEMATIC - MATH 119

Federal University of Technology - EE - 100

LISTA 3 - Prof. Jason Gallas, IFUFRGS3 de Dezembro de 2005, as 16:24 ` Exerccios Resolvidos de Optica Fsica i iJason Alfredo Carlson Gallas, professor titular de fsica te rica, i oDoutor em Fsica pela Universidade Ludwig Maximilian de Munique, Alemanh

Cornell - PHYS - 1116

A Comparison between the Newtonian Universe and RealityPhysics 1116, Spring 2009 As strange as Einsteins theory of relativity may seem to you, it is the way the real world actually works. It is in fact the Newtonian universe that is a fictional construct

Cornell - PHYS - 1116

Cornell - PHYS - 1116

Cornell - PHYS - 1116

Cornell - PHYS - 1116

Cornell - PHYS - 1116

Cornell - PHYS - 1116

Cornell - PHYS - 1116

Cornell - PHYS - 1116

Cornell - PHYS - 1116

Cornell - PHYS - 1116

Cornell - PHYS - 1116

Cornell - PHYS - 1116

Cornell - PHYS - 1116

An Inventory of Forces Encountered in Physics 1116Below is a list of the forces that you will encounter in Physics 1116, with a basic description of how they act. You might be interested to know that, with the exception of gravity, all the other forces a

Cornell - PHYS - 1116

Cornell - PHYS - 1116

Cornell - PHYS - 1116

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