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"Math 16A (Autumn 2005) Thomas Scanlon University of California, Berkeley Week 10 Thomas Scanlon Math 16A (Autumn 2005) University of California, Berkeley Dierential equation for exponential functions Fact Fix a real number k. y = f (t) is a solu"
"MATH 16B, SPRING 2004 PRACTICE FOR THE FINAL EXAM There are more questions listed here than you will actually see on the nal. The general distribution of the questions accurately reects the topics to be covered. 1. Find the point (x, y, z) which max"
"Section 9.6: Improper integrals We have considered only integrals of the form a f (x)dx where a b are real numbers and f is a function which is dened and continuous on the interval [a, b] := {x | a x b}. Sometimes, it makes sense to consider integ"
"Math 16A Week 3 HW Solutions Section 1.4 18. lim 26. lim x2 1 x1 x1 = lim x1 (x1)(x+1) (x1) = lim (x + 1) = 2 x1 1 2 x9 (x9) does not exist. 1 72x f (3) = lim f (3+h)f (3) h h0 1 1 72(3) = lim 72(3+h) h h0 36. f (x) = = lim = h0 h 1(12h)"
"General method If R := {(x, y) | a x b, g(x) y h(x)} then b h(x) f (x, y)dxdy = R a ( g(x) f (x, y)dy)dx To compute the iterated integral, rst nd F (x, y) for which F y = f (x, y) so that h(x) g(x) y=h(x) f (x, y)dy = F (x, y)|y=g(x) = F (x,"
"Section 7.7: Double integrals If R is some region in the plane and f (x, y) is a (continuous) function taking positive values, then the volume of {(x, y, z) | (x, y) in R and 0 z f (x, y)} is the integral f (x, y)dx dy R More generally, if f (x, "
"MATH 113: ABSTRACT ALGEBRA (AUTUMN 2007) SOLUTIONS TO MIDTERM # 1 1. Let G be a group. The center of G, Z(G), is by definition the set of elements of G which commute with every element of G, Z(G) := {g G : (x G)gx = xg}. Prove that Z(G) is a subgr"
"Section 10.2: Separation of variables The method of separation of variables applies to dierential equations of the form y = p(t)q(y) where p(t) and q(x) are functions of a single variable. 1 Example Find the general solution to the dierential equat"
"MATH 113: INTRODUCTION TO ABSTRACT ALGEBRA AUTUMN 2007 MIDTERM 2 PRACTICE PROBLEMS 1. Compute 35,789,345 in Z70 . 2. Prove or disprove: If G is a group and K G and N G are two normal subgroups which are isomorphic to each other, N K, then G/K G/N "
"MATH 113: ABSTRACT ALGEBRA (AUTUMN 2007) MIDTERM # 2 SOLUTIONS 1. (10 points) Compute 58,238,390,323 in Z33 . Solution: 33 = 3 11. Hence, Z33 Z2 Z11 so that Z Z Z which = 33 = 3 11 has (3 1) (11 1) = 20 elements. Dividing, we see that the re"
"Math 16A (Autumn 2005) Thomas Scanlon University of California, Berkeley Week 13 Thomas Scanlon Math 16A (Autumn 2005) University of California, Berkeley Introduction to Riemann sums The antiderivatives introduced in the last section can be used"
"PART 3 THERMODYNAMICS TEMPERATURE AND HEAT Solution Temperature dierences on the Fahrenheit and Celsius scales are related by TF = (9/5) TC , so ( 9 ) 5 (10 C ) = 18 F. (Note that a temperature dierence and a temperature reading are not the same, even tho"
"CHAPTER 17 SOUND AND OTHER WAVE PHENOMENA How many miles away did the ash occur? Neglect the travel time for the light (why?) ActivPhysics can help with these problems: Activities 10.3, 10.4, 10.5, 10.6, 10.8, 10.9 Sections 17-1 and 17-2: Sound Waves and "
"CHAPTER 4 MOTION IN MORE THAN ONE DIMENSION Problem 3. An object is moving in the x direction at 1.3 m/s when it is subjected to an acceleration given by a = 0.52 m/s2 . What is its velocity vector after 4.4 s of acceleration? ActivPhysics can help with t"
"PROBLEM 4.1 The boom on a 4300-kg truck is used to unload a pallet of shingles of mass 1600 kg. Determine the reaction at each of the two (a) rear wheels B, (b) front wheels C. SOLUTION WA = mA g = (1600 kg ) 9.81 m/s 2 = 15696 N ( ) or WA = 15"
"CHAPTER 12 ROTATIONAL MOTION Problem 4. A 25-cm-diameter circular saw blade spins at 3500 rpm. How fast would you have to push a straight hand saw to have the teeth move through the wood at the same rate as the circular saw teeth? ActivPhysics can help wi"
"CHAPTER 18 FLUID MOTION density of the compressed air? (b) How large a volume would the same gas occupy at typical 3 atmospheric density of 1.2 kg/m ? Section 18-1: Describing Fluids: Density and Pressure Problem 1. The density of molasses is 1600 kg/m . "