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Calculus III

by: Braeden Lind

Calculus III MA 242

Braeden Lind
GPA 3.93

Larry Norris

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Larry Norris
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This 5 page Study Guide was uploaded by Braeden Lind on Thursday October 15, 2015. The Study Guide belongs to MA 242 at North Carolina State University taught by Larry Norris in Fall. Since its upload, it has received 48 views. For similar materials see /class/223700/ma-242-north-carolina-state-university in Mathematics (M) at North Carolina State University.

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Date Created: 10/15/15
Study Guide for Test 2 MA 242 on campus MA 242601 and MA 242651 LK Norris You should be able to work problems of the following type 1 Determine the largest set on which a given function of 2 or more variables is continuous In doing so you will need to use the following facts Section 112 a Polynomial functions are continuous everywhere b Rational functions are continuous at all points where the denominator polynomial is non zero c The composition of a continuous function with a continuous function is continu ous More precisely7 if ay is continuous at xmyo and g is continuous at fz0y07 then the composite function g o f is continuous at xmyo d The sum or difference of continuous functions is a continuous function e The product of continuous functions is a continuous function 3 Use the following theorem plus the theorems listed above to determine if a function ay is differentiable or not at a point Section 114 A function ay is differentiable at a point P0 xmyo if the X and y partial derivatives and exist near P0 and and are continuous at P0 9 Be able to compute rst and second partial derivatives of multivariable functions Sec tion 113 4 Be able to write down the chain rule formulas for the following situations a Given fyz where z gt and y ht and z lt are given functions of the independent variable t7 compute W an e mi m m ampm b Given ay z where z 90 t and y hrt and z lrt are given functions of the independent variables r and t7 compute w an g 37quot 7 3m 3r 3g 37quot 32 37quot an an atquot 6176t y at 62 6t 5 Know how to compute the directional derivative DgfP0 of a multivariable function f at a point P0 in a speci ed direction 6 I 00 Know how to compute the gradient Vfy7 z of a function ay z7 and know the geometrical signi cance of 1 the magnitude of Vfxyz and 2 the direction of the gradient vector See theorem 157 page 804 in your textbook Be able to nd 1 the tangent plane to a level surface of a function of 3 variables7 and 2 the tangent plane to the graph of a function of 2 variables Be able to nd all critical points of a function fy of two variables7 and then be able to use the second derivative test to determine if the critical points correspond to local maxima7 local minima or saddle points of the function Section 117 Study Guide for Test 3 MA 242601 and MA 242651 The test will cover the following sections of Chapter 121 2 3 4 5see below 7 and 8 In addition it will contain material from Chapter 9 section 7 on cyclindrical and spherical coordinates 1 Chapter 12 section 1 Double integrals over rectangles a You should know the general de nition 5 of the double integral of a function fxy over a rectangular region R b You should know the de nition at the top of page 842 for the volume below the graph of a function and above a region R in the xy plane c You should know the de nition of the average value of a function on a rectangle given on page 844 d You should know the properties of double integrals given on page 847 of your text book e You should realize that the above items will be generalized in section 123 where we will no longer require the region of integration to be a rectangle 2 Chapter 12 section 2 lterated integrals and Fubini7s Theorem a Be able to compute iterated integrals double and triple such as those given in problems 3 10 on page 853 and problems 3 6 on page 890 At most you will be required to use 77substitution77 to evaluate such integrals b You should know Fubini7s theorem page 850 and be able to apply it to problems like those worked in the text and the examples I worked for you in class 3 Chapter 12 section3 Double integrals over general regions a You should know how to use the two basic theorems 3 and 5 for evaluating double integrals over type I and type ll regions b You should be able to decompose a general region into a set of subregions each of type I or type ll See problems 41 and 42 at the end of the section c You should be able to compute volumes below graphs of functions fxy and above a general region in the xy plane d You should be able to nd the volume between the graphs of two functions fxy and gxy by reducing this problem to one you have already solved For example the volume of the region between the paraboloids z 2 y2 and z 18 7 2 7y2 would be found as follows These two paraboloids intersect in the circle of radius 9 centered on the origin in the xy plane You nd this 77curve of intersection by solving the two equations simultaneously eliminating Z Let the region inside this circle be denoted D Then the volume between the two paraboloids would be given by the double integral of 18 7 2 7 y2 over D MINUS the double integral of 2 7 y2 over the region D e You should be able to 77reverse the order of integration77 on a given iterated inte gral To do this you Use the limits on the given iterated integral to write down the description of the region in set notation Use the set notation to sketch the region If the set notation indicates that the region is type l7 then use the sketch to rewrite it as type ll7 and conversely i lt Set up the iterated integral in the opposite order f Chapter 127 section 4 Double integrals in polar coordinates i You should know and be able to use theorems 2 and 3 to set up and eval uate double integrals in polar coordinates This involves using the trasforma tion equation x rcos0 and y rsin0 See the examples worked in the textbook and the examples I worked for you in class g Chapter 127 section 5 Applications of double integrals Below are the applications you are responsible for i Volume below the graph of a function and above a general region in the xy plane ii Average value of a function over a general region in the xy plane iii Area of a general region in the xy plane In this case the integrand of the double integral will be 1 iv Densities lf pxy is the mass or charge density of a region D in the xy plane7 then the total Mass or charge of the region is the double integral of pxy over the region D h Chaper 127 section 7 Triple integrals in Cartesian coordinates i ii You should know and be able to apply the three versions 67 7 and 8 of Fubini7s theorem for triple integrals See the examples worked in the book and the many exmaples I worked for you in class iii Know the formula 12 for the volume of the 3 dimensional region using triple integration 4 Chapter 127 section 8 Triple integrals in cylindrical coordinates a See Chapter 9 section 7 for the de nition of cylindrical and spherical coordinates b Know the transformation equations Cylindrical coords z r cost97 y r sin0 and z z and Spherical z psin cost97 y psin sin0 and z pcos in order to be able to transform an integrand given in terms of xy and Z to either of these coordinate systems c Be able to use the above to set up a triple integral as a triple iterated integral in either cylindrical or spherical coordinates See the examples worked in the text book and the many examples I worked for you in class 03 q 01 a T 00 Study Guide for test 1 MA 242601 and MA242651 LK Norris NOTE The material in sections 96 and 97 will NOT be covered on this test Know the de nition of Cartesian coordinates in space VECTORS Know how to work with vectors This includes a adding and subtracting multiplying by a scalar dot and cross products and vector projections b Know and be able to use Theorem 932 and the theorem in the LAST box on page 668 of your textbook Lines and planes Section 95 Know the equations for lines and planes in space Be able to work problems like those worked out in the textbook and in lecture In particular you should be able to work problems like problems 2 28 on pages 683 684 Vector functions and space curves Section 101 a Know the de nition of a vector valued function Be able to determine the do main of a vector valued function b Know how to determine if a vector valued function of t has a limit at a point to and whether or not a vector valued function of t is continuous at a point to c NOTE You will NOT be asked to sketch any curves on the test Section 102 derivatives and integrals of vector functions a Be able to compute derivatives and integrals of vector valued functions b Know the properties in Theorem 3 on page 714 c Be able to compute the unit tangent vector T for a given curve Section 103 Arc length and curvature of a curve a Be able to compute the arc length of simple curves as in Example 1 page 718 and problems 1 4 on page 723 b Know the de nitions of curvature of a curve and unit normal vector of a curve and be able to compute both quantities for a given curve c NOTE I will NOT ask you about the unit binormal B vector for a curve Section 104 Acceleration a Be able to compute the tangential LT and normal aN components of acceleration b Be able to compute the curvature amp of the curve


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