Math 21a. Multivariable CalculusReview Guide for Midterm IISpring 2008Thomas W. JudsonHarvard UniversityApril 10, 2008Midterm DetailsThe second midterm will focus on Chapters 11 and 12, but you will be responsible for any previousmaterial covered in the course The midterm exam will be on Monday, April 21 at 7-9 PM in ScienceC. There will also be a course-wide review session on Thursday, April at 6-7:30 PM in Science D.We plan to videotape the review session, and you should be able to access the video by clicking onLecture Videos at the course website.Studying and Reviewing• You can find copies of old midterms and solutions by clicking on the Exams page of the coursewebsite. Please keep in mind that these midterms may cover slightly different material.• You should also try working some of the problems in the review sections of Chapters 11 and12. Solutions will be posted in the on the Exams page of the course website.• Be sure to take advantage of the TF office hours, CA se ctions, and the MQC.Topics for Midterm II• To understand functions of several variables and be able to represent these functions usinglevel sets. (Section 11.1)• To understand and be able to apply the concept of a limit of a function of several variables.(Section 11.2)1• To understand and be able to apply the definition of continuity for a function of severalvariables. (Section 11.2)• To understand and be able to apply the definition of a partial derivative. (Section 11.3)• To be able to compute partial derivatives. (Section 11.3)• To understand and be able to apply Clairaut’s Theorem. If f is defined on a disk D thatcontains the point (a, b) and the functions fxyand fyxare continuous on D, thenfxy(a, b) = fyx(a, b).(Section 11.3)• To understand the idea of a partial differential equation, and to be able to verify solutions topartial differential equations. (Section 11.3)• To understand the concept of a tangent plane to a surface z = f (x, y) and to be able tocompute the equation of tangent planes (Section 11.4)• To understand and be able to find a linear approximation to a function z = f(x, y). (Section11.4)• To understand the concept of a tangent plane to a parametrically defined surfacer(u, v) = x(u, v) i + y(u, v) bfj + z(u, v) kand to be able to compute the equation of tangent planes. (Section 11.4)• To understand and be able to find the differential to a function z = f (x, y),dz =∂z∂xdx +∂z∂ydyTo be able to use the differential to estimate maximum error. (Section 11.4)• To understand and be able to apply the chain rule for functions of se veral variables. (Section11.5)• To understand and be able to implicitly differentiate functions. (Section 11.5)• To understand and be able to apply the definition of the directional derivative. (Section 11.6)• To understand and be able to apply the definition of the gradient of a function f. (Section11.6)• To understand that |∇f (x)| is the m aximum value of the directional derivative, Duf(x).(Section 11.6)• To be able to use the gradient to find the tangent plane to a level surface. (Section 11.6)• To understand and be able to find the local extrema and saddle points of a function z =f(x, y). (Section 11.7)• To understand and be able to apply the Second Derivative Test. (Section 11.7)2• To understand and be able to find the absolute maximum and minimum of a function z =f(x, y). (Section 11.7)• To understand and be able to apply the Method of Lagrange Multipliers to solve constrainedoptimization problems. To find the minimum and maximum values of f (x, y, z) subject tothe constraint g(x, y, z) = k. (Section 11.8)1. Find all values of x, y, z, λ such that∇f(x, y, z) = λ∇g(x, y, z)and g(x, y, z) = k.2. Evaluate f at all of the points (x, y, z) from step (1). The largest of these values will bethe maximum value of f and the smallest the minimum value of f .• To understand and be able to apply the definition of the double integral of a function f overa region R,ZZRf(x, y) dA.(Section 12.1)• To understand and be able to apply the midpoint rule for the double integral of a function fover a region R. (Section 12.1)• To understand and to be able to compute the value ofZZRf(x, y) dA.when R is a rectangle in the xy-plane. In particular, to be able to evaluate the integral as aniterated integral and to be able to apply Fubini’s Theorem. (Section 12.2)• To understand and be able to apply the properties of double integrals. (Section 12.2)• To understand and to be able to compute the average value of a function z = f (x, y) over aregion R in the xy-plane. (Section 12.2)• To understand and be able to compute the double integral of a function f over a generalregion D,ZZdf(x, y) dA.(Section 12.3)• To understand and be able to compute the double integral of a function f over a type I regionR,ZZDf(x, y) dA =ZbaZg2(x)g1(x)f(x, y) dy dx,whereD = {(x, y)|a ≤ x ≤ b, g1(x) ≤ y ≤ g2(x)}.(Section 12.3)3• To understand and be able to compute the double integral of a function f over a type IIregion R,ZZDf(x, y) dA =ZdcZh2(y)h1(y)f(x, y) dx dy,whereD = {(x, y)|c ≤ y ≤ d, h1(y) ≤ x ≤ h2(y)}.(Section 12.3)• To understand and be able to apply the properties of double integrals.(a)ZZD[f(x, y) + g(x, y)] dA =ZZDf(x, y) dA +ZZDg(x, y) dA(b)ZZDcf(x, y) dA = cZZDf(x, y) dA(c) If f (x, y) ≥ g(x, y) for all (x, y) ∈ D, thenZZDf(x, y) dA ≥ZZDg(x, y) dA.(d) If D = D1∪ D2, where D1and D2do not overlap except possibly on their boundaries,thenZZD[f(x, y)] dA =ZZD1f(x, y) dA +ZZD2f(x, y) dA(e)ZZDdA is the area of D, A(D).(f) If m ≤ f (x, y) ≥ M for all (x, y) ∈ D, then m · A(D) ≤ZZDf(x, y) dA ≤ M · A(D).(Section 12.3)• To understand and be able to change to polar coordinates in a double integral,ZZDf(x, y) dA =ZβαZbaf(r cos θ, r sin θ) r dr dθ.(Section 12.4)• To understand and be able to compute the surface area of a parameterized surfacer(u, v) = x(u, v)i + y(u, v)j + z(u, v)kwhere (u, v) are in a domain D.A(S) =ZZD|ru× rv| dA,whereru=∂x∂ui +∂y∂uj +∂z∂uk rv=∂x∂vi +∂y∂vj +∂z∂vk(Section 12.6)4• To understand and be able to compute the surface area of the graph of a function z = f(x, y),A(S) =ZZDs1 +∂z∂x2+∂z∂y2dA.(Section 12.6)• To understand and be able to compute the surface area of a surface of revolution S obtainedby rotating a curve y = f (x), a ≤ x ≤ b, about the x-axis, where f(x) ≥ 0 and f0iscontinuous,A(S) = 2πZbaf(x)p1 +
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