calc iv - exams

Note: More information, with possible changes, will be added as the exam dates near.

exam 1: fri sep 29 (in class)

this exam will cover all of chapter 14, excluding 14.5 (the chain rule). while not necessarily a comprehensive list of things you may be expected to do on the exam, here are the main points you should be comfortable with (though there may not be time to ask about all of them on the exam):

answer true/false questions about basic theory.
find/sketch domain, range, graphs for simple functions of 2 variables. you may also be asked about level curves.
determine partial derivatives, gradients of functions of 2 or 3 variables, and use this to write down equations for the tangent plane of a graph z=f(x,y) at a point. you may need to use a tangent plane to do linear approximation to approximate the value of a 2-variable function at a point.
compute directional derivatives of functions of 2 or 3 variables, be able to determine the direction of fastest increase or decrease of a function of 2 or 3 variables (e.g., convert the gradient to convert to a unit vector)
find local/absolute minima/maxima, as well as saddle points, of functions of 2 variables. (think: critical points and second derivatives test.) expect to have to deal with regions with boundary points, where you may need to:
use lagrange multipliers to maximize/minimize a function of 2 or 3 variables subject to a constraint. unless explicitly stated that you need to use lagrange multipliers, you are also allowed to solve such problems by the "substitution/reduction" method (e.g., as in example 5 in sec 14.7, though it may not be possible/easy to do so).

there are review problems at the end of chapter 14 (pp 1021--1024) for more practice problems. some suggested problems from there are:

concept check: 1, 3, 5, 6, 7(a), 8, 14, 15, 16, 17, 19
true-false quiz: 5, 7, 8, 9, 10, 12
exercises: 1, 3, 4, 13, 20, 25(a), 33, 43, 45, 48, 52, 56, 60, 63

exam 2: wed nov 8 (in class)

the second exam will cover all of chapter 15, except for section 15.4. while not necessarily a comprehensive list of things you may be expected to do on the exam, here are the main points you should be comfortable with (though there may not be time to ask about all of them on the exam):

compute double and triple integrals in cartesian (rectangular) coordinates. this includes setting up, and possibly changing the order of, iterated integrals.
compute areas of regions in the plane, and surface areas and volumes of regions in 3-space.
be able to go back and forth between cartesian coordinates and polar coordinates or cylindrical and spherical coordinates, including translating integrals between these different coordinate systems.
compute double integrals using general change of variables

here are some suggested review problems from the chapter 15 review in the book (pp. 1101-1104). i recommend you attempt these before clas on wed apr 5, and i can try to answer any questions you have about them wed in class or during office hours. (standard disclaimer: these exercises do not necessarily cover everything you will be tested on.)

concept check: 1(abc), 2(a), 3, 6, 9, 10(ab)
true-false: 1, 2, 3, 4, 5, 9
exercises: 3, 5, 9, 10, 13, 15, 16, 19, 24, 25, 30, 31, 37, 38, 55

final exam: th dec 14 (4:30-6:30pm)

the final exam will be cumulative, covering (most of) chapters 14, 15 and 16 of the book. you should expect several true/false questions, as well as a few (2-4?) problems on each of the 3 chapters covered in this course. a list of the main topics covered on the final exam is:

all the topics listed in the bullet points above for exams 1 and 2
vector fields: be able to draw them (in R^2), determine if they are conservative (in R^2 or R^3), compute curl and div (in R^3), and know basic facts about div and curl
line integrals: be able to compute the various kinds of line integrals (ds, dx, dy, dz and vector field line integrals), both directly as well as using the fundamental theorem and green's theorem
surface area: be able to compute surface area in the plane with double integrals as in ch 15 as well as with green's theorem, and areas of more general surfaces such as graphs of functions or parametric surfaces as in sec 16.6
surface integrals (scalar fields): be able to parameterize surfaces and compute surface integrals of functions
surface integrals (vector fields): expect 1 problem related to surface integrals of vector fields, i.e. flux, such as computing flux, directly or with the divergence theorem, or using stokes' theorem to compute a line integral in R^3 as a surface integral. there will not be more than one problem about flux, and it may be that this problem can be solved without stokes' or the divergence theorem also, but possibly such a solution will be more difficult. on the other hand, there may be other problems not directly about flux in R^3 for which you can use stokes' theorem or the divergence theorem.

since there was no homework on the last 3 topics (flux, stokes, divergence), here are a few practice problems for you on this material:

in addition, here are some suggested problems from the ch 16 review (pp 1188-1190):

concept check: 2a, 3abde, 4ab, 5-10, 11bc, 12abc, 13cd, 14-16
true-false: 1-13
exercises: 3, 4, 9, 11, 13, 17, 18, 25, 27, 29, 30, 33, 34

(see the practice problems for the first two exams for chapters 14 and 15)

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