MAT 267 TEST 2 REVIEW
NOTE: Below the Examples and hw refer to examples and homework from the textbook, ww refers to webwork hw problems.
CHAPTER 11
11.1 Know how to:
· Sketch the graph of basic surfaces (Examples 3,4, hw 18).
· Find the domain of two and three variable functions (Examples 1,2,10, ww 1, 2, hw 4, 8 ).
· Know how to graph level curves (Examples 6, 7, 8, 9, ww 6) and be able to match graphs of 3 dimensional surfaces and their contour maps (ww 4,5, hw 21, 22).
· Know how to find equations for the level surfaces of a three variable function (Example 11) .
11.3 Know how to
· Find partial derivatives of two and three variable functions algebraically (Examples 1, 3,5, ww 1-8), and how to estimate their value given the contour plot of the function (hw 2). Understand the geometrical meaning of the partial derivative with respect to x (rate of change in the direction of the positive x-axis) and y (rate of change in the direction of the positive y-axis) (Example 2, hw 2,3,4).
· Find higher order partial derivatives (Examples 6 and 7, ww 7, 8).
· Differentiate implicitly if z is defined implicitly as a function of x and y (Example 4, ww 9). (Note that in section 11.5 a very convenient formula for implicit differentiation is derived).
11.4 Be able to
· Find the equation of the tangent plane to a surface z = f( x, y) at a given point :
· Find linear approximation (local linearization) of a two variable function f(x,y) at a given point :
and use it to approximate the function at a point close to .
· Know the definition of differentials (Example 3, webwork 6) and be able to use them to estimate errors (Examples 4 and 5, ww 7, 8).
11.5
· Know the chain rule and how to draw tree diagrams to derive all possible cases. (Examples 1,3,4,5, ww 1-5)
· Be able to find derivatives for two and three variable functions defined implicitly:
For an implicit function of the form F(x, y) = 0: (Example 8, ww 6)
For an implicit function of the form F(x, y, z) = 0: (Example 9, ww 7)
· Practice on word problems involving the chain rule (related rates) (Example 2, ww 8, 9).
11.6
· Know how to find the directional derivative of a function f in the direction of unit vector u :(Examples 1,2,3,4, ww 1- 4, hw 22, 27) and understand its meaning as a rate of change of f in the direction of u.
· Understand the meaning of the gradient (direction of maximum rate of change), and its magnitude (value of maximum rate of change). (Examples 5, 6, ww 5-10, hw 26). Remember that the gradient is always orthogonal to the level curves (or level surfaces for a 3D function) (ww 13, hw 28).
· Know how to find the equation of a tangent plane and the normal line at a given point for a surface defined implicitly (the normal vector is the gradient);
If the surface is defined implicitly by F(x, y, z) = 0, then the tangent plane at at has equation: (Example 7, ww 11, 12).
11.7
· Know how to find the critical points of a two variable functions (Examples 1, 2).
· Be able to use the second derivative test to classify the critical points and know its conclusion regarding local minima and maxima (Example 3, ww 1, 2, 4, 5, 6).
Discriminant:
Second Derivative Test: assume
D(a, b) / / Classification of (a, b)+ / + / Local minimum
+ / − / Local maximum
− / Saddle point
0 / inconclusive
· Be able to determine the type of critical point from the contour plot (ww 3, hw 2)
· Practice on applied problems (Examples 4, 5, ww 9, 10, 11)
· To find absolute minima and maxima on a closed and bounded region:
1. First find the critical points inside the region,
2. Then express the boundaries as one variable function(s) and find the extrema for those.
3. Evaluate the function at the points found in steps 1 and 2; the largest value will be the absolute
maximum and the smallest value will be the absolute minimum.
(Example 6, ww 7, 8)
Chapter 11 Review problems recommended.
Concept check: 1,2, 5-18.
EXERCISES: 1, 2, 5, 7, 8, 11, 13, 15, 17, 23, 25, 27, 31, 32, 33, 35, 38, 40, 41, 42, 43, 44, 45, 47, 49, 52
Answers to even Exercises above
CHAPTER 12 : SECTIONS 12.1 - 12.3
12.1
· Understand the definition of the double integral as the limit of a double Riemann sum.
· Know how to divide a rectangular region into sub-rectangles, and how to calculate double Riemann sums given a formula for the function (Examples 1, 3, ww 1), given a contour map (hw 5) or given a table (hw 6).
· Be able to evaluate iterated double integrals (Example 4, ww 4-10).
· Understand the statement of Fubini's Theorem (Examples 5 and 6).
· Know the geometrical interpretation of the double integral of f (x, y) over a region R (signed volume of the solid between the graph of f and the region R) (Example 7, ww 2, 9).
· Understand that the integral of 1 over a region R gives the area of R.
· Know how to find the average value of a two variable function over a rectangular region (ww 10).
· Know how to find double integrals over type 1 regions (vertical slices) and over type 2 regions (horizontal slices). Remember that you cannot simply interchange the integral terms in these cases (Examples 1, 2, 3, 4, ww 1-3).
· If a region can be interpreted both as type 1 and type 2, then you can use this fact to change the order of integration. In cases like these, be able to rewrite the region as the other type and be able to change the limits of integration accordingly. Remember that outside limits MUST always be constant (Examples 2 , 3, 5, ww 5-11).
· Practice on finding the volume of given solids using double integrals (hw 22 and 24, ww 4).
· Know how to change a double integral into polar coordinates. Remember that you need to multiply the function by r (ww 1,2).
· Understand when it is convenient to use polar coordinates rather than rectangular: regions involving circles or sectors of circles, (Example 1, hw 1- 4, 26) volumes involving cones, paraboloids and spheres (Examples 2,3, hw 14, 16, 18, ww 3-7)
Chapter 12: Review Problems recommended:
Concept check (page 722): 1,2,3
EXERCISES (page 723): 1,2,3,5, 9, 10,13,14,15,17,18,19, 21, 22
Answers to even problems above.