Review for Exam 3
Sections 14.7, 15.1-15.5, 15.7.
50 minutes.
5, 6, problems, similar to homework problems.
No calculators, no notes, no books, no phones.
No green book needed.
Triple integral in cylindrical coordinates (Sect. 15.7)
Example
Use cylindrica
Tangent planes and linear approximations (Sect. 14.6)
Review: Dierentiable functions of two variables.
The tangent plane to the graph of a function.
The linear approximation of a dierentiable function.
Bounds for the error of a linear approximation.
The d
Review for Exam 2
Tuesday Recitations: Sections 13.1-13.3. 14.1-14.6.
Thursday Recitations: Sections 13.1-13.3. 14.1-14.7.
50 minutes.
From ve 10-minute problems to ten 5-minutes problems.
Problems similar to homework problems.
No calculators, no notes, n
Local and absolute extrema, saddle points (Sect. 14.7)
Review: Local extrema for functions of one variable.
Denition of local extrema.
Characterization of local extrema.
First derivative test.
Second derivative test.
Absolute extrema of a function in a do
Double integrals (Sect. 15.1)
Review: Integral of a single variable function.
Double integral on rectangles.
Fubini Theorem on rectangular domains.
Examples.
Review: Integral of a single variable function
Denition
The denite integral of a function f : [a,
Integrals in cylindrical, spherical coordinates (Sect. 15.7)
Integration in cylindrical coordinates.
Review: Polar coordinates in a plane.
Cylindrical coordinates in space.
Triple integral in cylindrical coordinates.
Next class:
Integration in spherical c
Double integrals on regions (Sect. 15.2)
Review: Fubinis Theorem on rectangular domains.
Fubinis Theorem on non-rectangular domains.
Type I: Domain functions y (x).
Type II: Domain functions x(y ).
Finding the limits of integration.
Review: Fubinis Theore
Areas and double integrals. (Sect. 15.3)
Areas of a region on a plane.
Average value of a function.
More examples of double integrals.
Areas of a region on a plane
Denition
The area of a closed, bounded region R on a plane is given by
A=
dx dy .
R
Remark:
Double integrals in polar coordinates (Sect. 15.4)
Review: Polar coordinates.
Double integrals in disk sections.
Double integrals in arbitrary regions.
Changing Cartesian integrals into polar integrals.
Computing volumes using double integrals.
Review: Po
Review for Exam 2
Sections 13.1-13.3. 14.1-14.6.
50 minutes.
5 problems, similar to homework problems.
No calculators, no notes, no books, no phones.
No green book needed.
Section 14.6
Example
(a) Find the linear approximation L(x, y ) of the function
f (
Triple integrals in Cartesian coordinates (Sect. 15.5)
Triple integrals in rectangular boxes.
Triple integrals in arbitrary domains.
Volume on a region in space.
Triple integrals in rectangular boxes
Denition
The triple integral of a function f : R R3 R i
Triple integrals in Cartesian coordinates (Sect. 15.5)
Review: Triple integrals in arbitrary domains.
Examples: Changing the order of integration.
The average value of a function in a region in space.
Triple integrals in arbitrary domains.
Review: Triple
Directional derivatives and gradient vectors (Sect. 14.5)
Directional derivative of functions of two variables.
Partial derivatives and directional derivatives.
Directional derivative of functions of three variables.
The gradient vector and directional de