Reading Assignments

RA 13.2 (Due September 12, before 9:30am)

• Read Section 13.2.
• Answer the following questions:
  • What is the difference between a vector and a scalar? Give two examples of scalar quantities and two examples of vectorial quantities.
  • How do you add two vectors geometrically? How do you add them algebraically?
  • What is the difference between a "geometric vector" and an "algebraic vector"? How are the two concepts related?

RA 13.3 (Due September 12, before 9:30am)

• Read Section 13.3.
• Answer the following questions:
  • The dot product of two vectors is positive. What can you say about the two vectors?
  • Let a and b be nonzero vectors. are the scalar projection of b onto a and the vector projection of b onto a related?
  • Which of the following actions will result in doubling the dot product of two vectors:
    • double the length of both vectors
    • double the length of the first vector
    • double the length of the second vector
    • double the angle between the vectors?
    Justify your answer!

RA 13.4 (Due September 14, before 9:30am)

• Read Section 13.4.
• Answer the following questions:
  • Three vectors a, b, and c are given. Which of the following are scalars and which are vectors?
    • the dot product of a and c
    • the cross product of c and b
    • the cross product of a and (the cross product of b and c)
    • the dot product of c and (the cross product of b and a)
  • What happens if you apply the "right-hand rule" but, incorrectly, use your left hand?
  • Name two quantities that can be computed or expressed using cross products.

RA 13.5 (Due September 19, before 9:30am)

• Read Section 13.5.
• Answer the following questions:
  • How many points in $R^3$ determine a line? What happens if you have fewer/more than that number of points? If you know the coordinates of those points, how do you find the equation of the line?
  • How many points in $R^3$ determine a plane? What happens if you have fewer/more than that number of points? If you know the coordinates of those points, how do you find the equation of the plane?
  • How do you find the equation of a plane if you know a point it passes through and two directions parallel to the plane? Is there enough information?

RA 13.6 (Due September 21, before 9:30am)

• Read Section 13.6.
• Answer the following questions:
  • How do you recognize the equation of a quadric surface? Give an example of a quadric surface, and an example of a surface that is not a quadric surface.
  • What is a trace of a quadric surface, and what kind of curves can occur as traces of quadric surfaces?
  • List all types of quadric surfaces that are symmetric with repect to origin (see Table 1 on page 872). What do their equations have in common?

RA 13.7 (Due September 26, before 9:30am)

• Read Section 13.7.
• Answer the following questions:
  • What type of coordinates would a pilot flying a plane use to specify the position of the plane with respect to a fixed point on the ground? Why?
  • In Cartesian coordinates, the position of a point is given by a UNIQUE TRIPLE of numbers (distance ahead, distance to the left, distance above). What is the situation for cylindrical coordinates?
  • Describe the surfaces given in spherical coordinates by
    • $\rho$ = constant
    • $\theta$ = constant
    • $\phi$ = constant

RA 14.1 (Due September 28, before 9:30am)

• Read Section 14.1.
• Answer the following questions:
  • How many parameters does one need to describe a space curve?
  • How many parametrizations does a space curve have?
  • What do you think is harder: to represent a curve geometrically given its parametrization, or trically? Justify.

RA 14.2 (Due September 28, before 9:30am)

• Read Section 14.2.
• Answer the following questions:
  • How do you compute the derivative of a vector function? What does the derivative represent?
  • How do you compute a definite integral of a vector function? What is the result of such a computation?
  • What do the differentiation rules 3, 4, 5 in Theorem 3 have in common? What is the corresponding rule for scalar functions?

RA 15.1 (Due October 10, before 9:30am)

• Read Section 15.1
• Answer the following questions:
  • Give two examples of functions of several variables, different from the examples given in the text.
  • What is the graph of a function of two variables? In what space R^k do we represent it? In what space do we represent the graph of a function of three variables?
  • What is a contour line for a function of two variables? In what space R^k do we represent contour lines? How many contour lines are there? What is the analogue for a function of three variables?

RA 15.2 (Due October 10, before 9:30am)

• Read Section 15.2
• Answer the following questions:
  • True or false: If a function of two variables has a limit as the point (x,y) approaches (0,0), then the limit is unique.
  • For functions of one variable, there is a notion of "side-limit." Is there an analogue of that for functions of several variables?
  • Can a function be continuous at a point where it is not defined? Why, or why not?

RA 15.3 (Due October 12, before 9:30am)

• Read Section 15.3
• Answer the following questions:
  • How are partial derivatives related to derivatives of functions of one variable?
  • What is the geometric interpretation of the partial derivative of a function of two variables at a point P(a,b)?
  • Let $P$ be a polynomial function in x and y. Why is it true that the second partial derivatives $P_{xy}$ and $P_{yx}$ are equal?

RA 15.4 (Due October 12, before 9:30am)

• Read Section 15.4
• Answer the following questions:
  • What is the one-variable analogue of the tangent plane to the graph of a function at a given point?
  • Why is it helpful to know the linearization of a function at a point?
  • What is the relationship between the increment $\Delta z$ and the differential $dz$?

RA 15.5 (Due October 19, before 9:30am)

• Read Section 15.5.
• Answer the following questions:
  • What do the differentiation rules 3, 4, 5 in Theorem 3 have in common? What is the corresponding rule for scalar functions?
  • What is the the meaning of the Implicit Function Theorem?
  • How do you construct a tree diagram and how would you use it to aply the chain rule?

RA 15.6 (Due October 24, before 9:30am)

• Read Section 15.6.
• Answer the following questions:
  • Explain, in your own words, the meaning (NOT the formula!) of the directional derivative of a function at a point. In what direction is the directional derivative minimal?
  • What is the gradient of a function, and how is it related to level curves/surfaces?
  • How can you use the gradient vector at a point to find the equation of the tangent plane to an implicitly defined surface?

RA 15.7 (Due October 26, before 9:30am)

• Read Section 15.7.
• Answer the following questions:
  • How do you find the critical points of a function of two variables? What is the geometric argument supporting your method?
  • What points do you have to consider if you want to determine the minimum value of a function on the closed disk of radius 1 centered at the origin?
  • True or false: Using the second derivative test at a critical point, we can always determine whether the critical point corresponds to a local min/max or to a saddle point.

RA 15.8 (Due October 31, before 9:30am)

• Read Section 15.8.
• Answer the following questions:
  • For what kind of problems would you use the method of Lagrange multipliers? Give an example, different from the ones presented in the book.
  • What do you think is the hardest part in solving an optimization problem using Lagrange multipliers?
  • Some constrained optimization problems can be solved without using the method of Lagrange multipliers. How? When is it easier to use Lagrange multipliers?

RA 16.3 (Due November 14, before 9:30am)

• Read Section 16.3.
• Answer the following questions:
  • Consider the vowels A,E,I,O, and U. Imagine that they are written with a brush and that each line is 1 in wide (for example, O looks like a disk with a hole in it). Which of the five regions are plane regions of type I? Which are of type II?
  • When is it possible to change the order of integration, and why is it helpful sometimes to do so?
  • How can you reconstruct the region over which a double integral is computed by knowing the limits of integration of the iterated integrals?

RA 16.4 (Due November 16, before 9:30am)

• Read Section 16.4.
• Answer the following questions:
  • What is the geometric description (in rectangular coordinates) of a region that looks "rectangular" in polar coordinates? (That is, a region given by $r_1 < r < r_2$, $theta_1 < theta < theta_2$.) What is the area of such a region?
  • What kind of regions are suitable for changing to polar coordinates when double integrals?
  • The formula for computing double integrals using polar coordinates is similar to what rule/method for computing definite integrals of functions of one variable?

RA 16.5 (Due November 16, before 9:30am)

• Read Section 16.5.
• Answer the following questions:
  • Give three examples of quantities that are computed using double integrals.
  • What is the relationship between the units for the quantity expreesed by a function f and the units for the quantity expressed by the double integral of f over a region?
  • Suppose the joint density function of two random variables X and Y is constant and equal to 1/5 over a region that contains the rectangle [0,1] x [0,2]. What is the probability that X is between 0 and 1 and Y is between 0 and 2?

RA 16.7 (Due November 21, before 9:30am)

• Read Section 16.7.
• Answer the following questions:
  • What are the "ingredients" of a triple Riemann sum, and how are triple Riemann sums used to define a triple integral?
  • How do you construct an iterated integral from a triple integral? Is it always possible to express a triple integral as an iterated integral? Why, or whay not?
  • A triple integral is sometimes computed as a single integral followed by a double integral (as in formulas 6, 10, and 11). What is the geometric interpretation of this method? Is it possible to evaluate a triple integral as a double integral followed by a single integral? Give a geometric description.

RA 16.8 (Due November 28, before 9:30am)

• Read Section 16.8.
• Answer the following questions:
  • What is the most important factor when deciding whether to use rectangular, spherical, or cylindrical coordinates?
  • The term $dx dy dz$ is replaced by $\rho^2 \sin \phi d\rho d\theta d\phi$. What is the meaning of the term $\rho^2 \sin \phi$?
  • From rectangular to cylindrical coordinates we replace $dx dy dz$ by $r dr d\theta dz$. The factor $r$ is exactly the same as the change in polar coordinates. How do you explain that?

RA 16.9 (Due November 30, before 9:30am)

• Read Section 16.9.
• Answer the following questions:
  • Why would you use a change of variables to compute a double or a triple integral?
  • What are the three things that are changed when performing a change of variables for a double/triple integral?
  • What is the hardest thing to do when computing a double/triple integral through a change of variables?