Math 221: Honors Calculus III

Assignment 2 due August 26:
Problems 1-4, 1-6, 1-7, 1-9 Solutions and Grader's comments

Assignment 3 due August 29:
Problems 1-10, 1-14, and:
Show there do not exist nonzero vectors x,y,z in R² which are pairwise orthogonal,
i.e., x is orthogonal to y, y is orthogonal to z, and z is orthogonal to x.
Solutions and Grader's comments

Assignment 4 due August 31:
Problems 1-15,1-17,1-20,1-21,1-22 Solutions

Assignment 5 due September 2:
Problems 1-23, 1-24, 1-29, 1-31
Show that any plane P is convex, i.e., for any distinct points v and w on P, the segment [v,w] lies completely in P.
Solutions and Grader's comments

Assignment 6 due September 7:
Problems 1-33, 1-35, 1-36, 1-37, 1-38
Solutions

Assignment 7 due September 9:
Problems 2-1, 2-4, 2-5, 2-7
Let f and g be continuous functions from R to R^m. Show that f+g is also continuous.
Show that the function from R to R³ given by f(t)=(t,t²,t³) is continuous at t=0.
Note:For problems like 2.4, feel free to use the differentiation formulas you learned in Calculus I.
Solutions

Assignment 8 due September 12:
Problems 2-8, 2-9, 2-15, 2-16 Grader's comments

Assignment 9 due September 14:
Problems 2-22, 2-23, 2-30, 2-32, 2-36 Grader's comments and Solutions

Plot of f(x,y)=x²y/(x⁴+y²) view 1, alternate view 2, alternate view 3,

Assignment 10 due September 16:
Problem 2-40, and
Let L be a linear function from Rⁿ to R. Show that for any vectors x and y and any scalar b we have:
    L(x+y)=L(x)+L(y)     L(bx)=bL(x)

Assignment 11 due September 19:
Problems 2-42, 2-43, 2-44, 2-45
2-43 will be graded as Extra Credit Grader's comments

Assignment 12 due September 21:
Problems 2-47,2-50 Solutions

September 23 and 26 classes are cancelled
Assignment 13 due September 28:
Problems 2-55, 2-57, 2-60, 2-62, 2-64

Assignment 14 due September 30:
Problems 2-66, 2-68, 2-73
Give an example of a nonzero 2 x 2 matrix A such that A²=0.
Grader's comments

Assignment 15 due October 6:
Problems 2-78, 2-80, 2-81, 2-82, 2-85
Sorry for posting this late; I'll accept this anytime before noon on Thursday.
Solutions

Assignment 16 due October 12:
Problems 2-87, 2-88, 2-90, 2-95, 2-96, 2-99, 2-103 Grader's comments

Assignment 17 due October 14:
Problems 3-3, 3-9, 3-12 Solutions

Putnam exam information

Assignment 18 due October 17:
Problems 3-13, 3-14, 3-15 Grader's comments

Assignment 19 due October 19:
Problems 3-16, 3-17 Solutions

Assignment 20 due October 21:
Problems 3-18-b, 3-19 Grader's comments

Assignment 21 due October 24:
Problems 3-20, 3-24, 3-27, 3-28

Assignment 22 due October 26:
Problems 3-29, 3-30, 3-32 Grader's comments

Assignment 23 due October 28:
Problems 3-35, 3-36

Assignment 24 due October 31:
Problems 3-38, 3-39, 3-40, 3-42 Grader's comments

Assignment 25 due November 2:
Problems 3-43, 3-46, 3-55, 3-56

Assignment 26 due November 4:
Problems 3-47, 3-49 Grader's comments

Assignment 27 due November 7:
Find a linearly independent set of four vectors in R⁴ containing (1,1,0,0), (0,1,1,0), (0,0,1,1).
Problems 4-2, 4-3, 4-5, 4-6 Grader's comments

Assignment 28 due November 9:
Problems 4-8, 4-9, 4-10

Assignment 29 due November 11:
Problems 4-14, 4-18, 4-20cdefgh
Note that O(n) designates the n x n orthogonal matrices

Assignment 30 due November 14:
Problems 4-23, 4-24, 4-25, 4-26 Grader's comments

Assignment 31 due November 16:
Problems 4-31, 4-32, 4-33, 4-34

Assignment 32 due November 18: Grader's comments
1)Let f and g be real-valued functions which are C² in a neighborhood of 0 in Rⁿ. Suppose that 0 is a critical point of both f and g.
a)Show that 0 is a critical point of the product fg.
b)Suppose f and g are nondegenerate at 0. Give an example to show that fg might be degenerate.
c)Consider the Hessians H(f) and H(g) at 0. Suppose that H(f) is positive definite, H(g) is positive semidefinite, and f(0) and g(0) are positive. Show that fg has a strict local minimum at 0.
2)Show there is no real-valued function f which is C¹ in a neighborhood of the origin and satisfies x²+f(x)⁵=0 for all x. Show there is such a function satisfying x²+f(x)⁵=1 for all x in a neighborhood of the origin.

Assignment 33 due November 21:
Problems 5-1, 5-6, 5-7, 5-9, 5-13

Assignment 34 due November 23:
Problems 5-15, 5-17, 5-20, 5-24, 5-27 Grader's comments

Assignment 35 due November 28:
Problems 5-31, 5-32 Grader's comments

Assignment 36 due November 30:
5-33, 5-34 Grader's comments

Assignment 37 due December 2:
6-3, 6-6

Research experience for undergraduates information
National Science Foundation, American Mathematical Society, and Rice VIGRE program