MAT 480 Topology Problem Set 1 Due 2/1/2019 Spring 2019 1. Let a (x1, z2, 23) and y - (1, 2, va) be elements of R3. Define d(x,y) = maxlx1-vil, la 2-U21, las-val). Prove that d defines a metric on R3 2. Let u(u1, u2, ..., un) and - (v, V2, ..., Vn) be vectors in Rn. Prove the Cauchy-Schwarz Inequality lin 히 lizil . Ilull in six easy steps: (a) Explain why lu.卵く间2·间2 implies lu. 히 (b) Explain why, for any real z, 阅·间. (c) Rewrite the right-hand side of the inequality above as a quadratic polynomial az2 + baz + c. What are the values of the coefficients a, b, and c? (d) Let A-b2 4ac. Explain why A s 0. Hint: (b) implies 0 az2 ba + c; what do you know about the graph of such a quadratic polynomial'? (e) Use (d) to show lž.2I2. l12 (f) Conclude that the Cauchy-Schwarz Inequality is true. 3. Given a metric space (X, d), a point a E X, and some e> 0, we can define the "open" e-ball centered at as Similarly, the "closed" e-ball is defined as B(r, e)(v E 지 d(x, y) e). (a) Let de be the Euclidean metric on R2. Sketch a picture of B(0,0), 1) in (R2, dB) (b) Let dr be the Taxi-cab metric on IR2. Sketch a picture of B((0,0), 1) in (R2, dr). (c) True or false, with explanation: given B((0,0),1) in (R2, dE) and any point V E B((0,0), 1), there is always some 6> 0 so that (d) Do we have the same answer if we replace B(0, 0), 1) with 瓦(0,0), 1) in (c)? Explain.

4. Find the interpolating polynomial y = ax² + bx+c that passes through all three of the following data points: (1, 13), (2, 15), and (3, 19). You may use a computer or calculator to perform elimination in this problem only.

2 2) 2x-3y=-1 y = x - 1 4) -3x-3y = 3 y=-5x-17 6) y = 5x-7 34 -3x - 2y = -12 8) -7x-2y=-13 x-2y = 11 2x-3y- 6x-37-19

G Gmail Bb Pearson's MyLab & Mastering x Do Homework - L05 Homewor x AG 2021-2022 Aggie Gentz Book x Award History + A mathxl.com/Student/PlayerHomework.aspx?homeworkld=591061310&questionld=17&flushed=false&cld=6387569&back=https://www.mathxl.com/. I Apps Is Mesh Contrast Sh.. s Frill Neck Semi Sh.. S Sheer Mesh Long.. C Kinky Curly Wefte. R dress O money in marriage A Other Bookmarks Math132-003 (Calculus II - 2021S) Online - Clemence Makala Langley & | 02/18/21 2:50 PM Homework: L05 Homework: Trig Substitution Save Score: 0 of 1 pt 25 of 27 (23 complete) v HW Score: 79.63%, 21.5 of 27 pts 8.4.68-Setup & Solve Question Help ▼ Evaluate the following integral. dx V(x - 1)(7 - x) Rewrite the radicand by completing the square and simplifying. -S dx dx V(x- 1)(7- x) Enter your answer in the answer box and then click Check Answer. parts remainina Check Answer Clear All javascript:doExercise(25);