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10 Problems in all
Section 1.0#8
8. The gure below shows the composite develop- mental skill level of chessmasters at different ages as determined by their performance against other chessmasters. (From \"Rating Systems for Human Abilities,\" by W.H. Batchelder and RS. Simpson, 1988. UMAP Module 698.) (a) At what age is the \"typical" chessmaster play- ing the best chess? (b) At approximately what age is the chessmas ter's skill level increasing most rapidly? (c) Describe the development of the "typical\" chessmaster's skill in words. (d) Sketch graphs that you think would reason ably describe the performance levels versus age for an athlete, a classical pianist, a rock singer, a mathematician and a professional in your major field. 2000 oonposilc dcvclopmcnlal skill lcvcl 'n chess skill level 20 30 40 SD 60 age (years) In problems 1417, lim f(x) = L and the function 1H! f and a value for e are given graphically. Find a length for 5 that satises the limit definition for the given function and value of e. \f12. Find the one and twosided limits of x) as x > 0, 1 and 2, if f(x) is dened as: x ifx0 h Y In 1718, describe the behavior at each integer of the function y = f (x) in the gure provided ,using one of these phrases: \f20. Show that liIIbsin(31c) does not exist. (Sugges- 1} tion: Let f(x) = sin(%) and let an = i so T1131\" that an) = sin(%) = sin(mt) = 0 for every n. Thenpickbn '1 sothatf(bn)=sin(%)= _ ZmHg sin(2mr + g) = sin(%) = 1 for every n.) 8. Two students claim that they both started with the points x = 1 and x = 9 and applied the Bisection Algorithm to the function graphed below. The rst student says that the algorithm converged to the root near as = 8, but the second claims that the algorithm will converge to the root near 3: = 4. Who is correct? 22. (a) (b) A square sheet of paper has a straight line drawn on it from the lower left corner to the upper right corner. Is it possible for you to start on the left edge of the sheet and draw a \"connected" line to the right edge that does not cross the diagonal line? Prove: If f is continuous on the interval [0, l] and 0 3 f(x) 3 1 for all x, then there is a number c with 0 3 c g 1 such that f(c) = c. (The number (3 is called a \"fixed point\" of f because the image of c is the same as c: f does not \"move\" c.) Hint: Dene a new function g(x) = f(x) x and start by considering the values g(0) and g0). (C) What does part (b) have to do with part (a)? (d) Is the theorem in part (b) true if we replace the closed interval [0, 1] with the open interval (0, 1)? 12. You need to cut four pieces of wire (exactly the same length after the cut) and form them into a square. If the area of the square must be within 0.06 inches of 100 inches, then each piece of wire must be within how many inches of 10? MATH 140 Quiz 2 Professor: Dr. C. Libis Section 1.0 #8 Section 1.1 #12 Section 1.1 #16 Section 1.2 #12 Section 1.2 #18 Section 1.2 #20 Section 1.3 #8 Section 1.3 #22 Section 1.4 #12 Section 1.4 #16
