PROBLEM SET # PROBLEMS SOLUTIONS 1 Due in Ses #5 1. Prove Thms I.6 and I.11 on p. 18 2. Do exercises 5 and 6 on p. 36 3. Prove by induction: (a+b)^n = \sum_{k=0}^n C_n^k a^k b^(n-k), where C_n^k = (n!)/{(k!)(n-k)!} Solution Set 1(PDF) 2 Due in Ses #8 1. Do Ex. 6, p. 28 2. Do Ex. 7, p. 64 3. Prove that the integral in Ex. 11, pp. 71 is independent of the partition, and do parts a, b, c of the exercise. Solution Set 2(PDF) 3 Due in Ses #11 1. Ex. 22b, p. 83 2. Ex. 16, p. 94 3. Ex. 10, p. 114 Solution Set 3(PDF) 4 Due in Ses #17 1. Ex. 6, p. 155 2. Ex. 5, p. 149 3. Let f(x) be defined for all nonnegative x, and suppose that it is continuous, strictly increasing and bounded on its domain. Let M be the supremum of the values of f(x), x nonnegative. (a) Show that f(x) takes on every value between f(0) and M, but does not take on the value M (b) Show that f(x) is uniformly continuous for all nonnegative x. Solution Set 4(PDF) 5 Due in Ses #19 1. Derive the formula for the derivative of f(x)=x^{1/3} (third power root of x), for nonnegative x, directly from the definition. 2. Differentiate f(x) = ((tan^2(x) -1)(tan^4(x) +10tan^2(x) +1))/(3 tan^3(x)), assuming 0< x < 90 degrees. 3. Sketch the graph of f(x)=(x^4 - 3)/x. Find critical points, zeros, asymptotes, intervals of monotonicity, convexity, and points of inflection. Solution Set 5 (PDF) 6 Due in Ses #22 1. Ex. 17 on p. 208 2. Show that for any nonzero number k and any numbers a and b, there is at most one function f(x) defined for all real numbers and satisfying the conditions: (a) f''(x) = -k^2 f(x) for all x (b) f(0) =a, f'(0)=b (Hint: If there are two such functions f(x) and g(x), consider u(x) = f(x/k) - g(x/k) and v(x) = u'(x), and show that u(x)=0, v(x)=0). Guess the unique function that satisfies the conditions. 3. Ex. 18, 19 on p. 216 Solution Set 6(PDF) 7 Due in Ses #25 1. Ex. 30, p. 224 - derive the formula 2. Ex. 27 and 30, p. 249 3. Ex. 40 on p. 258 (suggestion: trig. substitution and by parts) and find the primitive of f(x) = 1/(x sqrt(x^2 +3)) (suggestion: trig. substitution). Solution Set 7(PDF) 8 Due in Ses #28 Problem Set 8 (PDF) Solution Set 8 (PDF) 9 Due in Ses #34 1. Ex. 15, p. 399 2. Ex. 3, 12 on p. 402 3. Ex. 11, 22 on p. 409 Solution Set 9 (PDF) 10 Due in Ses #38 1. (a) Between the curves y=1/x^3 and y=1/x^2 and to the right of x=1 are constructed infinitely many segments parallel to the y-axis at an equal distance from each other. Will the sum of the lengths of these segments be finite? (b) The same question as in (a) with the curve y= 1/x^2 replaced by the curve y= 1/x. 2. Ex. 8, 9 on p. 415 3. Ex. 14 on p. 420 4. Ex. 12 on p. 430 5. Ex. 5 on p. 438 Solution Set 10(PDF)