CHALLENGE PROBLEMS ■ CHALLENGE PROBLEMS: CHAPTER A Click here for answers Click here for solutions S ; Three mathematics students have ordered a 14-inch pizza Instead of slicing it in the traditional way, they decide to slice it by parallel cuts, as shown in the figure Being mathematics majors, they are able to determine where to slice so that each gets the same amount of pizza Where are the cuts made? dx x7 Ϫ x The straightforward approach would be to start with partial fractions, but that would be brutal Try a substitution Evaluate y Ϫ x7 Ϫ s Evaluate y (s Ϫ x ) dx 14 in FIGURE FOR PROBLEM A man initially standing at the point O walks along a pier pulling a rowboat by a rope of length L The man keeps the rope straight and taut The path followed by the boat is a curve called a tractrix and it has the property that the rope is always tangent to the curve (see the figure) (a) Show that if the path followed by the boat is the graph of the function y f ͑x͒, then y pier f Ј͑x͒ L (x, y ) (b) Determine the function y f ͑x͒ (L, 0) O dy ϪsL Ϫ x dx x x A function f is defined by f ͑x͒ FIGURE FOR PROBLEM y cos t cos͑x Ϫ t͒ dt ഛ x ഛ 2 Find the minimum value of f If n is a positive integer, prove that y ͑ln x͒n dx ͑Ϫ1͒n n! Show that y ͑1 Ϫ x ͒n dx 2n ͑n!͒2 ͑2n ϩ 1͒! Hint: Start by showing that if In denotes the integral, then Ikϩ1 2k ϩ Ik 2k ϩ ; Suppose that f is a positive function such that f Ј is continuous (a) How is the graph of y f ͑x͒ sin nx related to the graph of y f ͑x͒? What happens as n l ϱ? (b) Make a guess as to the value of the limit lim Thomson Brooks-Cole copyright 2007 nlϱ y f ͑x͒ sin nx dx based on graphs of the integrand (c) Using integration by parts, confirm the guess that you made in part (b) [Use the fact that, since f Ј is continuous, there is a constant M such that f Ј͑x͒ ഛ M for ഛ x ഛ 1.] If Ͻ a Ͻ b, find lim tl0 ͭy ͮ Խ 1͞t ͓bx ϩ a͑1 Ϫ x͔͒ t dx Խ ■ CHALLENGE PROBLEMS tϩ1 x ; 10 Graph f ͑x͒ sin͑e ͒ and use the graph to estimate the value of t such that xt f ͑x͒ dx is a maxi- mum Then find the exact value of t that maximizes this integral y Խ Խ 11 The circle with radius shown in the figure touches the curve y 2x twice Find the area of the region that lies between the two curves 12 A rocket is fired straight up, burning fuel at the constant rate of b kilograms per second Let v v͑t͒ be the velocity of the rocket at time t and suppose that the velocity u of the exhaust gas is constant Let M M͑t͒ be the mass of the rocket at time t and note that M decreases as the fuel burns If we neglect air resistance, it follows from Newton’s Second Law that y=| 2x | FIGURE FOR PROBLEM 11 FM x dv Ϫ ub dt where the force F ϪMt Thus M dv Ϫ ub ϪMt dt Let M1 be the mass of the rocket without fuel, M2 the initial mass of the fuel, and M0 M1 ϩ M2 Then, until the fuel runs out at time t M2 b, the mass is M M0 Ϫ bt (a) Substitute M M0 Ϫ bt into Equation and solve the resulting equation for v Use the initial condition v ͑0͒ to evaluate the constant (b) Determine the velocity of the rocket at time t M2 ͞b This is called the burnout velocity (c) Determine the height of the rocket y y͑t͒ at the burnout time (d) Find the height of the rocket at any time t 13 Use integration by parts to show that, for all x Ͼ 0, 0Ͻy ϱ sin t dt Ͻ ln͑1 ϩ x ϩ t͒ ln͑1 ϩ x͒ ; 14 The Chebyshev polynomials Tn are defined by Tn͑x͒ cos͑n arccos x͒ n 0, 1, 2, 3, (a) What are the domain and range of these functions? (b) We know that T0͑x͒ and T1͑x͒ x Express T2 explicitly as a quadratic polynomial and T3 as a cubic polynomial (c) Show that, for n ജ 1, Tnϩ1͑x͒ 2x Tn͑x͒ Ϫ TnϪ1͑x͒ (d) (e) (f) (g) (h) (i) Use part (c) to show that Tn is a polynomial of degree n Use parts (b) and (c) to express T4 , T5 , T6 , and T7 explicitly as polynomials What are the zeros of Tn ? At what numbers does Tn have local maximum and minimum values? Graph T2 , T3 , T4 , and T5 on a common screen Graph T5 , T6 , and T7 on a common screen Based on your observations from parts (g) and (h), how are the zeros of Tn related to the zeros of Tnϩ1 ? What about the x-coordinates of the maximum and minimum values? Thomson Brooks-Cole copyright 2007 (j) Based on your graphs in parts (g) and (h), what can you say about xϪ1 Tn͑x͒ dx when n is odd and when n is even? (k) Use the substitution u arccos x to evaluate the integral in part (j) (l) The family of functions f ͑x͒ cos͑c arccos x͒ are defined even when c is not an integer (but then f is not a polynomial) Describe how the graph of f changes as c increases CHALLENGE PROBLEMS ANSWERS Thomson Brooks-Cole copyright 2007 S Solutions About 1.85 inches from the center f ͑͒ Ϫ͞2 ͑b baϪa ͒1͑͞bϪa͒eϪ1 11 Ϫ sinϪ1 (2͞s5 ) ■ ■ CHALLENGE PROBLEMS SOLUTIONS E Exercises By symmetry, the problem can be reduced to finding the line x = c such that the shaded area is one-third of the area √ c√ of the quarter-circle The equation of the circle is y = 49 − x2 , so we require that 49 − x2 dx = 31 · 41 π(7)2 x ⇔ √ 49 − x2 + 49 c sin−1 (x/7) = 49 π 12 c [by Formula 30] ⇔ √ 49 − c2 + 49 sin−1 (c/7) = 49 π 12 This equation would be difficult to solve exactly, so we plot the left-hand side as a function of c, and find that the equation holds for c ≈ 1.85 So the cuts should be made at distances of about 1.85 inches from the center of the pizza The given integral represents the difference of the shaded areas, which appears to be It can be calculated by integrating with respect to either x or y, so we √ find x in terms of y for each curve: y = − x7 ⇒ x = − y and √ y = − x3 ⇒ x = − y7 , so − y7 − √ √ − x3 − − x7 dx But this 1 − y dy = √ √ − x7 − − x3 dx = equation is of the form z = −z So Recall that cos A cos B = 12 [cos(A + B) + cos(A − B)] So π f (x) = = = π cos t cos(x − t) dt = t cos x + cos x + sin(2t − x) sin(−x) − π π [cos(t + x − t) + cos(t − x + t)] dt = = π cos x + sin(−x) = π sin(2π − x) − π [cos x + cos(2t − x)] dt sin(−x) cos x The minimum of cos x on this domain is −1, so the minimum value of f (x) is f (π) = − π2 In accordance with the hint, we let Ik = integrate Ik+1 by parts with u = − x2 − x2 k+1 k dx, and we find an expression for Ik+1 in terms of Ik We ⇒ du = (k + 1) − x2 k (−2x), dv = dx ⇒ v = x, and Thomson Brooks-Cole copyright 2007 then split the remaining integral into identifiable quantities: Ik+1 = x(1 − x2 )k+1 = (2k + 2) 1 + 2(k + 1) − x2 k x2(1 − x2 )k dx − − x2 dx = (2k + 2)(Ik − Ik+1 ) CHALLENGE PROBLEMS ⇒ Ik+1 = So Ik+1 [1 + (2k + 2)] = (2k + 2)Ik I0 = = 2k + Ik Now to complete the proof, we use induction: 2k + 20 (0!)2 , so the formula holds for n = Now suppose it holds for n = k Then 1! Ik+1 = = 2(k + 1)22k (k!)2 2(k + 1) 2(k + 1)22k (k!)2 2k + 22k (k!)2 2k + = Ik = = · 2k + 2k + (2k + 1)! (2k + 3)(2k + 1)! 2k + (2k + 3)(2k + 1)! [2(k + 1)]2 22k (k!)2 22(k+1) [(k + 1)!]2 = (2k + 3)(2k + 2)(2k + 1)! [2(k + 1) + 1]! So by induction, the formula holds for all integers n ≥ < a < b Now [bx + a(1 − x)]t dx = Now let y = lim t→0 b ut a (b−a) bt+1 − at+1 (t + 1)(b − a) du [put u = bx + a(1 − x)] = 1/t Then ln y = lim t→0 ut+1 (t + 1)(b − a) b = a bt+1 − at+1 (t + 1)(b − a) bt+1 − at+1 ln This limit is of the form 0/0, t (t + 1)(b − a) so we can apply l’Hospital’s Rule to get bt+1 ln b − at+1 ln a b ln b − a ln a b ln b a ln a bb/(b−a) = − = − − ln e = ln − bt+1 − at+1 t+1 b−a b−a b−a eaa/(b−a) ln y = lim t→0 Therefore, y = e−1 bb aa 1/(b−a) An equation of the circle with center (0, c) and radius is 11 x2 + (y − c)2 = 12 , so an equation of the lower semicircle is √ y = c − − x2 At the points of tangency, the slopes of the line and semicircle must be equal For x ≥ 0, we must have √ x y0 = ⇒ √ = ⇒ x = − x2 ⇒ 1−x x2 = 4(1 − x2 ) ⇒ 5x2 = ⇒ x2 = ⇒ x= √ √ √ = 45 The slope of the perpendicular line segment is − 21 , so an equation of the line √ √ √ √ √ √ segment is y − 54 = − 21 x − 52 ⇔ y = − 21 x + 15 + 54 ⇔ y = − 21 x + 5, so c = and √ √ an equation of the lower semicircle is y = − − x2 Thus, the shaded area is and so y = 2 √ (2/5) Thomson Brooks-Cole copyright 2007 √ (2/5) √ √ √ x√ 30 − − x2 − 2x dx = x − − x2 − sin−1 x − x2 2 √ − − 2(0) · √ − sin−1 √ =2 2− 5 5 2 = − sin−1 √ = − sin−1 √ 5 ■ ■ CHALLENGE PROBLEMS 13 We integrate by parts with u = −1 , dv = sin t dt, so du = and ln(1 + x + t) (1 + x + t)[ln(1 + x + t)]2 v = − cos t The integral becomes ∞ I= = lim b→∞ ⎛ − cos t sin t dt = lim ⎝ ln(1 + x + t) b→∞ ln(1 + x + t) − cos b + + ln(1 + x + b) ln(1 + x) ∞ where J = ∞ Thomson Brooks-Cole copyright 2007 − ⎞ cos t dt ⎠ (1 + x + t)[ln(1 + x + t)]2 − cos t dt +J = ln(1 + x) (1 + x + t)[ln(1 + x + t)]2 − cos t dt Now −1 ≤ − cos t ≤ for all t; in fact, the inequality is strict (1 + x + t)[ln(1 + x + t)]2 except at isolated points So − − b b 1