n and C and D be positive constants so that nn 1
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1 Math Activity 0 (Due by end of class August 6). The graph of the equation y is called an astroid. a) Find the length of this curve. {Hint: One-fourth of the curve is given by the graph of y for 0.} b) Find the surface area of the solid of revolution generated by revolving this curve about the -ais.. a) Let n and C and D be positive constants so that nn n the curve y C n D from to. b) Let n and 6CD. Find the length of D in the previous formula to find a function whose length on Find as many functions f with a continuous derivative for, so that f length of the graph of f on, is given by there?, is and the d. How many such functions are
2 . Find all continuously differentiable functions f with f 0 0 so that the length of its graph over the interval 0, is always. 5. The graph of y f passes through the origin. The length of the curve from, f, 0, is t e dt. Find all such functions, f Suppose that it is known that the length of the curve y f lnsec a tan a for every 0a. If the curve y f such functions, f? B 0,0 to on the interval 0 a is passes through the origin, find all 7. a) Find the arclength of the curve y A on the interval ab,, where A and B are positive constants with AB and ab, 0. b) Let B, a, b in the formula on,. y A and determine the length of its graph B 8. a) Find the length of the curve ay b y ln b 8a b from y b to y b, for ab, 0. b) Let a in the previous formula to find a function of y whose length on, 9. Find the length of the curve y t dt from to 6. b b is 7.
3 0. Find the length of the curve ln sin y from to.. By finding an upper bound on the integrand of the arclength formula, find a decent upper bound on the length of the graph of the function f cos on the interval 0,.. By comparing the integrands of the arclength formulas, decide which curve has a longer length on the interval,, sin or?. Find the -coordinate of the point on the graph of two points on the graph: 0,0 and,. f that is halfway between the
4 . Find the surface area of the solid formed by revolving y Consider y and double the result: y y d. 0 y y on the interval 0, y about the -ais. 5. Two lanes of a running track are modeled by the semiellipses as shown. The equation for lane is y 00, and the equation for lane is y 50. The starting point for 5 5 lane is at the negative -intercept 500,0. The finish points for both lanes are the positive -intercepts. Where should the starting point be placed on lane so that the two lane lengths will be equal (running clockwise)? Start of lane Start of lane
5 6. Let P be a point on the parabola y other than the origin. The normal line to the parabola at P will intersect the parabola at another point Q. Find the coordinates of the point P so that the length of the parabola from P to Q is as small as possible. For a point P with -coordinate of, you can show that the -coordinate of Q is. You need to minimize L t dt for 0. L From the geometry of the problem, there should be a minimum at which the derivative is zero, so Q P. Multiplying by yields. Epanding inside the radical on the right 5 yields. Squaring both sides gives 5, or 6. Dividing by and epanding the square leads to 6. Epanding the right side leads to 6 6. See if you can solve it from here. 7. By comparing the integrands of the arclength formulas, decide which curve is longer: f, or g,? on on 8. a) Consider the curve y f, 0, f 0 a Suppose that the length of the curve from 0, a to, constant C. Find f and the permissible values of C. b) Is it possible that the length of the curve from 0, a to, n? Eplain., and f is continuously differentiable. f is given by C for some f is given by n, for
6 9. Suppose that f is a continuously differentiable function with f 0 0, and the length of the graph of f from 0,0 to, f for 0 is given by e f. Find all such possible functions, f. 0. Consider the function ;0 f ; ;, whose graph is given below. a) Show that f is continuous on 0,. b) Find the length of the graph of f on 0,. {Hint: From symmetry, find the length on 0, and quadruple it.} f n g, with n a positive integer, on the n interval 0, n, whose graph is given below. c) Consider the new function g, where n n Find the length of the graph of g on the interval 0, n. {Hint: Find the length on 0, n and quadruple it.}
7 d) Consider the new function h, where the graph of h is n adjacent copies of the graph of g on the interval 0,, whose graph is below. h has a continuous derivative as well. n n Find the length of the graph of h on the interval 0,. n e) Since lim h 0 for all in the interval n lim length of h. Show that this is not the case. n 0,, you might think that. Find the length of the graph of the function f on the interval 0,. {Hint: After you set up the integral, use the substitution tan.}. Let f be a function with a continuous derivative, and for 0 graph of f between the point be the length of the line segment joining the point a h, f a h. Find lim L h h0 g h {Hint: L h g h h, let a, f a and the point a h, f a h,. h f a h f a ah a f d simple fraction, and then divide top and bottom by h.} g h be the length of the. For 0 a f a and the point h, let Lh. Differentiate top and bottom, simplify into a
8 . Let C be the portion of the parabola y a that lies inside the unit circle Find the value of a that maimizes the arc length of C. y. a a a a {Hint: The arc length is given by a a 0 a d, so you can integrate and then differentiate with respect with to a.}. A projectile is launched from the ground with an initial speed of V at an angle with the horizontal. Assume that the -ais is the horizontal ground and y is the height above the ground. Neglecting air resistance and letting g be the acceleration of gravity, it can be shown that the trajectory of the projectile is given by y k y, where ma g V sin k and y. V cos ma g a) The high point of the trajectory occurs at ma a,0 and a,0, then find a. b) From symmetry, the length of the trajectory is given by 0, y. If the projectile is on the ground at a 0 k d. Evaluate this integral and epress the result in terms of V, g, and. c) For fied values of V and g, show that the launch angle that maimizes the length of the trajectory satisfies the equation sinln sec tan. d) Approimate the optimal launch angle in radians to four decimal places.
9 5. If f is a continuously differentiable decreasing function on the interval 0, with f 0 and f 0, then use geometry to find the best possible numbers l and m so that. l f d m 0 length of the curve
10 6. a) A dog spots a rabbit running in a straight line. Fortunately for the rabbit, the dog is specially trained to maintain a constant distance from the rabbit. Assuming that the dog always moves towards the rabbit, and the distance between them is always 0 feet, find an equation to describe the path of the dog. {Hint: 00 0 This leads to the differential equation dy 00 and initial condition d y 0 0 for the dog s path.} b) Now assume that an untrained dog starts at the point 9,0, the rabbit starts at 0,0 and has constant speed, and the dog moves toward the rabbit at twice the speed of the rabbit. Find an equation to describe the dog s path. {Hint: 0, rt y,
11 dy y rt d and 9 rt y u du dy Solving for t in each and equating leads to 9. y d y u du d y Differentiating leads to y d or dw dy w, where w with w 9 0 and y 9 0. d d } c) How far does the rabbit run in part b) before the dog catches it?
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