assignment__12 - AP CALULUS BC ASSIGNMENT#12 L 2016-17 DUE...

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Unformatted text preview: AP CALULUS BC ASSIGNMENT #12 - L 2016-17 DUE: Friday, March 31 TEACHER: Mrs. Dopkin You may work in groups but all members must hand in his/ her own paper. Be certain that all members of your group understand the solutions to each question because a member of your group will be required to present one of the problems. To expedite the correcting of this project, each problem needs to be completed neatly on a separate page. This might mean recopying your work to make it a neater presentation. SHOW ALL WORK. Indicate clearly the methods you use because you will be graded on the correctness of your methods as well as the accuracy of your final answers. Please clearly indicate the final answer with a box or circle around it. It is not advised that you wait until the last night to begin this. It is designed to take a good amount of time. SECTION I: A graphing calculator is required for some problems or parts of problems on this section. If you choose to use decimal approximations, your answer should be correct to three decimal places. ‘. 1. Let function f be continuous and decreasing, with values as shown in the table: - (3) Use the trapezoid method to estimate the area between f and the x—axis on the - t L interval 2.5 S x S 5.0. (b) Find the average rate of change of f on the interval 2.5 S x S 5.0. w .1mmPMruww<wwe“g<WMmmswwm_ e_ _ were“:..l...m,.q..._«.:,l..,e...n»meme“. r3 .e._..4x_.._ .. ,..—..—..—.-.e;.;-g (0) Estimate the instantaneous rate of change of f at x = 2.5. '(d) If g(x) = f fix), estimate the slope of g at x = 4. 2. An object starts at point (1,3) and metres along the parabola y = x2 + 2 for 0 S t S 2, with the horizontal component of its velocity given by 3:? = 2 4 4 . t t + (a) Find the object’s position at t = 2. (b) Find the object’s speed at t: 2. . (c) Find the distance the object traveled during this interval. SECTION II: A calculator may NOT be used on this section. —1 " ! I i .1 3. Given a function f such that fl3) = 1 and f{")(3) = (5%??? ' n (a) Write the first four nonzero terms and the general term of the Taylor series for f around it = 3. Ch) Find the radius of convergence of the Taylor series. (c) Show that the third—degree Taylor polynomial approximates fl4) to within 0 .01. 4. - The curve y = 8 sin (%) divides a first quadrant rectangleginto regions A and B, as shown in the figure. (a) RegionA is the base of a solid. Cross sections of this solid perpendicular to the x-axis are rectangles. The height of each rectangle is 5 times the length of its base in region A. Find the volume of this solid. . (b) The other region, B, is rotated around the y—axis to form a dififereut solid. Set up but do not evaluate an integral for the volume of this solid. - (i2) 5. Abungee jumper has reached a point in her exciting plunge where the taut cord is 100 feet long with a 1/2—inch radius, and stretching. She is still 80 feet above the ground and is now falling at 40 feet per second. You are observing her jump from a spot on the ground 60 feet from the potential point of impact, as shown in the diagram above. _ (a) Assuming the cord to be a cylinder with volume remaining constant as the cord stretches, at what rate is its radius changing when the radius is 1/2”? (b) From your observation point, at what rate is the angle of elevation to the jumper changing when the radius is 11’2”? 6. The figure aboVe shows the graph off, whose domain is the closed interval [—2,-6}. Let F(x) = I! f(:) dt . (a) Find HQ) and F(6). (b) For what value(s) of x does F (x) 0? (c) 011 what interva1(s) 1s F increasing? (d) Find the maximum value and the minimum value of F . (e) At What value(s) of 15 does the graph of F have points of inflection? Justify your answer. _ ...
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