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Excercises 1-7

Excercises 1-7 - MIT OpenCourseWare http/ocw.mit.edu 18.01...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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18.01 EXERCISES SUnit .D iffej i tiaiti oi•7-. ...... .................... ............... 1A. Graphing 1A-1 By completing the square, use translation and change of scale to sketch a) y= z -' 2 - 1 b) y = 3X 2 + 6z + 2 1A-2 Sketch, using translation and change of scale a)y = 1+ I + 21 b) y = -- (Z - 1)2 1A-3 Identify each of the following as even, odd , or neither z 3 + 3z a) + b) sinz tan z c) tan d) (1 + z) 4 1 + X 2 e) Jo(z ), where Jo(z) is a function you never heard of 1A-4 a) Show that every polynomial is the sum of an even and an odd function. b) Generalize part (a) to an arbitrary function f(z) by writing f(s) + f (-) f( C) + f(-z) A ) 2 2 Verify this equation, and then show that the two functions on the right are respectively even and odd. c) How would you write as the sum of an even and an odd function? z+a 1A-5. Find the inverse to each of the following , and sketch both f(z) and the inverse function g(z). Restrict the domain if necessary. (Write y = f(z) and solve for y; then interchange x and y.) a) 2-1 ) b)z 2 + 2 2T +3 1A-6 Express in the form A sin (x + c) a) sinx + Vcos x b) sinx - cos 1A-7 Find the period , amplitude., and phase angle,-and use these to sketch a) 3 sin (2 - 7r) b) -4cos (x + r/2) 1A-8 Suppose f(z) is odd and periodic. Show that the graph of f(z) crosses the x-axis infinitely often. @Copyright David Jerison and MIT 1996, 2003
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E. 18.01 EXERCISES 1A-9 a) Graph the function f that consist of straight line segments joining the points (-1, -1), (1, 2), (3, -1), and (5,2). Such a function is called piecewise linear. b) Extend the graph of f periodically. What is its period? c) Graph the function g(r) = 3f((z/2) - 1) - 3. 1B. Velocity and rates of change 1B-1 A test tube is knocked off a tower at the top of the Green building. (For the purposes of this experiment the tower is 400 feet above the ground, and all the air in the vicinity of the Green building was evacuated, so as to eliminate wind resistance.) The test tube drops 16t 2 feet in t seconds. Calculate a) the average speed in the first two seconds of the fall b) the average speed in the last two seconds of the fall c) the instantaneous speed at landing 1B-2 A tennis ball bounces so that its initial speed straight upwards is b feet per second. Its height s in feet at time t seconds is.given by a = bt - 16t a) Find the velocity v = ds/dt at time t. b) Find the time at which the height of the ball is at its maximum height. c) Find the maximum height. d) Make a graph of v and directly below it a graph of a as a function of time. Be sure to mark the maximum of s and the beginning and end of the bounce. e) Suppose that when the ball bounces a second time it rises to half the height of the first bounce. Make a graph of s and of v of both bounces, labelling the important points.
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