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MIT 18 01 - Graphing

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable CalculusFall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.18.01 EXERCISES SUnit .Diffej 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= 3X2+ 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 z3 + 3za) + b)sinz tan z c) tan d) (1+ z)4 1 + X2 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)z2 + 2 2T +3 1A-6 Express in the form Asin (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, 2003E. 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. (You will have to decide how long the second bounce lasts and the initial velocity at the start of the bounce.) f) If the ball continues to bounce, how long does it take before it stops? 1C. Slope and derivative 1C-1 a) Use the difference quotient definition of derivative to calculate the rate of change of the area of a disk with respect to its radius. (Your answer should be the circumference of the disk.) b) Use the difference quotient definition of derivative to calculate the rate of change of the volume of a ball with respect to the radius. (Your answer should be the surface area of the ball.) 10-2 Let f(z) = (z -a)g(z). Use the definition of the derivative to calculate that f'(a) = g(a), assuming that g is continuous. 1C-3 Calculate the derivative of each of these functions directly from the definition. a) f(x) = 1/(2z + 1) b)f(z) = 2W2 +5x + 4 c) f 2) = 1/(z2 + 1) d) f() = 1/V Se) For part (a) and (b) find points where the slope is +1, -1, 0.1. DIFFERENTIATION 1C-4 Write an equation for the tangent line for the following functions a) f(zx) = 1/(2z + 1) at ,z= 1 b)f(z) = 2X2 + 5x + 4 at x = a c) f () = 1/(z2 + 1). at x = 0 d) f(x) = 1/V at z = a 1C-5 Find all tangent lines through the origin to the graph of y = 1 + (z -1)2. 1C-68 Graph the derivative of the following functions directly below the graph of the function. It is very helpful to know that the derivative of an odd function is even and the derivative of an even function is odd (see 1F-6). a) semicircle b) pirabola c) odd function d) even function e)periodic; period = ? r '* \-4 t 1D. Limits and continuity 1D-1 Calculate the following limits if they exist. If they do not exist, then indicate whether they are +oo, -oo or undefined. • q a) limb) lim4: c) lim 4s' 2-oX2 - 1 z-+2 2 +1 z--2 x + 2 d) lim 422 e) 4A2 f) lim4: ~ 2-42+ 2 --zlim+2-2-z Z-+0o 2 - 2 4@2 za + 22 + 3 g) lim - 2 4---4 i) lim +2-+oo2 j) lim•--+oo 322 -2z + 4 =-+2 z2 - 4 1D-2 For which of the following should one use the one-sided limit? Evaluate it. a) lim t b) lim 1 c) li 1 d) lim Isinz e) 'limn I 2-o0 -+1 X- 1 2-I (X- 1)4 ,.wO --+O2; 1D-3 Identify and give the type of the points of discontinuity of each of the following: 2-2 1 z>0a) x2-22- 4 b) -sin x d) f (x) = xa, z<0 e) f'(z),for the f(s) in d) f) (f(()) 2, where f(s) ýd •i 1D-4 Graph the following functions. 4x2 1a) 2-2 (See 1D-lefg.) b)2X2 +2x+2E. 18.01 EXERCISES racr+b z> 1ID-5 Define f(z) = •a b, < 1 a) Find all values of a,bsuch that f<x) is continuous. b) Find all values of a,b such that f'(x) is continuous. (Be careful!) 1D-6 For each of the following functions, find all values of the constants a and b for which. the function is differentiable. a)f( = Z+ +1 , x>; b) f()={2+4+1, >l;a)f(x) +b, X< 0. '+, ab) <1.a 1D-7 Find the values of the constants a, b and c for which the following function is differentiable. (Give a and b in terms of c.) f(W)= " +4x+ 1, x2 1; axa + b, a < 1. 1D-8 For each of the following functions, find the values of the constants a and b for which the function is continuous, but not differentiable. t·,a)(X) f +b, X>0; b) f (x)= a+b, a>O; sin 2, a …


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MIT 18 01 - Graphing

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