Ex:44 3.2 The Derivative as a Function13539.40.Theory and ExamplesIn Exercises 49-52,a. Find the derivative f'(x) of the given function y = f(x).b. Graph y - f(x) and y - f'(x) side by side using separatesets of coordinate axes, and answer the following questions.c. For what values of x, if any, is f' positive? Zero? Negative?y- fa)y = f(x)P(1, 1)y- 2x -1P(1, 1)y =xd. Over what intervals of x-values, if any, does the functiony - f(x) increase as x increases? Decrease as x increases?How is this related to what you found in part (c)? (We willsay more about this relationship in Section 4.3.)50. y = -1/x52. y - x/4y VxIn Exercises 41 and 42, determine if the piecewise-defined function isdifferentiable at the origin.49. y = -x51. y-x'/3S2r - 1,lư + 2x + 7, x<041. f(x) =53. Tangent to a parabola Does the parabola y = 2r? - 13x + 5have a tangent whose slope is - 1? If so, find an equation for theline and the point of tangency. If not, why not?54. Tangent to y = V Does any tangent to the curve y = Vcross the x-axis at x=-1? If so, find an equation for the line andthe point of tangency. If not, why not?55. Derivative of -f Does knowing that a function f(x) is differ-entiable at x = xo tell you anything about the differentiability ofthe function -f at x- N? Give reasons for your answer.42. gx) =/3 x<0Differentiability and Continuity on an IntervalEach figure in Exercises 43-48 shows the graph of a function over aclosed interval D. At what domain points does the function appear to bea. differentiable?b. continuous but not differentiable?c. neither continuous nor differentiable?56. Derivative of multiples Does knowing that a function g() isdifferentiable at !-7 tell you anything about the differentiabil-ity of the function 3g at i- 7? Give reasons for your answer.Give reasons for your answers.57. Limit of a quotient Suppose that functions g() and h(r) aredefined for all values of I and g(0) = h(0) = 0. Canlim, o (g(1))/(h(1)) exist? If it does exist, must it equal zero?Give reasons for your answers.43.44.y- f)D: -35xs2y = f)D: -25S358. a. Let f(x) be a function satisfying |f(x)| sx for -1 Sxs1.Show that f is differentiable at x-0 and find f'(0).to 123b. Show thatf(x) =0,X = 045.46.is differentiable at x=0 and find f'(0).T 59. Graph y = 1/(2Vx) in a window that has 0 sx S 2. Then, onthe same screen, graphy = f)D: -3sxs3y -fx)D: -25xS 3Vr + h - Vifor h = 1,0.5, 0.1. Then try h --1, -0.5, -0.1. Explain whatis going on.T 60. Graph y = 3x in a window that has -2 s x S 2,0 sys 3.Then, on the same screen, graph247.48.(x + h) - xуy- f)D: -1SS 2D: -3 sxS34for h = 2, 1, 0.2. Then try h = -2, -1, -0.2. Explain what isgoing on.61. Derivative of y = |x| Graph the derivative of f(x) = |x].Then graph y = (\x| - 0)/(x - 0) = \x|/x. What can youconclude?-10

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Description: Ex:44 3.2 The Derivative as a Function13539.40.Theory and ExamplesIn Exercises 49-52,a. Find the derivative f'(x) of the given function y = f(x).b. Graph y - f(x) and y - f'(x) side by side using separatesets of coordinate axes, and answer the follow