Can you help me with the following questions. Thanks so much! :)

Use the limit definition of derivative and show that the derivative of the function f(x) = x|x| exists at x = 0. That is, f 0 (0) exists.

Section 3.2: 75, 77, 80 (Use Desmos for these problems and print out an output)

Section 3.4: 7

SECTION 3.2 The Derivative as a Function 145 GU in Exercises 75-80, zoom in on a plot of f at the point (a. f(a)) 83. Calculate the subtangent of and state whether or not f appears to be differentiable at x = a. If it is nondifferentiable, state whether the tangent line appears to be vertical or ((x) = x' + 3x aux = 2 does not exist. 84. Show that the subtangent of f(x) = el is everywhere equal to 1. 75. f(x) = (x - likx]. a=0 76. f(x) = (x - 3)50. a = 3 85. Prove in general that the subnormal at P is If'(x)f(x)I. 77. f(x) = (x - 3)1/3. a = 3 78. f(x) = sin(x'/3). a = 0 86. Show that PO has length If(x)IVI + S'(x)--. 79. f(x) = | sinx]. a =0 80. f(x) = lx - sinxl. a = 0 81. Find the coordinates of the point P in Figure 19 at which the tangent y - f (x)/ line passes through (5. 0). f(x) - 9 -12 94 P - (x.f(x)) Tangent line R FIGURE 20 FIGURE 19 87. rove the following theorem of Apollonius of Perga (the Greek math- ematician born in 262 BCE who gave the parabola, ellipse, and hyperbola 82. (GU Plot the derivative f' of f(x) = 23 - 10x-' for x > 0 and their names): The subtangent of the parabola y = x at x = a is equal observe that f'(x) > 0. What does the positivity of f'(x) tell us about the to a/2. graph of f itself? Plot f and confirm this conclusion. 88. Show that the subtangent to y = x at x = a is equal to a/3. Exercises 83-86 refer to Figure 20. Length OR is called the subtangent at 89. Formulate and prove a generalization of Exercises 87 and 88 for P. and length RT is called the subnormal. y=x". Further Insights and Challenges 90. Two small arches have the shape of parabolas. The first is the graph of f(x) = 1 - x2 for -1 5 x S I and the second is the graph of g(x) = 4 - (x - 4)2 for 2 5 x 5 6. A board is placed on top of these arches so it rests on both (Figure 21). What is the slope of the board? Hint: Find the tangent line to y = f(x) that intersects y = g(x) in exactly one point. FIGURE 22 92. Let f be a differentiable function, and set the function g(x) = f(x + c). where c is a constant. Use the limit definition to show that g'(x) = f'(x + c). Explain this result graphically. recalling that the graph of g is obtained by shifting the graph of f c units to the left (if c > 0) or FIGURE 21 right (if c