For this process we'll use the function f(2)= 2²-2 • For a = 2, and the interval [2, 3], find each of the following without actually calculating the equation of the tangent line, explaining how you do so, and illustrating them on your graph (using screenshot annotation tools, for example, or some other method). Zoom in enough to make sure each the quantities are shown clearly! ο Δ.Σ. • dx • Ay • dy • Now, find the equation of the tangent line at a = 2, and the equation of the secant line for the interval [2, 3]. Add these two lines to your graph, and illustrate the quantities Az, dx, Ay. dy again. Explain the relationships between these quantities and the tangent line and secant line. • For the same function, without calculating anything, sketch an illustration of the Newton's Method process for finding the ₁ and ₂ approximations of the zero of the function, using the seed value * = 2. (Draw in the lines that would produce the next two approximations after this initial value.) Explain how each line produces the next approximation. • Now use the Newton's Method formula to find ₁ and 2₂. How do they compare to your illustration? • Because you already found the tangent line at x = 2 in an earlier problem, use it to verify the value you got using the formula. • Find the zero using technology. How close was to the goal? What is the exact value of this zero? Is this a useful way of approximating this value? Explain. • Sketch the graph of f(x)=x²-2, with area shaded under the curve and above the z-axis over [2, 3]. • Write a definite integral which represents the area of the shaded region. • Draw a right endpoint rectangle approximation of the area with = 4 and describe what each of the following are in your illustration: · Δε for i=1,2,3,4: • The area estimate given by this approximation. (If you use technology to compute this, explain what computations the technology is doing.) • Draw a left endpoint rectangle approximation of the area with n = 8. • How do Az and the change? • If you start with only one of your eight rectangles, on the left side of your interval, and add them in one at a time, until all of the approximating rectangles are present, how much area do you add at each step? (Don't actually calculate this estimate for eight rectangles, just describe it in terms of the information you have, and explain how you would calculate it.) • How is this related to Part 1 of the Fundamental Theorem of Calculus? • Explain how each of the following are related to each other for this function, and how they are each related to the question of how much area is under the curve over this interval. Be sure to write what each of them are. • Definite integral; • Indefinite integral; ▪ General antiderivative; • Particular antiderivative. • Use Part 2 of the Fundamental Theorem of Calculus to find the exact area. (You can compare this with the value given by the Desmos calculator linked above, but please compute it by hand.) • Explain how the ideas of "accumulation" and "net change" are at work when we use Part 2 of the FTC. Explain the differences between Ay. dy, and "net change" in these computations.

W. A. Mozart’s opera Die Zauberflöte is scheduled to premiere at Theater auf der Wieden. If the stage can be described by the function y  =  3x3 ln(17x) with  1/17 < x < 9, what is the area of the stage? Answer correct to 3 decimals.