One Function, Many Perspectives For this entire process we'll use the function f(x)=x²-2. Linear Approximation and Differentials 1. Graph f(x)=x²-2 using technology. For a = 2, and the interval [2, 3], illustrate (don't calculate for now, just illustrate!) Ax, dx, Ay, and dy 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! 2. Now actually calculate Ax, dx, Ay, and dy, without actually finding the equation of the tangent line. In other words, if finding the equation of the tangent line is one process we could use to calculate one or more of these quantities, use another process in the videos/notes to find them without finding the tangent line. 3. 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 Ax, dx, Ay, dy again. 4. Explain the relationships between these quantities and the tangent line and secant line. Newton's Method 1. 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.) 2. Explain how each line produces the next approximation. 3. Now use the Newton's Method formula to find ₁ and 2. • 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 ₁ value you got using the formula. • Find the zero using technology. o How close was 2 to the goal? o What is the exact value of this zero? Is this a useful way of approximating this value? Explain. Area Estimates 1. Sketch the graph of f(x)=x²-2, with area shaded under the curve and above the x-axis over [2, 3]. 2. Write a definite integral which represents the area of the shaded region. 3. Draw a right endpoint rectangle approximation of the area with n = 4. • What is Ax for this estimation? • What are x for i = 1, 2, 3, 4? 4. Find the area estimate given by this approximation. (If you use technology to compute this, explain what computations the technology is doing.) 5. Draw a right endpoint rectangle approximation of the area with 72 = 8. • How do Ax and the change? The Fundamental Theorem of Calculus (FTC) and Antiderivatives 1. 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.) 2. How is this related to Part 1 of the Fundamental Theorem of Calculus? 3. Find the actual area using Part 2 of the Fundamental Theorem of Calculus. (You can compare this with the value given by the Desmos calculator we used in the collaborative assignment, but please also compute it by hand also to show your process, to show you understand the connection between integrals, antiderivatives, and areas.) 4. Explain how the ideas of "accumulation" and "net change" are at work when we use Part 2 of the FTC. 5. Explain the differences between Ay, dy, and "net change" in these computations.