Find the actual absolute errors when we are three units from x = = 2 (i.e. a = −1 and a 5). How do these values compare to the output of the error bound formula: Mxa2, when using the smallest possible M? What is happening to the size of the error as the distance between a and x = 2 grows? (ii) Here, we will look at how the size of f" (x)| impacts the error, which also connects to the size of M. Consider x = 2 and a = 1, along with the following quadratic polynomials: y = (x y = y = - 1)² + 3(x − 1) +5 (x 1)²+3(x-1) + 5 3 (x − 1)² + 3(x − 1) +5 - The quadratics have been written in a form that emphasizes that we are interested in the neighbourhood around a = 1 (which we call the centre). (This will be very important when you encounter Taylor polynomials in MAT187.) Show that L(x) = 3(x − 1) + 5, for all three quadratics (which is simply the last two terms in their algebraic expressions). Note: You may find it more efficient to show a more general result: L(x) = c(x-a) +d when f(x)=6(x − a)² + c(x − a) +d, and then apply it to the specific quadratics above. For each of the quadratics above, determine the smallest value of M such that |y" | ≤ M. How do these functions compare graphically? How does the shape of their graphs connect to the magnitude of their second derivatives? Find the actual absolute errors when x = 2 and a = 1. How do these values compare to the output of the error bound formula |x - a 2 when using the smallest possible M? Compare the size of the errors for the quadratics, keeping in mind the comparison between the values of their |y" (or equivalently, the associated M values). 12. Suppose an object of mass m is attached to the end of a spring hanging from the ceiling. The mass is at its equilibrium position y = 0 when the mass hangs at rest. Suppose you push the mass to a position yo units above its equilibrium position and release it. As the mass oscillates up and down (neglecting any friction in the system), the position y of the mass after t seconds is k y = yo cos " M where k > 0 is a constant measuring the stiffness of the spring (the larger the k the stiffer the spring) and y is positive in the upward direction. (a) Describe what represents in natural language. (b) Find d dy dt (c) How would the velocity be affected if the experiment were repeated with four times the mass on the end of the spring? (d) How would the velocity be affected if the experiment were repeated with a spring having four times the stiffness (k is increased by a factor of 4)? (e) The period T is the time required for the mass to complete one oscillation. State the expression for T in terms of m and k. (f) Assume that k is constant and calculate d dm' What does this expression represent in natural language? (g) Give a physical explanation of why dT dm is positive. 9. Let f(x) be a continuous function on the interval [0, 10] with the following properties: |f(x)|≤ 5. ⚫ The slope of the secant line between x = 0 and x 2 is -1. The slope of the secant line between x = 8 and x = 10 is -4. Is it possible that f(x) could be strictly decreasing on the interval [0,10]? If yes, give an example of such f(x) (either by using a formula or by drawing a sketch). If no, explain why it is impossible. = 10. Isabelle is a bee keeper who wants to sell honey at the local farmers market. Let y H(d) be the amount of honey, in pounds, that Isabelle will sell in a month if she charges d dollars per pound of honey. The functions H(d) and H'(d) are defined and differentiable for all d≥ 0. Some values are given in the table below. d H(d) 5.00 5.75 6.50 7.25 8.00 52 59 46 38 H' (d) -10.4 -9.1 -7.8 -11 29 -12.2 Assume that between each pair of consecutive values of d given in the table H(d) is strictly decreasing, and that H'(d) is either always increasing or always decreasing. (a) Estimate the amount of honey, in pounds, Isabelle will sell if she charges $6.30 per pound. (b) Is your estimate from the previous part an overestimate, an underestimate, neither, or there is not enough information to decide? Explain your answer. 11. In this question, we will try to develop a better understanding of the error bound in the linear approx- imation to a function f(x) near x = a.: Let I be an interval containing a and suppose that |f" (x)|≤ M for some M≥ 0 and for all x € I. Then, |ƒ(x) – L(x)| ≤ 2 |x − a|² for all x € I. M (a) Determine if the following statements are true or false. Include a picture or statement to support your answer. (i) M represents the maximum value of |f"(x)| on the interval I. (ii) The larger the magnitude of the second derivative near xa, the faster the function will deviate from the tangent line at (a, f(a)), resulting in a larger absolute error for x values that are farther from a. M (iii) If we were to sketch the graph of the error function, g(x) = f(x) — L(x), it is guaranteed to be bound below by the quadratic y -(x - a)² and bound above by the quadratic (x-a)², as long as we are on the interval I. Y = - = (iv) |x − a| = |Ax and as x gets farther from a, the larger the error bound gets and the harder it is for us to know how close our approximation is to the exact value. We will now consider cases with quadratic polynomials as our function f. This will allow us to look at both the bound for the errors and the function's actual values. When we do this, we will experiment with values for the distance between x and a and also the value of M. We will see how both seem to affect the error term, which will help convince us of the error bound. We encourage you to use Desmos to help you visualize your answers. (i) Consider the value of the function f(x) = x² + x + 3 at x = 2. ⚫ Determine the linear approximation at a general point (a, f(a)). • Find the actual absolute errors when we are one unit from x = 2 (i.e. when a = 1 and a = = 3). How do these values compare to the output of the error bound formula: M|xa|2, when using the smallest possible M?