(c) (1) Find the Taylor polynomial P2(x) for the function f(x) =/x at x = 9. (ii) Use it to approximate v10, writing your answer as a single fraction. (iii) Use Taylor's Theorem to find an upper bound for the error in this approximation. (Write your answer as either a fraction or as a decimal to 1 significant figure.)

[7 0] Consider A = 5 7 and find AAT and ATA and AATA or explain why these products do not exist.

The roots of a degree 4 polynomial f (x) are 0, 0, 1, −1. Suppose f (2) = 1. Find a formula for f (x). The roots of a degree 3 polynomial f (x) are 1, −1 and 12 Suppose f (0) = −2. Find a formula for f(x).

In a hospital, a drug is given to a patient. Let f(t) be the amount of drug (in mg) in the patient's body t minutes after it is administered. The nurses want to predict how the body will absorb the drug (the shape of the graph y = f(t)) to be able to prepare the next course of action. One of them remembers about Taylor polynomials from their Calculus course and explains that they allow to predict the shape of a function by making very accurate measurements of the function at one point. They decide to do this and they find the following details about the patient's absorption of the drug: • f(15) = 140 • f'(15) = -10 • f" (15) = 12 1 000 Use these details to approximate the average amount of drug in the patient's body over the first 2 hours: 120 1 f(e) dt 120 Important: Remember that the measurements were made at t = 15 minutes.