CAN YOU PLEASE SOLVE QUESTION 1A, 1B, 1C AND SHOW THE STEPS THAT LEAD TO THE ANSWER? THANKS IN ADVANCE.
ATH 100 / MATH 109 u Using Derivatives Using Derivatives... 1. ... to evaluate limits: (a) Explain why the limit lim e" +e-2 *-+00 et -e- is in a suitable form la suitable form for applying l'Hospital's rule. (b) Attempt to evaluate lime" +e** et - era by applying l'Hospital's rule. What went wrong? (You will need to use l'Hospital's rule more than once to see the problem). (c) Without using l'Hospital's rule, show that lim e +e** 2et -e-a = 1. 2. ... to optimize things: Of all the lines tangent to y = 243, find the equation of the one with the largest slope. 3. ... to approximate things: (a) Approximate In(1.01) by using an appropriate linearization, and by using one iteration of Newton's method on an appropriate function. (b) Explain why we cannot use Newton's method to find a root of f(x) = 13 - 2x + 2 with initial guess x0 = 1. Illustrate your explanation with a sketch. Note: To see the problem you will need to do more than one iteration of Newton's method. 4. ... to prove things: (a) Prove that arb ++1 has exactly one zero on [-2, 2). (b) Suppose that a runner completes a 3 mile race in 16 minutes. Assume that her initial speed is 0 and her speed at the end of the race is also 0. Prove that she was running at exactly 11 miles per hour at least two times during the race.
