- Question 1 Which of the following statements is the Extreme Value Theorem? If f attains a local maximum and a local minimum on an interval [a, b), then f is continuous. If f is a continuous function on any interval of R, then f attains a global maximum and a global minimum. If f is a continuous function on R, then f attains a global maximum and a global minimum. If f is a continuous function on an interval [a, b), then f attains a global maximum and a global minimum. O If f is a function on an interval [a, b), then f attains a global maximum and a global minimum. O If f attains a global maximum and a global minimum on an interval [a, b], then f is continuous. Note: A statement of the form "if A is true, then B is true" is false if you can come up with even one example where A is true but B is not true. Such an example is called a counter-example. After you have identified the Extreme Value Theorem from this list, sketch counter-examples to show that in fact, each of the other statements is false (and therefore are not theorems at all).Question 10 We want to compute the limit below with I'Hospital's Rule lim cot(8) sin(I) I-+0 a) What is the indeterminate type of this limit? 00 - 00 0 0* 00 0 9 0 10 0 b) To be able to use I'Hospital's Rule, we rewrite the limits in terms of cosine and sine functions only, to get lim cot(8z) sin(x) = lim A(3) z-+0 1-+0 B(I) where [A(x), B(=)] = FORMATTING: Enter your answer as [A(x), B(x)], including the square brackets and with a comma (,) between the terms. For this question, you must use strict scientific calculator notation: multiplication is written *; for example, you must write 3 sin(3x) as 3*sin(3*x). c) What is the indeterminate type of the limit lim A(I) - found in (b)? I-+0 B(I) 0 09 0 00- 00 0 0 0+ 00 0 20 d) According to I'Hospital's Rule, lim A(I) = lim A] (I) -+0 B(I) 1-+0 B1(I) for [A1 (x), B1 (x)] = FORMATTING: Type your answer in the form [A] (I), Bj (I)], include the square brackets and write a comma between A1 (x) and Bj (z). For this question, you must use strict scientific calculator notation: multiplication is written * ; for example, you must write 3 sin(3x ) as 3*sin(3*x). e) Conclude that lim cot(8r) sin(x) = lim A1 (I) 1-+0 B1(I) FORMATTING: Give the exact answer.Question 11 Consider the limit lim (xe3/x - x) a) What type of indeterminate form is this? 0 0 . 00 0 0/0 0 00 - 00 0 00/00 0 00 0100 b) Compute the limit, using I'Hospital's Rule in the appropriate manner, if it applies. lim C-700Question 12 Consider the limit 2 15m {cos{23))1/z . z>0+ a] This limit is an indeterminate form of what type? 0000 on\" 0100 0000 Ooofoo 00m b] Compute the limit using l'Hpspital's Rule in the appropriate manner, it it applies. 2 $th = m Question 2 Consider the function f : R - R. defined by f (x) =24 (1-2)3. (a) Provide a list of all the critical numbers of f . Separate the values by a semi-colon should there be more than one. Input the word "none" (without quotes or capitals) should there be none. (b) Provide a list of all the local minima of f (following the same instruction). C = (c) Provide a list of all the local maxima of f (following the same instruction).Question 3 Consider the function given by Determine the absolute maximum value and absolute minimum value of f over the interval [3, 5] . FORMATTING: Give your answer with an accuracy of at least 3 decimal places. Minimum value = Number Maximum value = NumberQuestion 4 Find the point on the parabola y = I that is the closest to the point (10, 13/2) on the Cartesian plane. a) Write the formula for the square of the distance between a point (I, I ) on the parabola and the point (10, 13/2). Answer: h(I) = b) Compute the derivative of h. Answer: h (I) = c) Find the critical points p1 P3 Click for List e) What is the point of the parabola which is the closest to [10, 13/2] and what is this distance? Don't forget that h(I) represents the square of the distance. Give your answers with an accuracy of two decimal places. Answer: The point is [x, y) = P (You must use square brackets instead of parentheses to denote vectors). The distance is NumberQuestion 5 We are interested in a function f. but the only graph we have is of its derivative. f, given in the following figure. 30 20- 10- 5 3 3 -10 15 5/3 10 3 10 -20- 30 Graph of y = f (I) Which of the following statements about f are true? (1) f is increasing on the intervals (-co, -5) U (0, 5) (2) f is decreasing on the intervals (-oo, -5) U (0, 5) (3) f is concave up on the interval (-5/3 \\/3, 5/3 v/3) (4) f is concave down on the interval (-5/3 \\/3, 5/3 (/3) (5) The critical points of f are I = -5, x = 0 and I = 5 (6) The inflection points of f are I = -5, I =0 and I = 5 O (5), (6) O (2), (4), (5) O (2), (6) O (1), (3), (5) O (1), (6) O (3) only O (2), (3), (6) O (1), (4), (5)Question 6 Consider the function f which is given on its domain (0, oo) by In(x) f(x) = 21/3 Find the critical number c of f on its domain and enter its exact value in the first column of the table below. The second column consists of several drop-down menus. Do the following: for each of the given intervals of (0, oo), select the phrase that best describes the behaviour of the function f on that interval; for the critical number, select the phrase that best describes its nature; . for each of 0 and oo, select the phrase that best describes the corresponding limiting behaviour of f. behavior 0 Click for List O C Click for List Click for ListQuestion 7 We wish to sketch the graph of the function f(x) = (I + 3) . Find all the critical numbers of f and of f. There is only one in the domain of f. In the first column of the table below, list both this exact value and the value x = -3 at which the function is not defined, in ascending order : a b Click for List Click for ListQuestion 8 We wish to sketch the graph of the function f(z) = x - 71+9 (x - 3)2 . Find all the critical numbers of f and of f (that is, all potential extrema and/or inflection points). There are two of them; enter their exact values, together with the value x = 3 at which f is undefined, in the first column of the table below (in ascending order : a c Click for List Click for ListQuestion 9 We want to compute the limit below with the I'Hospital's Rule if it applies. 1 -e4x lim x-+0 sin (91) a) What is the indeterminate type of the limit? 0 0 0/0 0/0 0 0 00 b) According to I'Hospital's Rule, 1 - e4x lim A(I) lim x-+0 sin (91) X-+0 B(x) where [A(x), B(z)] = FORMATTING: Enter your answer as [A(x), B(I)], including the square brackets and with a comma (,) between the terms. For this question, you must use strict scientific calculator notation: multiplication is written * ; for example, you must write 2x as 2*x. c) Conclude that lim 1-etc A(x) lim x-+0 sin (92) x-+0 B(x) FORMATTING: Give the exact