Calculus

This is a take in exam. You may consult different sources of information including barely not curb to class notes, homework and/or textbook. You may also collaborate with your classmates but you moldiness write your own solutions. It is fairly obvious when a pupil is Just copying the work from an external source if I control a solution has just been copied I may give you a warning for Academic Dishonesty (ADD for short).Solutions to some of these problems are available elsewhere, if you snuff it to come cross one of them you should make your best effort to sympathize it, thence write your own using your ideas and understanding of the topics. Question 2 3 4 5 6 7 8 9 10 Total Points 25 20 200 Score disport do not answer the questions in the limited space provided wasting disease scratch paper and attach it to this cover scallywag. Name Signature rogue 1 of 6 Please go on to the next page Questions (10 puts) 1. I. Use Rollers theorem to prove that f x ex. root . 013 xx 2 has at most one real Hint If has both root (say a and b) then FAA Feb. O. What does Rollers theorem say in this situation? (1 5 puts) it. let f be incessant on a, and differentiable on a, b . Show that there exists c a, b such that the tangent at c, FCC is parallel to the secant through a, FAA and b, Feb. . In other words, show that FAA equality (1) is known as the Mean Value Theorem formula. Hint Apply Rollers theorem on a, to the function G x Feb. FAA Feb. Keep in mind that a, FAA , b and Feb. are constants. 2.True or false. (5 puts) I. Iffy O, f is n any concave up nor concave down around x a. I. It. If is continuous on a, b and c iii. If f is continuous but not necessarily differentiable on O, then the absolute maximum and the absolute minimum get rid of exist. V. If f is differentiable on a, b then it is also continuous on a, b and the absolute maximum and absolute minimum exist. V. If x a corresponds to an inflection point morose , then f ii a around x a. A, b is a local max imizes then fix O. O and f ii x changes distinguish 3. hydrogen is puff on a leash that passes through a pulley on a MM t pole and is attached to a wagon. befool that the rope is attached to a loop on the wagon 2 Ft off the ground. Let x be the distance in the midst of the loop and the pole (see figure 1). (10 puts) I. adjust a formula for the facilitate of the wagon in terms of x and the rate at which Henry lulls the rope. We say that x a is a root (or is a adjust) off x , if FAA O. We say thatch is a local maximizes if f c is a local maximum. Page 2 of 6 Henry stick out 1 Henry pulling the wagon from problem 3. 10 puts) it. Find the speed of the wagon when it is 12 Ft from the pole, assuming that Henry pulls the rope at a rate of 1. 5 Ft sec. (25 puts) 4. Olav Adagio -a reason student of mine- was asked to sketch the chart of a function. Unfortunately Olav often forgets things. fortuitously for you, he wrote down some statements. Regarding the function f x , he wrote * It is only defined on , and it is continuous. * It is strictly positive, except at x 2 and x O where its value is zero. *f 2 2, f 3 1, and f 4 1. 1 when x O.Regarding if x , he wrote XSL * On the legal separation (-2, 1) it exists only at those points where g x is healthy defined. Moreover, it is positive when g is positive negative when g is negative and zero when g is zero. * On the interval (1,2) it is identically equal to zero. * On the interval (2,4) it is negative. Lastly, regarding f ii x , he wrote * On (-2, 1) it exists whenever h x signs on this interval. Is well defined. They also have opposite On (2,4) it changes sign from negative to positive at x 3. Help Olav sketch the graph off .Make sure to clearly identify the local and global intense as well as the inflection points. 5. A component of telegram 24 CM long is given to you. You can choose to either come forth it into two pieces or leave it the way it is. If you decide to pass over it, one piece must be bent in to the Page 3 of 6 shape of a square, while the remaining one must be bent into the shape of a circle. If you decide not to cut it, you can bend it into either shape. (5 puts) I. Denoting by x the length of the piece of the wire that will be bent into the shape of circle, obtain an flavour for the area enclosed by the wire.Make sure that the formula works unheeding of whether or not the wire is bent into one or two pieces. (20 puts) it. Find the maximum area that can be enclosed by the wire. Explain how this area can be obtained by specifying the dimensions (ii. , length of sides and/or radius) of the objects to be constructed. The following facts might come in handy If a square has perimeter.