This is a take-home exam. You may consult different sources of information including but not limited to class notes, homework and/or textbook. You may also collaborate with your classmates but you must write your own solutions. It is fairly obvious when a student is Just copying the work from an external source; if I deem 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 happen to come cross one of them you should make your best effort to understand it, then 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: Please do not answer the questions in the limited space provided; use scratch paper and attach it to this cover page. Name: Signature: Page 1 of 6 Please go on to the next page… Questions (10 puts) 1. I. Use Roller’s theorem to prove that f x ex. root . 013 xx 2 has at most one real Hint: If has two roots (say a and b) then FAA Feb. O. What does Roller’s theorem say in this situation? (1 5 puts) it. Let f be continuous 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 Equation (1) is known as the Mean Value Theorem formula. Hint: Apply Roller’s 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 neither 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 off 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 off , then f ii a around x a. A, b is a local maximizes then fix O. O and f ii x changes sign 3.
Henry is pulling on a rope that passes through a pulley on a MM t pole and is attached to a wagon. Assume that the rope is attached to a loop on the wagon 2 Ft off the ground. Let x be the distance between the loop and the pole (see figure 1). (10 puts) I. Find a formula for the speed 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 zero) off x , if FAA O. We say thatch is a local maximizes if f c is a local maximum. Page 2 of 6 Henry Figure 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 former student of mine- was asked to sketch the graph of a function. Unfortunately Olav often forgets things. Luckily 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 interval (-2, 1) it exists only at those points where g x is well 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 extreme as well as the inflection points. 5. A piece of wire 24 CM long is given to you. You can choose to either cut it into two pieces or leave it the way it is. If you decide to cut it, one piece must be bent into 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 expression for the area enclosed by the wire.
Make sure that the formula works regardless 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.