MAT Calculus for Engineers I EXAM #3

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1 MAT 65 - Calculus for Engineers I EXAM #3 Instructor: Liu, Hao Honor Statement By signing below you conrm that you have neither given nor received any unauthorized assistance on this exam. This includes any use of a graphing calculator beyond those uses specically authorized by the School of Mathematical and Statistical Sciences and my instructor. Furthermore, I agree not to discuss this exam with anyone until the exam testing period is over. In addition, my calculator's memory and menus may be checked at any time and cleared by any testing center proctor or School of Mathematical and Statistical Sciences instructor. Signature Date Instructions:. The exam consists of three parts: multiple choice, worth 60%, free response (show your work), worth 40%, and extra credit true or false questions, worth 5 points. Please read each problem carefully.. There are 0 multiple choice questions worth 6 points each. Please circle your answer choice on the exam AND ll in the answer grid on the last page. 3. Provide complete and well-organized answers in the free response section. 4. Answers in the free response section without supporting work will be given zero credit. Partial credit is granted only if work is shown. 5. No calculators with Qwerty keyboards or ones like the Casio FX-, TI-89, TI-9, or TI-nspire that do symbolic algebra may be used. 6. Proctors reserve the right to check calculators. 7. Please request scratch paper from your instructor. 8. The use of cell phones is prohibited. TURN YOUR CELL PHONE OFF! Do not allow your cell phone to ring while you are taking the exam. Do not use the calculator on your cell phone. If a proctor sees you using a cell phone, they will take your exam and you will be reported to the Dean of Students for cheating. 9. If you read the instructions, please place a check mark here for one point!

2 PART I - Multiple choice. Please circle your answer choice on your exam and ll in the answer page on the last page. Show your work on the space provided to earn partial credits. x + cos(3x). Use L'Hospital's Rule to evaluate lim x 0 e x Answer is B. lim x 0 sin(3x)(3) 4e x = 4. Use L'Hospital's Rule to help evaluate exactly lim x 0 ( + x) /x. First we evaluate Then it follows that lim ln( + x) = lim x 0 x x 0 +x =. lim x 0 ( + x)/x = e lim x 0 x ln(+x) = e = e. Answer is D. 3. Use L'Hospital's Rule to help evaluate exactly lim (sin x) ln x. As x 0 +, ln x and sin x 0, so this is a 0 type. Then we use the following trick Answer is C. lim ln x sin x = lim = lim = lim x cos x sin x sin x x cos x sin x cos x cos x x sin x = 0 4. Which of the following is critical number of the function f(x) = xe x? Hence f = 0 x = ±. Answer is D. f (x) = ( x )e x Summer 03 c School of Mathematical & Statistical Sciences - Arizona State University

3 5. The continuous function f(x) = x 3 x +x attains the absolute maximum value on the closed interval [0, ]. At which x value does this occur? f (x) = 3(x )(x 7) So critical number is x = while x = 7 / [0, ] so we don't consider it. We nd among f(0) = 0, f() =, f() = 0 the largest value, so the absolute max is f() = 0 and occurs at x =. Answer is B. 6. Consider the function f(x) = e x x on the interval [, ]. Find a point c in the interval which satises the conclusion of the mean value theorem for this f(x) and [a, b]. To 4 decimal place accuracy this c = Answer is D. f (c) = e c = e e ( ) e e c = ln The function f(x) = 3x 5 5x 3 is decreasing on an interval [, a] and then is increasing on [a, ). Find a. f (x) = 5x (x ) So critical numbers are x = 0, ±. Note that when x (, 0), f (x) < 0; when x (0, ), f (x) < 0; when x (, ) we have f (x) > 0. Although f (0) = 0 we still have that f is decreasing on [, ] and increasing on [, ]. Answer is B. 8. The function f(x) = has a single inection point in the interval (0, ). x + Find it. f (x) = x (x + ), f (x) = 6x (x + ). 3 So f = 0 gives x = ± 3. Also note that f changes sign at x = 3 we have the single inection point is at x = 3. Answer is B. 9. Find the general antiderivative of f(x) = x cos x 3x x 6 sec (x) + 5. f(x) = cos x 3 x 6 sec x + 5. Summer 03 c School of Mathematical & Statistical Sciences - Arizona State University

4 So the general antiderivative is F (x) = sin x 3 ln x 6 tan x + 5x + C. Answer is A. 0. Find f(x) such that f (x) = 3x + 5 sin(x) 7e x + and f(0) = 4. The general antiderivative is f(x) = x 3 5 cos x 7e x + x + C. Since f(0) = 4 we have f(0) = + C = 4 and C = 8. Answer is D. Summer 03 c School of Mathematical & Statistical Sciences - Arizona State University 3

5 PART II - Free response. Show all your work.. Let f(x) = x x (a) ( point) Where does f(x) fail to be continuous? x =. (b) (3 points) Calculate the rst and second derivatives of f. f (x) = x x (x ) f (x) = (x ) 3. (c) (3 points) Find all critical numbers and all possible inection points of f. f = 0 x = 0, Also f is undened at x = so critical numbers are 0,,. f = 0 has no solution and although f is undened at x = we note that x = is not in the domain, so there is no possible inection point of f. (d) (3 points) Describe the intervals where f is increasing/decreasing, and concave up/down. We use critical numbers to divide the real line into subintervals, and arrive at the following table Interval f f f f (, 0) + Inc CD (0, ) Dec CD (, ) Dec + CU (, ) + Inc + CU. According to U.S. Postal regulations, a rectangular box is oversized if the sum of its height and its girth (perimeter of its base) exceeds 08 inches. Find the maximum volume (in cubic inches) that a box can have if () it has a square base, and () it's not oversized. Let the side of the square base be x and the height of box be h. Then by assumptions And the volume of the box is denoted by 0 4x + h 08. V = x h. Summer 03 c School of Mathematical & Statistical Sciences - Arizona State University 4

6 For a given x, to maximize V one would choose As a result, we need to maximize with 0 x 7. Taking derivative of f(x) we have Critical numbers are x = 0, 8. h = 08 4x. f(x) = x (08 4x) f (x) = 6x x Now to nd absolute max of f(x) over [0, 7], we nd f(0) = 0 f(8) = 664 f(7) = 0 Hence the absolute max volume is 664 cubic inches and at x = 8 inches. 3. Algebraically nd intervals of concavity and the inection points for the function f(x) = x 3 x 3. f (x) = 6x Possible inection point is at x = /3 where f (/3) = 0. Observe that So the function Interval f f (, /3) CD (/3, ) + CU curve changes concavity at x = /3. Hence inection point is (, 3 f()) = (, 83 ). The function is concave down on (, /3) and concave up on (/3, ). 4. Let f(x) = ( 3x)e x. Find f (x) and algebraically determine () all critical numbers, () intervals on which f is increasing or decreasing and (3) local maximum and local minimum values of f. f (x) = 3e x (x + ). Summer 03 c School of Mathematical & Statistical Sciences - Arizona State University 5

7 Critical numbers x =. Observe that when x (, ) we have f > 0 so f is increasing, and for x (, ) we have f < 0 so f is decreasing. As a result, by rst derivative test, at x = we have local maximum f( ) = 3e. There is no local minimum. Summer 03 c School of Mathematical & Statistical Sciences - Arizona State University 6

8 PART III - Extra credits. True or false questions. point each.. If x = c is a critical number of f, then f has a local maximum or local minimum at x = c. False. Counterexample f(x) = x 3.. If f(x) is continuous on [a, b], then f(x) attains an absolute maximum at a number c in [a, b]. True. By extreme value theorem. 3. If f (x) is continuous on [a, b], then f(x) attains an absolute maximum at a number c in [a, b]. True. Dierentiable implies continuous over [a, b]. 4. If f (c) = 0, then (c, f(c)) is an inection point. False. f(x) = x 4. Concave up for all x 0 when f (0) = If f is increasing on [0, ], then f () > 0. False. f(x) = (x ) 3 with f = 3(x ). Increasing over [0, ], with f () = 0. Summer 03 c School of Mathematical & Statistical Sciences - Arizona State University 7

9 Last Page Answer to multiple choice part Question number Answers: A, B, C, D, or E (Please use capital) Total point Multiple choice points Show work points Extra credits Summer 03 c School of Mathematical & Statistical Sciences - Arizona State University 8