MA FINAL EXAM INSTRUCTIONS VERSION 01 DECEMBER 9, Section # and recitation time

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1 MA 6500 FINAL EXAM INSTRUCTIONS VERSION 0 DECEMBER 9, 03 Your name Student ID # Your TA s name Section # and recitation time. You must use a # pencil on the scantron sheet (answer sheet).. Check that the cover of your question booklet is GREEN and that it has VERSION 0 on the top. Write 0 in the TEST/QUIZ NUMBER boxes and blacken in the appropriate spaces below. 3. On the scantron sheet, fill in your TA s name (NOT the lecturer s name) and the course number. 4. Fill in your NAME and PURDUE ID NUMBER, and blacken in the appropriate spaces. 5. Fill in the four-digit SECTION NUMBER. 6. Sign the scantron sheet. 7. Blacken your choice of the correct answer in the spaces provided for each of the questions 5. Do all your work on the question sheets. Show your work on the question sheets. Although no partial credit will be given, any disputes about grades or grading will be settled by examining your written work on the question sheets. 8. There are 5 questions, each worth 8 points. The maximum possible score is 8 5 (for taking the exam) = 00 points. 9. NO calculators, electronic device, books, or papers are allowed. Use the back of the test pages for scrap paper. 0. After you finish the exam, turn in BOTH the scantron sheets and the exam booklets.. If you finish the exam before 8:55, you may leave the room after turning in the scantron sheets and the exam booklets. If you don t finish before 8:55, you should REMAIN SEATED until your TA comes and collects your scantron sheets and exam booklets.

2 Questions. Let f(x) = (x + ) 3. Evaluate lim x ( f(x) f() x ). A. 7 (correct) 8 C. 3 D. E.. Evaluate. lim x sin( x 0 + x ). A. 0 (correct) C. D. π E. The limit does not exist.

3 3. Let h(x) be the function defined by { cos π x if 0 < x < 3 x a if 3 x < π Determine the value of a so that the function h is continuous for all 0 < x < π. A. 9 7 (correct) C. 8 D. 5 E. There is no such value of a. 4. Let y = ln(tan x). Then dy dx = sec A. x (correct) tan x tan x sec x C. sec x D. tan x E. sec x tan x 3

4 5. Compute ( ) x + x lim x 0 + x 4. x A. 4 (correct) C. D. E. The limit does not exist. 6. Given f(x) = x3 x 3 +, find the formula for its inverse function f (x). A. f (x) = 3 x 3 x+ f (x) = 3 x+ 3 x C. f (x) = x3 + x 3 D. f (x) = 3 x +x E. f (x) = 3 +x x (correct) 4

5 7. Assume that y is defined implicitly as a differentiable function of x by the equation Find dy dx at the point (, ). xy + xy + x = 6. A. (correct) 6 C. 6 D. E Use the linear approximation of the function f(x) = 4 x at a = 6 to estimate the number A (correct) C D E

6 9. Find the absolute maximum Max and absolute minimum Min of the function f(x) = x 6 x 3 on the interval [, ]. A. Max = 48, Min = (correct) Max = 48, Min = 0 C. Max = 3, Min = D. Max = 3, Min = 0. E. Max = 0, Min = 0. Find the interval where the function f(x) = 4x 3 6x + 3x + takes the value 3. A. (, ) (, 0) C. (0, ) D. (, ) (correct) E. (, 3) 6

7 . Evaluate lim x ln x. x 0 + A. 0 (correct) C. D. E.. Which of the following is/are true about the function f(x) = x 3 3x 9x? () The function f is increasing on the interval (, 3). () 5 is a local maximum value of f. (3) The graph of f is concave down when x <. A. () and (3) only (correct) () and () only C. () and (3) only D. (3) only E. All are true 7

8 3. Choose the one which describes best the graph of the function A. f(x) = sin x + cos x on [0, π) (π, π]. C. D. (correct) E. 8

9 4. Assume 3 f (x) 6 for all values of x. What are the (a) minimum possible value, and (b) maximum possible value of f(8) f()? A. (a) 4 (b) 48 (b) 8 (b) 36 (correct) C. (a) 9 (b) 6 D. (a) 6 (b) 0 E. (a) (b) 5. Water is withdrawn from a conical reservoir, 4 feet in diameter and 4 feet deep (vertex down) at the constant rate of ft 3 /min. How fast is the water level falling when the depth of the water in the reservoir is 3 feet? A. 4 9π ft/min (correct) 9π ft/min C. 9π 4 ft/min D. 9π ft/min E. π 4 ft/min 9

10 6. The points on the ellipse x + y = 5 that are closest to the point (, 0) are: A. (, ± ) (correct) ( 3, ± ) C. (, ± 3 ) D. (, ± ) E. ( 3, ±) 7. Evaluate d dx (arcsin(4x + ) ). A. (4x+) (4x+) (4x+) C. (4x+) 4 D. E. 4(4x+) (4x+) 4 8(4x+) (4x+) 4 (correct) 0

11 8. Let f(x) = x ln x. Evaluate f (x). (ln x ) A. (ln x)x (ln x)x (ln x ) (correct) C. (ln x)x ln x (ln x ) D. x E. x x 9. The area under the graph of y = 9e 3x ( + e 3x ) 5 and above the y axis between x = 0 and x = 3 is: A. ( + e 9 ) ( + e9 ) 6 48 C. D. (+e 9 ) 6 3 (correct) (+e 9 ) 6 E. 3 4 ( + e9 ) 6

12 0. Compute π 6 0 sin(t) cos(t) dt. A. (correct) 3 C. D. π E. 3. The half-life of a certain material is 0 years. If we start with a sample of 00 grams, how much remains after 50 years? A. 5 grams (correct) 5 grams C. 50 grams D. 50 grams E grams

13 . Evaluate at x =. ( ) d x (t + 3) 0 dt dx 0 A C D. 9 0 E (correct) 3. The directrix of the parabola y = x is: A. y = 7 8 (correct) y = C. y = D. x = E. x = 4 3

14 4. Find an equation of the ellipse with the following conditions: (a) the foci are (, ) and (4, ), and (b) one of the vertices is ( 4, ). A. x 5 + y 4 = C. D. E. (x ) 5 + (y+) 3 = (x ) 5 + (y+) 4 = (correct) (x+) 5 + (y ) 3 = (x+) 5 + (y ) 4 = 5. One of the foci for the hyperbola 9x 4y 36x 8y 4 = 0 is: A. ( + 3, ) (correct) ( + 3, ) C. ( 3, 0) D. (, + 3) E. (, + 3) 4

MA FINAL EXAM INSTRUCTIONS VERSION 01 DECEMBER 12, Section # and recitation time

MA FINAL EXAM INSTRUCTIONS VERSION 01 DECEMBER 12, Section # and recitation time MA 1600 FINAL EXAM INSTRUCTIONS VERSION 01 DECEMBER 1, 01 Your name Student ID # Your TA s name Section # and recitation time 1. You must use a # pencil on the scantron sheet (answer sheet).. Check that