University of Toronto Scarborough Department of Computer & Mathematical Sciences Final Exam MATA31 { Calculus I for Computer & Mathematical Sciences Examiner: Z. Shahbazi Date: April 22, 2016 Time: 3 Hours FIRST NAME: LAST NAME(S): STUDENT NUMBER: TUTORIAL NUMBER: You will lose 2 marks if you do not write the tutorial number. SIGNATURE: DO NOT OPEN THIS BOOKLET UNTIL INSTRUCTED TO DO SO NOTES: There are 14 numbered pages in the test including the blank pages 13, and 14. It is your responsibility to ensure that, at the start of the exam, this booklet has all its pages and that all pages are handed in. Answer all questions. Explain and justify your answers. If you need more space use the back of the page. Calculators and other electronic aids are NOT allowed. FOR MARKERS ONLY Question Marks 1 / 10 2 / 25 3 / 25 4 / 20 5 / 20 TOTAL /100 MATA31 page 1 1 (10 marks) Prove that sum of two rational numbers is a rational number. Is the converse of this statement true Why MATA31 page 2 2. (25 marks) 2.(a) (7 marks) Use denition of the limit to prove that lim x!5 (x2) = 25: MATA31 page 3 2.(b) (15 marks) Evaluate the following ve limits. 1: lim x!3+ p x2 2x 3 x 3 2: lim x!1 ( p xpx+ 3) 3: lim x!0 x3 cos( 10 x4 ) 4: lim x!1 sin(lnx) x 1 5: lim x!1 x 1 x You can use next page for showing the details related to this question. MATA31 page 4 This page is an extra space for question 2. MATA31 page 5 2. (c) (3 marks) Let f(x) = 8<: 2x2 1; if x < 2 A; if x = 2 x3 2Bx; if x > 2 For what values of A and B is f continuous at x = 2. MATA31 page 6 3. (25 marks) 3. (a) (7 marks) Use the denition of derivative to nd f 0(x), if f(x) = 1 x . MATA31 page 7 3. (b) (18 marks) Compute the derivative of the following functions: 1: f(x) = e3x(ln(x2 + 1)) 2: f(x) = sin(arctan(x)) 3: f(x) = xx 4: f(x) = (x 1)12(2x+ 1)5 (x2 2)100 5: f(x) = ln(x+ 3); if x < 1 4x2 1; if x 1 6: sin(xy) = x2 + y3 You can use next page for showing the details related to all parts of this question. MATA31 page 8 This page is the extra space for showing your work for question 3. MATA31 page 9 4. (20 marks) 4. (a) (10 marks) State the Rolle's Theorem and then prove it. MATA31 page 10 4. (b) (10 marks) Show that equation x3 + 9x2 + 33x 8 = 0 has exactly one real root. MATA31 page 11 5. (20 marks) 5 . (a) (10 marks) Find domain, critical values, in ection points, asymp- totes and symmetries of the function y = e x2 2 and then sketch its curve. MATA31 page 12 5. (b) (10 marks) The cost of the material for the top and bottom of a cylindrical can is 10 cents per square inch. The material for the rest of the can costs 5 cents per square inch. If the can must hold 500 cubic inches of liquid, what dimensions should be chosen to make the cheapest can MATA31 page 13 (This page is intentionally left blank.) MATA31 page 14 (This page is intentionally left blank.)
