MATH 1014 3.0 N (Peter Taylor's section) Sample questions of the type to expect on Test #1, though some are from old exams and may be a little harder. Also the front page of that test. MATH 1014 3.0 N (Peter Taylor's section) NAME: STUDENT #: TEST #1 Fri, Mar 27, 2009, 08.30 - 09.20 (50 min) Test covers: Stewart Chapter 5, Sections 6.1,6.2 and Appendices E and G. YOU ARE ALLOWED to bring in two sides of notes (8.5x11 inch paper). There will be no formulae given on the test paper. Do not ask invigilators for help unless you are convinced that there is an error or ambiguity in a question. No other aids except for a non-graphing calculator. ANSWER ALL QUESTIONS. WRITE YOUR ANSWERS ON THESE SHEETS - use backs and sheets at end if necessary - but clearly indicate where solutions are written. There are 5 questions. All questions carry equal marks (10 per question). EXPLAIN your methods and calculations with (a few) words wherever appropriate. All candidates must sign the attendance sheet and have photo ID available if required. --------------------------------------------SAMPLE QUESTIONS – note the test will have 5 questions only 1. Evaluate the following definite integrals. π/2 1 xdx a) ∫ 1 + x2 −1 d) 2. b) ∫ 0 sin x dx 4 ( 1+ cos x ) 1 c) I(δ) = 1/2 where 0 < δ < 1 δ For part (c) what is the limit of I(δ) as δ → 0? Evaluate the following definite integrals or explain why the integral is not defined over the interval specified. 2 a) 2 2 xdx ∫(x -2 2 2 +1 ) , b) dx ∫ (x - 1 ) 3 π/2 , c) a) USE THE COLLAPSING SUM, Sn = cosx ∫ (2 + sinx ) 4 dx -π/2 0 n 3. dx ∫x ∑ (i + 1) − i 3 3 n to prove that i =1 b) Evaluate the sum of the squares of the integers from 1 to 10. ∑i 1 2 = n(n + 1)( 2 n + 1) / 6 4. a) b) Consider the triangle formed by the lines y = 0, x = 0 and y = 1- x . Sketch the triangle and the solid of revolution formed by rotating it about the line x = 0. USE CALCULUS to show that the volume of the solid of revolution is π/3. Do not simply quote the volume formula. 5. a) b) c) 6. a) Consider the function f(x) = ln x. What is its natural domain and range. Where is the slope positive? Where is it concave down? Sketch the function. b) Prove that ln (ab) = ln a + ln b 7. a) Carefully sketch the curves y = ln x and y = exp(x) and the square 0 ≤ x ≤ 2, 0 ≤ y ≤ 2. The square should be divided into 3 pieces by the two curves. b) Show that Dx(xlnx - x) = ln x [Just use derivative rules, not the limit definition.] What is the value of a = ln[exp(-999.9)] and b = ln [1] Differentiate y = exp(-x3) Find the anti-derivative of f(x) = 1/(x lnx). You can assume x > 1. 2 Evaluate ∫ ln xdx - the area of one of the pieces of the square. c) 1 [Hint: Use part (b) and the (second) fundamental theorem of calculus.] d) Determine the areas of the other two pieces. 8. Consider the function f ( x) = 2 − x 2 / 2 , on the interval [0,2] a) Write out the Riemann sum R4(f) obtained by partitioning [0,2] into 4 equal subintervals and choosing sample points, xi = xi (i.e. the right hand end of the ith interval). b) The Riemann sum R4(f) approximates the area under f by four rectangles. Carefully sketch the graph of f together with these four rectangles. Is R4(f) larger or smaller than the 2 2 true area ∫ ( 2 − x / 2) dx ? Explain. 0 c) Using summation notation, write the Riemann sum Rn(f), for the partition, Pn, obtained by dividing [0,2] into n equal subintervals and choosing each xi = xi , and express 2 ∫ (2 − x 2 / 2) dx as a limit. (You do not have to evaluate this limit). 0 9. a) Simplify the expression exp(lnx2-ylnx) and evaluate it with x=e, y =1 b) Suppose that y = f(x) is continuous and strictly increasing on [0,1] and that f(0) = 0 and f(1) = 1. Draw a sketch to illustrate this curve on [0,1]. 1 If ∫ f (x )dx = 2 / 5 , find 0 10. 1 ∫f −1 ( y )dy , where f −1 is the inverse function of f. 0 Find the general antiderivatives, a) 1 ∫ (x + x ) 2 dx b) 1 ∫ (x + x )dx c) ∫ (cos x + sin x) 2 dx d) ∫ (sin x) 2 dx