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MÄLARDALEN UNIVERSITY School of Education, Culture and Communication Department of Applied Mathematics Examiner: Lars-Göran Larsson EXAMINATION IN MATHEMATICS MAA151 Single Variable Calculus, TEN1 Date: 2015-01-08 Write time: 3 hours Aid: Writing materials This examination is intended for the examination part TEN1. The examination consists of eight randomly ordered problems each of which is worth at maximum 3 points. The pass-marks 3, 4 and 5 require a minimum of 11, 16 and 21 points respectively. The minimum points for the ECTS-marks E, D, C, B and A are 11, 13, 16, 20 and 23 respectively. If the obtained sum of points is denoted S1 , and that obtained at examination TEN2 S2 , the mark for a completed course is according to the following S1 ≥ 11, S2 ≥ 9 and S1 + 2S2 ≤ 41 → 3 S1 ≥ 11, S2 ≥ 9 and 42 ≤ S1 + 2S2 ≤ 53 → 4 54 ≤ S1 + 2S2 → 5 S1 ≥ 11, S2 ≥ 9 and S1 + 2S2 ≤ 32 → E S1 ≥ 11, S2 ≥ 9 and 33 ≤ S1 + 2S2 ≤ 41 → D S1 ≥ 11, S2 ≥ 9 and 42 ≤ S1 + 2S2 ≤ 51 → C 52 ≤ S1 + 2S2 ≤ 60 → B 61 ≤ S1 + 2S2 → A Solutions are supposed to include rigorous justifications and clear answers. All sheets of solutions must be sorted in the order the problems are given in. 1. The sum of two non-negative numbers is 12. Which are the numbers if the product of the first and the square of the second is a maximum? Prove your conclusion. 2. Find the area of the region precisely enclosed by the three curves y = ln(x), 3. y = 0 and x = e . Find the inverse of the function f defined by f (x) = x+2 . x−5 Especially, state the domain and the range of the inverse. 4. For which a converges the series a + 4a3 + 16a5 + . . . ? Find the limit for these a. 5. Find the function f which is an antiderivative of tan (tangent) and which satisfies f (0) = 1. 6. Solve the initial value problem 7. Determine whether √ xy 0 + 12 y = lim x→0 √ − √x xe , y(4) = 0 . ln(1 + 2x) 1 − e3x exists or not. If the answer is no: Give an explanation of why! If the answer is yes: Give an explanation of why and find the limit! 8. 3 Let f (t) = t + t and g(t) = 4 arctan(t). Find an equation for the tangent line x = f (t) to the curve at the point for which the y-coordinate is equal to π. y = g(t)
