Test 1 - Gauthier Joseph – Exam 1 – Due Oct 2 2007...

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Unformatted text preview: Gauthier, Joseph – Exam 1 – Due: Oct 2 2007, 11:00 pm – Inst: JEGilbert 1 This print-out should have 18 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. The due time is Central time. 001 (part 1 of 1) 10 points Locate the points given in polar coordinates by P ‡ 4 , 5 6 π · , Q ‡ 1 , 1 2 π · R ‡ 4 , 1 3 π · , among 2 4- 2- 4 2 4- 2- 4 1. P : Q : R : 2. P : Q : R : correct 3. P : Q : R : 4. P : Q : R : 5. P : Q : R : 6. P : Q : R : Explanation: To convert from polar coordinates to Carte- sian coordinates we use x = r cos θ , y = r sin θ . For then the points P ‡ 4 , 5 6 π · , Q ‡ 1 , 1 2 π · R ‡ 4 , 1 3 π · , correspond to P : Q : R : in Cartesian coordinates. keywords: polar coordinates, Cartesian coor- dinates, change of coordinates, 002 (part 1 of 1) 10 points Find the vector v having a representation by the directed line segment--→ AB with respect to points A (- 1 , 1) and B (3 , 4). 1. v = h 4 ,- 3 i 2. v = h 2 ,- 5 i 3. v = h- 2 , 5 i 4. v = h 2 , 5 i 5. v = h- 4 , 3 i 6. v = h 4 , 3 i correct Explanation: Since--→ AB = h 3 + 1 , 4- 1 i , we see that v = h 4 , 3 i . keywords: vectors, directed line segment, 003 (part 1 of 1) 10 points Which one of the following could be the graph of the curve given parametrically by x ( t ) = t 3- 1 , y ( t ) = 2- t 2 ? Gauthier, Joseph – Exam 1 – Due: Oct 2 2007, 11:00 pm – Inst: JEGilbert 2 1. 2. 3. 4. 5. correct 6. Explanation: These examples illustrate the diversity of curves in 2-space. But simple properties such as (i) behaviour as t → ∞ , (ii) x and y-intercepts, (iii) passing through the origin or not, (iv) symmetry (even or oddness), can often be used to determine which graph goes with which function. For instance, three of the graphs above lie within a square cen- tered at the origin, suggesting that the other three are unbounded; on the other hand, only three pass through the origin; and some, but not all, have various symmetries such as even- ness, oddness, or symmetry about a line. In the case of the curve given parametrically by x ( t ) = t 3- 1 , y ( t ) = 2- t 2 , we see that lim t →∞ x ( t ) = ∞ lim t →∞ y ( t ) =-∞ . But only one of the graphs has these proper- ties, leaving as the only possible graph. Gauthier, Joseph – Exam 1 – Due: Oct 2 2007, 11:00 pm – Inst: JEGilbert 3 keywords: 2D-graph, parametric function, limit at infinity, symmetry, periodicity 004 (part 1 of 1) 10 points Find the slope of the tangent line to the graph of r = cos2 θ at θ = π/ 4. 1. slope =- 1 2. slope = 1 correct 3. slope = 7 3 √ 3 4. slope = 5 3 √ 3 5. slope = 1 5 √ 3 6. slope = 1 7 √ 3 Explanation: The graph of a polar curve r = f ( θ ) can expressed by the parametric equations x = f ( θ )cos θ , y = f ( θ )sin θ ....
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Test 1 - Gauthier Joseph – Exam 1 – Due Oct 2 2007...

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