Practice Exam 2 Solved - Create assignment, 57950, Exam 2,...

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Create assignment, 57950, Exam 2, Oct 25 at 1:12 pm 1 This print-out should have 28 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. The due time is Central time. CalC3g06a 49:05, calculus3, multiple choice, > 1 min, wording-variable. 001 Determine d 2 y/dx 2 when x 2 + y 2 =2 . 1. d 2 y dx 2 = 2 y 3 correct 2. d 2 y dx 2 = 2 y 3 3. d 2 y dx 2 = 2 y 2 4. d 2 y dx 2 = 2 y 2 5. d 2 y dx 2 = 1 y 3 Explanation: Di±erentiating implicitly with respect to x we see that 2 x +2 y dy dx =0 , so dy dx = x y . But then d 2 y dx 2 = d dx ³ x y ´ = µ y x dy dx y 2 = 1 y 2 ³ y + x 2 y ´ . Thus d 2 y dx 2 = 1 y 3 ³ x 2 + y 2 ´ = 2 y 3 . CalC3g16c 49:05, calculus3, multiple choice, < 1 min, wording-variable. 002 IF x = x ( t ) is defned implicitly by 5 e 3 x =3 tx +5 t, fnd the value oF x 0 ( t )at(1 , 0) . 1. x 0 = 5 12 2. x 0 = 1 3 3. x 0 = 5 18 4. x 0 = 1 2 5. x 0 = 5 12 correct Explanation: Di±erentiating 5 e 3 x 3 tx 5 t implicitly with respect to t we see that 15 e 3 x x 0 3( tx 0 + x ) 5=0 . Thus x 0 = 3 x 15 e 3 x 3 t , so at (1 , 0) x 0 = 5 12 . CalC3g52a 49:05, calculus3, multiple choice, > 1 min, wording-variable. 003
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Create assignment, 57950, Exam 2, Oct 25 at 1:12 pm 2 The points P and Q on the graph of y 2 xy +6=0 have the same x -coordinate x = 5. Find the point of intersection of the tangents to the graph at P and Q . 1. intersect at = ³ 12 5 , 24 5 ´ 2. intersect at = ³ 12 5 , 12 5 ´ 3. intersect at = ³ 24 5 , 12 5 ´ correct 4. intersect at = ³ 24 5 , 24 5 ´ 5. intersect at = ³ 12 5 , 4 5 ´ Explanation: The y -coordinate at P, Q will be the solu- tions of ( ) y 2 xy at x =5, i.e. , the solutions of y 2 5 y +6=( y 3)( y 2) = 0 . Thus P =(5 , 3) ,Q , 2) . To determine the tangent lines we need also the value of the derivative at P and Q . But by implicit di±erentiation, 2 y dy dx x dy dx y =0 . so dy dx = y 2 y x . Thus dy dx ¯ ¯ ¯ P =3 , dy dx ¯ ¯ ¯ Q = 2 . By the point-slope formula, therefore, the equation of the tangent line at P is y 3=3 ( x 5) , while that at Q is y 2= 2( x 5) . Consequently, the tangent lines at P and Q are y 3 x = 12 and y +2 x =1 2 respectively. These two tangent lines intersect at = ³ 24 5 , 12 5 ´ . StewartC5 03 07 37 49:05, calculus3, multiple choice, > 1 min, wording-variable. 004 Find an equation of the tangent line to the ellipse x 2 a 2 + y 2 b 2 at the point ( x o ,y o ). 1. x o x a 2 + y o y b 2 correct 2. x o x a 2 y o y b 2 = 1 3. x a 2 y b 2 = x o y o 4. x a 2 + y b 2 = 1 5. x o x a 2 + y o y b 2 = x o y o Explanation: x 2 a 2 + y 2 b 2 2 x a 2 + 2 yy 0 b 2 y 0 = b 2 x a 2 y
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Create assignment, 57950, Exam 2, Oct 25 at 1:12 pm 3 An equation of the tangent line at ( x o ,y o )is y y o = b 2 x o a 2 y o ( x x o ) Multiplying both sides by y o b 2 gives y o y b 2 y 2 o b 2 = x o x a 2 + x 2 o a 2 x o x a 2 + y o y b 2 = x 2 o a 2 + y 2 o b 2 Since ( x o o ) lies on the ellipse, we have x 2 o a 2 + y 2 o b 2 =1 x o x a 2 + y o y b 2 CalC3h01b 49:05, calculus3, multiple choice, > 1 min, wording-variable.
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This note was uploaded on 02/28/2010 for the course M 56200 taught by Professor Radin during the Fall '09 term at University of Texas at Austin.

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Practice Exam 2 Solved - Create assignment, 57950, Exam 2,...

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