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### Day22-1432class

Course: MATH 1432 11278, Spring 2012
School: U. Houston
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Word Count: 632

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1432 www.math.uh.edu/~pamb THINK: On Math each row place two letters that can be attached to the end of the word to the left and to the beginning of the word to the right to form a longer word in each case. When completed the six added letters will give a word reading downwards. THAT __ ORAL PRIM __ COVE __ HER OWL What is the word? POPPER 1. For what values of t does the curve given by the parametric...

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1432 www.math.uh.edu/~pamb THINK: On Math each row place two letters that can be attached to the end of the word to the left and to the beginning of the word to the right to form a longer word in each case. When completed the six added letters will give a word reading downwards. THAT __ ORAL PRIM __ COVE __ HER OWL What is the word? POPPER 1. For what values of t does the curve given by the parametric equations x = t 3 t 2 1 and y = t 4 + 2t 2 8t have a vertical tangent? a) 0 only b) 1 only 2 c) 0 and only 3 2 d) 0 , , and 1 3 2. Give the area of the region that is enclosed by the polar curve r = 1 + 2sin() and lies below the x-axis. a. 3 3 + 2 4 4 b. 3 + 2 2 c. 3 +2 d. 33 + 4 2 2 3. Give a parameterization for the line segment from (2, 1) to ( 4, 3). a. x(t) = 2 3t, y(t) = 1 + t, 0t1 b. x(t) = 4 + 6t, y(t) = 3 2t, 0t1 c. x(t) = 2 6t, 0t1 y(t) = 1 + 2t, d. x(t) = 4 + 3t, y(t) = 3 t, 0t1 4. Give the slope of the tangent line to the curve given parametrically by (cos(t), sin(2t)) at the point where t = /6. a. 2 b. 2 c. 3 d. 3 e. None of these. 5. Let L be the tangent line to the curve given parametrically by (cos(t), sin(2t)) at the point where t = /6. The line L strikes the y-axis at the point (0, a). a = ? 33 a. a = 4 b. a = something other than 33 4 Suppose a particle is traveling on a path represented by a portion of the xy plane. In this case, assume that the y-axis represents the up/down direction. Given this information, suppose the position of the particle is given at time t in seconds by r ( t ) = t , 16 t 2 + 4 0 t + 1 0 for t > 0 until the particle strikes the ( ) ground. Give the time when the particle strikes the ground, and determine the velocity and acceleration of the particle at this time. Arc Length and Speed Section 9.8 Velocity Speed and (magnitude of velocity) If the position of a particle at time t is given by r(t) = (x(t), y(t)) then the velocity is given by r(t) = v(t) = (x(t), y(t)) and the speed is given by speed = v ( t ) = x' ( t ) + y' ( t ) 2 2 A particle is traveling on an elliptic path in the xy-plane so that its position at time t is given by r ( t ) = ( 2 cos ( t ) ,3 sin ( t ) ) . 3 Give the position, velocity and speed of the particle at time t = /4. 2 1 3 3 2 1 1 1 2 3 4 2 3 4 If a curve C is given parametrically by (x(t), y(t)) for a t b and x(t) and y(t) are continuous functions, then the length of the curve is given by L (c) = b a x ' ( t ) + y ' ( t ) 2 2 dt Give an integral which represents the length of the curve given parametrically by r ( t ) = ( 2 cos ( t ) ,3 sin ( t ) ) for 0 t 2. 3 2 1 3 3 2 1 1 1 2 3 4 2 3 4 Give an integral which represents the length of the curve given parametrically by r ( t ) = ( 2 cos ( 3t ) ,3 sin ( 4t ) ) for 0 t 2. 3 2 1 3 3 2 1 1 1 2 3 4 2 3 4 Give a formula for the length of the curve given by the graph of f (x) = 2x +1 for 1 x 3. How can we find the length of the curve given by the graph of f (x) for a x b? L (c) = b a 1 + f ' ( x ) 2 dx Verify that the formula works for the previous example. Give a formula for the length of the curve given by the graph of f (x) = x2 for 1 x 1. How can we find the length of a polar curve? L (c) = ( ) + ' ( ) 2 2 d Summary: L (c) = b 1 + f ' ( x ) 2 dx Arc length (rectangular) x [a, b] a L (c) = b x ' ( t ) + y ' ( t ) dt Arc length (parametric) t [a, b] 2 2 a v( t ) = L (c) = x' ( t ) + y' ( t ) 2 ( ) + ' ( ) 2 2 2 d Speed (parametric) Arc length (polar) [, ]
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U. Houston - MATH 1432 - 11278
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U. Houston - MATH 1432 - 11278
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U. Houston - MATH 1432 - 11278
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U. Houston - MATH 1432 - 11278
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U. Houston - MATH 1432 - 11278
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U. Houston - MATH 1432 - 11278
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U. Houston - MATH 1432 - 11278
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U. Houston - MATH 1432 - 11278
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U. Houston - MATH 1432 - 11278
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U. Houston - MATH 1432 - 11278
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U. Houston - MATH 1432 - 11278
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U. Houston - MATH 1432 - 11278
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