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43-46. Arc length Find the arc length of the following curves. 12. 4V2 3/2, 2t ), for 1 < t < 3 3 for 1

43-46. Arc length Find the arc length of the following curves. 12. 4V2 3/2, 2t ), for 1 < t < 3 3 for 1

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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#46
a. Determine the initial speed of the basketball. b. Find the initial velocity v(0) at the moment it is released. c. Find the position function r(t) of the center of the basketball t seconds after th r(0) = (0, 8). d. Find the distance s(t) between the center of the basketball and the front of th after the ball is released. Assume the diameter of the basketball hoop is 18 in e. Determine the closest distance (in inches) between the center of the basketb basketball hoop. f. Is the assumption that the basketball does not hit the front of the hoop valid of a women's basketball is about 9.23 inches. (Hint: The ball hits the front o distance from the center of the ball to the front of the hoop is less than the 43-46. Arc length Find the arc length of the following curves. 43. r(t) = (t, V2;3/2, 2t ), for 1<t< 3 3 44. r(t) = (2t/2, t³), for 0 <t < 2 45. Tr(t) = (sin t, t+ cos t, 4t), for 0 < t < 2 46. r(t) = (t, In sec t, In (sec t+ tan t)), for 0 <t < 4 47. Velocity and trajectory length The acceleration of a wayward firework is given 0<t< 3. Suppose the initial velocity of the firework is v(0) = i. a. Find the velocity of the firework, for 0 <t< 3. b. Find the length of the trajectory of the firework over the interval 0 <t< 48-49. Arc length parameterization Find a description of the following curves that u 48. r(t) = (1+ 4t) i – 3t j, for t > 1 4/2, 49. r(t) = ( t2, 13/2, 2t 3 for t >0 50. Tangents and normals for an ellipse Consider the ellipse r(t) = (3 cos a. Find the tangent vector r', the unit tangent vector T, and the principal the curve. b. At what points does r' have maximum and minimum values? C. At what points does the curvature have maximum and minimum value (b). d Find the
Transcribed Image Text:a. Determine the initial speed of the basketball. b. Find the initial velocity v(0) at the moment it is released. c. Find the position function r(t) of the center of the basketball t seconds after th r(0) = (0, 8). d. Find the distance s(t) between the center of the basketball and the front of th after the ball is released. Assume the diameter of the basketball hoop is 18 in e. Determine the closest distance (in inches) between the center of the basketb basketball hoop. f. Is the assumption that the basketball does not hit the front of the hoop valid of a women's basketball is about 9.23 inches. (Hint: The ball hits the front o distance from the center of the ball to the front of the hoop is less than the 43-46. Arc length Find the arc length of the following curves. 43. r(t) = (t, V2;3/2, 2t ), for 1<t< 3 3 44. r(t) = (2t/2, t³), for 0 <t < 2 45. Tr(t) = (sin t, t+ cos t, 4t), for 0 < t < 2 46. r(t) = (t, In sec t, In (sec t+ tan t)), for 0 <t < 4 47. Velocity and trajectory length The acceleration of a wayward firework is given 0<t< 3. Suppose the initial velocity of the firework is v(0) = i. a. Find the velocity of the firework, for 0 <t< 3. b. Find the length of the trajectory of the firework over the interval 0 <t< 48-49. Arc length parameterization Find a description of the following curves that u 48. r(t) = (1+ 4t) i – 3t j, for t > 1 4/2, 49. r(t) = ( t2, 13/2, 2t 3 for t >0 50. Tangents and normals for an ellipse Consider the ellipse r(t) = (3 cos a. Find the tangent vector r', the unit tangent vector T, and the principal the curve. b. At what points does r' have maximum and minimum values? C. At what points does the curvature have maximum and minimum value (b). d Find the
Expert Solution
Step 1

(46).

The given curve is:

r(t)=t, lnsec t, lnsec t+tan t ;          0≤t≤π4

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