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(c) 7. Use Simpson's Rule with n= 6 to estimate the length of the are of the curve with equations x = 1², y = 1², z = 1², 0≤1≤3.

(c) 7. Use Simpson's Rule with n= 6 to estimate the length of the are of the curve with equations x = 1², y = 1², z = 1², 0≤1≤3.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 39RE
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EXERCISES 1. (a) Sketch the curve with vector function r()+cos mrj + sin mrk (b) Find r'(t) and r"(t). 2. Let r(t)-(√2-7. (e' - 1)/t, In(r + 1)). (a) Find the domain of r. (b) Find lim, r(t). (c) Find r'(1). 1-0 3. Find a vector function that represents the curve of intersec- tion of the cylinder x² + y² -16 and the plane x +-5. 4. Find parametric equations for the tangent line to the curve x=2 sin t, y=2 sin 2r, 2 sin 3r at the point (1.3, 2). Graph the curve and the tangent line on a com- mon screen. 5. If r(t)=1²i+r cos mrj + sin mrk, evaluate fr(t) dr. 6. Let C be the curve with equations x=2 r.y=2r - 1, = Int. Find (a) the point where C intersects the xz-plane, (b) parametric equations of the tangent line at (1, 1, 0), and (c) an equation of the normal plane to Cat (1, 1, 0). 7. Use Simpson's Rule with n= 6 to estimate the length of the arc of the curve with equations x = 1², y = 1³, z = 1¹, 0≤1≤3. 8. Find the length of the curve r(t) = (213/2, cos 2r, sin 2r), 0≤t≤ 1. 9. The helix r(t) = cos ti+ sin tj+ k intersects the curve r₂(t) = (1 + 1)i +2j+r³k at the point (1, 0, 0). Find the angle of intersection of these curves. 10. Reparametrize the curve r(t) = e'i + e' sin tj + e' cost k with respect to arc length measured from the point (1, 0, 1) in the direction of increasing t. (sin³t, cos³t, sin²t), 11. For the curve given by r(t) 0 ≤ ≤ 7/2, find (a) the unit tangent vector, (b) the unit normal vector, (c) the unit binormal vector, and (d) the curvature. 16. The figure shows the curve C traced b tion vector r() at time 1. (a) Draw a vector that represents the particle over the time interval 3- (b) Write an expression for the velo (c) Write an expression for the unit draw it. 12. Find the curvature of the ellipse x = 3 cos t, y = 4 sint at the points (3, 0) and (0,4). 13. Find the curvature of the curve y = x4 at the point (1, 1). 14. Find an equation of the osculating circle of the curve y=x²-x² at the origin. Graph both the curve and its osculating circle. lating plane of the curve 1 0 r(3) (3.2) 1200 1 17. A particle moves with position r(t)= t Inti+13+k. F acceleration of the particle. 18. Find the velocity, speed, and ing with position function r( the path of the particle and d acceleration vectors for = 19. A particle starts at the origin i-j+ 3k. Its acceleration Find its position function. 20. An athlete throws a shot at at an initial speed of 43 ft the ground. (a) Where is the shot 2 s (b) How high does the s (c) Where does the shot. 21. A projectile is launched from the floor of a tunne angle of elevation shou possible horizontal ran maximum range? 22. Find the tangential and tion vector of a partic! r(t) = 23. A disk of radius 1 is direction at a constar the center of the dis
Transcribed Image Text:EXERCISES 1. (a) Sketch the curve with vector function r()+cos mrj + sin mrk (b) Find r'(t) and r"(t). 2. Let r(t)-(√2-7. (e' - 1)/t, In(r + 1)). (a) Find the domain of r. (b) Find lim, r(t). (c) Find r'(1). 1-0 3. Find a vector function that represents the curve of intersec- tion of the cylinder x² + y² -16 and the plane x +-5. 4. Find parametric equations for the tangent line to the curve x=2 sin t, y=2 sin 2r, 2 sin 3r at the point (1.3, 2). Graph the curve and the tangent line on a com- mon screen. 5. If r(t)=1²i+r cos mrj + sin mrk, evaluate fr(t) dr. 6. Let C be the curve with equations x=2 r.y=2r - 1, = Int. Find (a) the point where C intersects the xz-plane, (b) parametric equations of the tangent line at (1, 1, 0), and (c) an equation of the normal plane to Cat (1, 1, 0). 7. Use Simpson's Rule with n= 6 to estimate the length of the arc of the curve with equations x = 1², y = 1³, z = 1¹, 0≤1≤3. 8. Find the length of the curve r(t) = (213/2, cos 2r, sin 2r), 0≤t≤ 1. 9. The helix r(t) = cos ti+ sin tj+ k intersects the curve r₂(t) = (1 + 1)i +2j+r³k at the point (1, 0, 0). Find the angle of intersection of these curves. 10. Reparametrize the curve r(t) = e'i + e' sin tj + e' cost k with respect to arc length measured from the point (1, 0, 1) in the direction of increasing t. (sin³t, cos³t, sin²t), 11. For the curve given by r(t) 0 ≤ ≤ 7/2, find (a) the unit tangent vector, (b) the unit normal vector, (c) the unit binormal vector, and (d) the curvature. 16. The figure shows the curve C traced b tion vector r() at time 1. (a) Draw a vector that represents the particle over the time interval 3- (b) Write an expression for the velo (c) Write an expression for the unit draw it. 12. Find the curvature of the ellipse x = 3 cos t, y = 4 sint at the points (3, 0) and (0,4). 13. Find the curvature of the curve y = x4 at the point (1, 1). 14. Find an equation of the osculating circle of the curve y=x²-x² at the origin. Graph both the curve and its osculating circle. lating plane of the curve 1 0 r(3) (3.2) 1200 1 17. A particle moves with position r(t)= t Inti+13+k. F acceleration of the particle. 18. Find the velocity, speed, and ing with position function r( the path of the particle and d acceleration vectors for = 19. A particle starts at the origin i-j+ 3k. Its acceleration Find its position function. 20. An athlete throws a shot at at an initial speed of 43 ft the ground. (a) Where is the shot 2 s (b) How high does the s (c) Where does the shot. 21. A projectile is launched from the floor of a tunne angle of elevation shou possible horizontal ran maximum range? 22. Find the tangential and tion vector of a partic! r(t) = 23. A disk of radius 1 is direction at a constar the center of the dis
Formula b Ax [*f(x)x² = ª (ƒ(²0) ƒ(x)x' 3 a +4ƒ(x₁) +2ƒ(x₂) + ··· +4ƒ(xn−1) + f(xn)) b-a where Ax n and x = a +iAx n = subdivisions of the function, it is an even number a = point at the start of the function graph b = point at the end of the function graph People also ask What is the formula of Simpson's one rule? What is Simpson's 1 3 rule formula? Which is the 2nd rule Simpson's formula? What does Simpson's 1 3 rd and 3 8 th rule mean? Q simpson rule formula — Private >
Transcribed Image Text:Formula b Ax [*f(x)x² = ª (ƒ(²0) ƒ(x)x' 3 a +4ƒ(x₁) +2ƒ(x₂) + ··· +4ƒ(xn−1) + f(xn)) b-a where Ax n and x = a +iAx n = subdivisions of the function, it is an even number a = point at the start of the function graph b = point at the end of the function graph People also ask What is the formula of Simpson's one rule? What is Simpson's 1 3 rule formula? Which is the 2nd rule Simpson's formula? What does Simpson's 1 3 rd and 3 8 th rule mean? Q simpson rule formula — Private >
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