Prove Formula 6 of Theorem 3.
> Graph the curve with parametric equations x = (1 + cos 16t) cos t y = (1 + cos 16t) sin t z = 1 + cos 16t Explain the appearance of the graph by showing that it lies on a cone.
> Graph the curve with parametric equations x = sin t, y = sin 2t, z = cos 4t. Explain its shape by graphing its projections onto the three coordinate planes.
> Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x = e-t cos t, y = e-t sin t, z = e-t; (1, 0, 1)
> Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x = ln (t + 1), y = t cos 2t, z = 2t, (0, 0, 1)
> Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. x = t? + 1, y= 4/t, z = e“-'; (2,4, 1) %3D
> At what points does the curve r(t) = t i + (2t - t2) k intersect the paraboloid z = x2 + y2?
> Find three different surfaces that contain the curve r(t) = t2 i + ln t j + s(1/t) k.
> Find three different surfaces that contain the curve r(td) = 2t i + et j + e2t k.
> Show that the curve with parametric equations x = sin t, y = cos t, z = sin2t is the curve of intersection of the surfaces z = x2 and x2 + y2 = 1. Use this fact to help sketch the curve.
> Show that the curve with parametric equations x = t cos t, y = t sin t, z = t lies on the cone z2 = x2 + y2, and use this fact to help sketch the curve.
> Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices. x = cos2 t, y = sin2 t, z = t ZA
> Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices. x = cos 8t, y = sin 8t, z = e0.8t, t > 0 ZA
> Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices. x = cos t, y = sin t, z = cos 2t ZA
> Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices. x = t, y = 1 / (1 + t2), z = t2 ZA II
> Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices. x = cos t, y = sin t, z = 1/ (1 + t2) ZA -y
> Match the parametric equations with the graphs (labeled I–VI). Give reasons for your choices. x = t cos t, y = t, z = t sin t, t > 0 ZA
> (a). Find the arc length function for the curve measured from the point P in the direction of increasing t and then reparametrize the curve with respect to arc length starting from P, and (b). find the point 4 units along the curve (in the direction of
> (a). Find the arc length function for the curve measured from the point P in the direction of increasing t and then reparametrize the curve with respect to arc length starting from P, and (b). find the point 4 units along the curve (in the direction of
> Find the derivative of the vector function. r (t) = sin2at i + tebt j + cos2ct k
> Find a vector equation and parametric equations for the line segment that joins P to Q. P (a, b, c), Q (u, v, w)
> Find a vector equation and parametric equations for the line segment that joins P to Q. P (0, -1, 1), Q ( 1 2 , 1 3 , 1 4 )
> Find a vector equation and parametric equations for the line segment that joins P to Q. P (-1, 2, -2), Q (-3, 5, 1)
> Find a vector equation and parametric equations for the line segment that joins P to Q. P (2, 0, 0), Q (6, 2, -2)
> Find the limit. lim ( te, 1 + t ,t sin 2t3 –
> Find the limit. 1 + t? lim -e-21 1- t2' tant, t
> Find the limit. -i+ /t + 8 j + t sin rt k In t lim
> Find the limit. 12 j+ cos 2t k sin't lim (e-3'i
> Find the derivative of the vector function. r (t) = t sin t i + et cos t j + sin t cos t k
> Find the domain of the vector function. 1 k t - 2 r(t) = cos ti + In tj +
> Find the domain of the vector function. г() — ( In(r + 1), - 2' V9 – t²
> Use the formula in Exercise 63(d) to find the torsion of the curve Exercise 63(d): (r' X r") · r" |r' X r"| (d) т r() = (. 늘?. ).
> Use the Frenet-Serret formulas to prove each of the following. (Primes denote derivatives with respect to t. Start as in the proof of Theorem 10.) (a) r" = s"T + K(s')²N (b) r' X r" = K(s')'B (c) r™ = [s" – k°(s')*]T + [3xs's" + k'(s')*]N + KT(s')°B
> Show that the curvature is related to the tangent and normal vectors by the equation dT KN ds
> Find the vectors T, N, and B, at the given point. r(t) = (r?.3r", t), (1.7, 1)
> Use the formula in Exercise 42 to find the curvature. Formula in Exercise 42: x = et cos t, y = et sin t |ty – yx|| K = [t? + y? ]/2
> Use the formula in Exercise 42 to find the curvature. Formula in Exercise 42: x = a cos wt, y = b sin wt |ty – yx|| K = [t? + y? ]/2
> Use the formula in Exercise 42 to find the curvature. Formula in Exercise 42: x = t2, y = t3 |ty – yx|| K = [t? + y? ]/2
> Use Theorem 10 to show that the curvature of a plane parametric curve x = f (t), y = g (f) is where the dots indicate derivatives with respect to t. |ty – yx|| K = [t? + y? ]/2
> Use a graphing calculator or computer to graph both the curve and its curvature function k (x) on the same screen. Is the graph of what you would expect? y = x-2
> Use a graphing calculator or computer to graph both the curve and its curvature function k (x) on the same screen. Is the graph of what you would expect? y = x4 - 2x2
> If u(t) = r(t) ∙ [r'(t) × r''(t)], show that u'(t) – r(t) ∙ [r'(t) × r'''(t)]
> If a curve has the property that the position vector r(t) is always perpendicular to the tangent vector r'(t), show that the curve lies on a sphere with center the origin.
> Show that if r is a vector function such that r'' exists, then d - [r() X г()] — г() X г"() dt
> Find an equation of a parabola that has curvature 4 at the origin.
> At what point does the curve have maximum curvature? What happens to the curvature as x →∞ ? y = ex
> Use Formula 11 to find the curvature. y = xex
> Use Formula 11 to find the curvature. y = tan x
> Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases. r(t) = cos t i - cos t j + sin t k
> Use Formula 11 to find the curvature. y = x4
> Prove Formula 3 of Theorem 3.
> Prove Formula 1 of Theorem 3.
> Evaluate the integral. t 1 k) dt 1 - t j+ V1 – 12
> Evaluate the integral. (sec?t i + t(t? + 1)°j + t² In t k) dt
> Evaluate the integral. /
> Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases. r(t) = t2 i + t4 j + t6 k
> Evaluate the integral. 1 i + t + 1 j+ t2 + 1 k dt 1² + 1
> Evaluate the integral. S (22 i+ (t + 1)/F k) đt 3/2
> Evaluate the integral. S (ti - r'j+ 3r°k) dt Jo
> Use Theorem 10 to find the curvature. r(t) = 6 t2 i + 2t j + 2t3 k
> Use Theorem 10 to find the curvature. r(t) = t i + t2 j + et k
> Use Theorem 10 to find the curvature. r(t) = t3 j + t2 k
> (a). Find the unit tangent and unit normal vectors T(t) and N(t). (b). Use Formula 9 to find the curvature. r(t) = (r, }r°, r²)
> Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Illustrate by graphing both the curve and the tangent line on a common screen. x = 2 cos t, y = 2 sin t, z = 4 cos 2t; ( 3 , 1, 2)
> Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. Illustrate by graphing both the curve and the tangent line on a common screen. x = t, y = e-t, z = 2t - t2; (0, 1, 0)
> Sketch the curve with the given vector equation. Indicate with an arrow the direction in which t increases. r(t) = 2 cos t i + 2 sin t j + k
> (a). Find the unit tangent and unit normal vectors T(t) and N(t). (b). Use Formula 9 to find the curvature. r(t) = (/2t, e', e-)
> Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. /t2 + 3, y= ln(t² + 3), z = t; (2, In 4, 1) X =
> Suppose you start at the point (0, 0, 3) and move 5 units along the curve x = 3 sin t, y = 4t, z = 3 cos t in the positive direction. Where are you now?
> Find the unit tangent vector T (t) at the point with the given value of the parameter t. r (t) = cos t i + 3t j + 2 sin 2t k, t = 0
> Let C be the curve of intersection of the parabolic cylinder x2 = 2y and the surface 3z = xy. Find the exact length of C from the origin to the point (6, 18, 36).
> Find the unit tangent vector T (t) at the point with the given value of the parameter t. r(t) = (r² – 21, 1 + 3t, + ), t= 2
> Find the derivative of the vector function. r (t) = t a × (b + t c)
> Find the derivative of the vector function. r (t) = a + t b + t2 c
> Find the length of the curve. r(t) = t2 i + 9t j + 4t3/2 k, 1 < t < 4
> Find the derivative of the vector function. 1 r(t) 1 +t -i + 1 + t j + k 1 + t
> Find the derivative of the vector function. r (t) = t2 i + cos (t2) j + sin2t k
> Find the derivative of the vector function. r(t) = (Vi – 2,3, 1/r²)
> The figure shows a curve C given by a vector function r (t). (a). Draw the vectors r (4.5) – r (4) and r (4.2) – r (4). (b). Draw the vectors (c). Write expressions for r' (4) and the unit tangent vector T (4). (d).
> Find parametric equations and symmetric equations for the line. The line through (-6, 2, 3) and parallel to the line 1 2 x – 1 3 y = z + 1
> If a = i - 2k and b = j + k, find a × b. Sketch a, b, and a × b as vectors starting at the origin.
> Find the cross product a × b and verify that it is orthogonal to both a and b. a = t i + cos t j + sin tk, b = i - sin t j + cos tk
> Find the cross product a × b and verify that it is orthogonal to both a and b. a = 1 2 i + 1 3 j + 1 4 k, b = i + 2 j – 3k
> Show that |a × b |2 = |a |2 | b |2 - (a ∙ b)2.
> (a). Let P be a point not on the line L that passes through the points Q and R. Show that the distance d from the point P to the line L is (b). Use the formula in part (a) to find the distance from the point P (1, 1, 1) to the line through Q (0, 6, 8)
> (a). Find all vectors v such that 1, 2, 1 × v = 3, 1, −5 (b). Explain why there is no vector v such that 1, 2, 1 × v = 3, 1, 5
> Find the unit vectors that are parallel to the tangent line to the parabola y − x2 at the point (2, 4).
> If a ∙ b = 3 and a × b = 1, 2, 2 , find the angle between a and b.
> A wrench 30 cm long lies along the positive y-axis and grips a bolt at the origin. A force is applied in the direction 0, 3, −4 at the end of the wrench. Find the magnitude of the force needed to supply 100 N ∙ m of torque to the bolt.
> Find the cross product a × b and verify that it is orthogonal to both a and b. a = 3i + 3j - 3k, b = 3i - 3j + 3k
> A particle moves according to a law of motion s = f(t), t ≥ 0, where t is measured in seconds and s in feet. (a) Find the velocity at time t. (b) What is the velocity after 1 second? (c) When is the particle at rest? (d) When is the part
> Use the scalar triple product to determine whether the points A (1, 3, 2), B (3, -1, 6), C (5, 2, 0), and D (3, 6, -4) lie in the same plane.
> Use the scalar triple product to verify that the vectors u = i + 5 j - 2 k, v = 3i - j, and w = 5i + 9 j - 4 k are coplanar.
> Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. P (3, 0, 1), Q (-1, 2, 5), R (5, 1, -1), S (0, 4, 2)
> Find the volume of the parallelepiped with adjacent edges PQ, PR, and PS. P (-2, 1, 0), Q (2, 3, 2), R (1, 4, -1), S (3, 6, 1)
> Find the volume of the parallelepiped determined by the vectors a, b, and c. a = i + j, b = j + k, c = i + j + k
> Find the volume of the parallelepiped determined by the vectors a, b, and c. a = 1, 2, 3 , b = −1, 1, 2 , c = 2, 1, 4
> A crane suspends a 500-lb steel beam horizontally by support cables (with negligible weight) attached from a hook to each end of the beam. The support cables each make an angle of 60° with the beam. Find the tension vector in each support cabl
> (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and (b) find the area of triangle PQR. P (2, -3, 4), Q (-1, -2, 2), R (3, 1, -3)
> (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and (b) find the area of triangle PQR. P (0, -2, 0), Q (4, 1, -2), R (5, 3, 1)
> (a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and (b) find the area of triangle PQR. P (0, 0, -3), Q (4, 2, 0), R (3, 3, 1)
