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 equations of the normal plane and osculating plane of the curve at the given point. x = ln t, y = 2t, z = t2; (0, 2, 1)
> Find the vectors T, N, and B, at the given point. r(t) = (cos t, sin t, In cos t), (1,0, 0)
> Consider the curvature at x = 0 for each member of the family of functions f (x) = ecx. For which members is k (0) largest?
> Two graphs, a and b, are shown. One is a curve y = f (x) and the other is the graph of its curvature function y = k (x). Identify each curve and explain your choices. y. a b
> Two graphs, a and b, are shown. One is a curve y = f (x) and the other is the graph of its curvature function y = k (x). Identify each curve and explain your choices. y b
> Plot the space curve and its curvature function k (t). Comment on how the curvature reflects the shape of the curve. r(t) = (te', e*, /Zt), -5<t<5
> Plot the space curve and its curvature function k (t). Comment on how the curvature reflects the shape of the curve. r(t) = (t – sin t, 1 – cos t, 4 cos(t/2)), 0<ts 87 %3D
> (a). Is the curvature of the curve C shown in the figure greater at P or at Q? Explain. (b). Estimate the curvature at P and at Q by sketching the osculating circles at those points. у. P
> At what point does the curve have maximum curvature? What happens to the curvature as x →∞ ? y = ln x
> Let’s consider the problem of designing a railroad track to make a smooth transition between sections of straight track. Existing track along the negative x-axis is to be joined smoothly to a track along the line y = 1 for x > 1. (a)
> The DNA molecule has the shape of a double helix (see Figure 3 on page 850). The radius of each helix is about 10 angstroms (1 Ã… = 10-8 cm). Each helix rises about 34 Ã… during each complete turn, and there are about 2.9 Ã
> Find the curvature and torsion of the curve x = sin h t, y = cos h t, z = t at the point (0, 1, 0).
> Graph the curve with parametric equations x = cos t, y = sin t, z = sin 5t and find the curvature at the point (1, 0, 0).
> Reparametrize the curve with respect to arc length measured from the point (1, 0) in the direction of increasing t. Express the reparametrization in its simplest form. What can you conclude about the curve? 2 2t r(t): i + t2 + 1 t2 + 1
> Show that the tangent vector to a curve defined by a vector function r(t) points in the direction of increasing t. [Hint: Refer to Figure 1 and consider the cases h > 0 and h Figure 1: r(t+h)– r(t) r(t+h) – r(t) h r'(t) P. /r(t) r(t+h) r(t) r(t+
> Prove Formula 5 of Theorem 3.
> Find r (t) if r' (t) = t i + e' t j + te' k and r (0) = i + j + k.
> Suppose u and v are vector functions that possess limits as t → a and let c be a constant. Prove the following properties of limits. (a) lim [u(t) + v(t)] = lim u(t) + lim v(t) a a (b) lim cu(t) = c lim u(t) a (c) lim [u(t) • v(t)]
> (a). Graph the curve with parametric equations (b). Show that the curve lies on the hyperboloid of one sheet 144x2 + 144y2 - 25z2 = 100. 27 x = sin 8t – sin 181 8 y = -% cos 8t + 39 cos 18t z = * sin 5t
> Two particles travel along the space curves Do the particles collide? Do their paths intersect? r, (1) = (t, t°, t³) r2(t) = (1 + 2t, 1 + 6t, 1 + 14t)
> If two objects travel through space along two different curves, it’s often important to know whether they will collide. (Will a missile hit its moving target? Will two aircraft collide?) The curves might intersect, but we need to know w
> Try to sketch by hand the curve of intersection of the circular cylinder x2 + y2 = 4 and the parabolic cylinder z = x2. Then find parametric equations for this curve and use these equations and a computer to graph the curve.
> Find a vector function that represents the curve of intersection of the two surfaces. The semiellipsoid x2 + y2 + 4z2 = 4, y > 0, and the cylinder x2 + z2 = 1
> Find a vector function that represents the curve of intersection of the two surfaces. The hyperboloid z = x2 - y2 and the cylinder x2 + y2 = 1
> Find a vector function that represents the curve of intersection of the two surfaces. The paraboloid z = 4x2 + y2 and the parabolic cylinder y = x2
> Find a vector equation for the tangent line to the curve of intersection of the cylinders x2 + y2 = 25 and y2 + z2 = 20 at the point (3, 4, 2).
> Find a vector function that represents the curve of intersection of the two surfaces. The cylinder x2 + y2 = 4 and the surface z = xy
> Show that the curve with parametric equations x = t2, y = 1 - 3t, z = 1 + t3 passes through the points (1, 4, 0) and (9, -8, 28) but not through the point (4, 7, -6).
> Graph the curve with parametric equations Explain the appearance of the graph by showing that it lies on a sphere. x = V1 – 0.25 cos? 10t cos t y = v1 – 0.25 cos² 10t sin t z = 0.5 cos 10t
> 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 (-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 6 of Theorem 3.
> 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
