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12.4E: Exercises for Section 12.4

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Determining Arc Length

In questions 1 - 5, find the arc length of the curve on the given interval.

1) r(t)=t2ˆi+(2t2+1)ˆj,1t3. This portion of the graph is shown here:

The parametric curve representing this problem

Answer
85 units 17.9 units

2) r(t)=t2ˆi+14tˆj,0t7. This portion of the graph is shown here:

CNX_Calc_Figure_13_03_203.jpg

3) r(t)=t2+1,4t3+3,1t0. This portion of the graph is shown here:

The parametric curve representing this problem

Answer
154(373/21) units 4.15 units

4) r(t)=2sint,5t,2cost,0tπ. This portion of the graph is shown here:

CNX_Calc_Figure_13_03_204.jpg

5) r(t)=etcost,etsint over the interval [0,π2]. Here is the portion of the graph on the indicated interval:

CNX_Calc_Figure_13_03_205.jpg

6) Set up an integral to represent the arc length from t=0 to t=2 along the curve traced out by r(t)=t,t4. Then use technology to approximate this length to the nearest thousandth of a unit.

7) Find the length of one turn of the helix given by r(t)=12costˆi+12sintˆj+32tˆk.

Answer
Length =2π units

8) Find the arc length of the vector-valued function r(t)=tˆi+4tˆj+3tˆk over [0,1].

9) A particle travels in a circle with the equation of motion r(t)=3costˆi+3sintˆj+0ˆk. Find the distance traveled around the circle by the particle.

Answer
6π units

10) Set up an integral to find the circumference of the ellipse with the equation r(t)=costˆi+2sintˆj+0ˆk.

11) Find the length of the curve r(t)=2t,et,et over the interval 0t1. The graph is shown here:

CNX_Calc_Figure_13_03_206.jpg

Answer
(e1e) units 2.35 units

12) Find the length of the curve r(t)=2sint,5t,2cost for t[10,10].

Unit Tangent Vectors and Unit Normal Vectors

13) The position function for a particle is r(t)=acos(ωt)ˆi+bsin(ωt)ˆj. Find the unit tangent vector and the unit normal vector at t=0.

Solution:
r(t)=aωsin(ωt)ˆi+bωcos(ωt)ˆjr(t)=a2ω2sin2(ωt)+b2ω2cos2(ωt)T(t)=r(t)r(t)=aωsin(ωt)ˆi+bωcos(ωt)ˆja2ω2sin2(ωt)+b2ω2cos2(ωt)T(0)=bωˆj(bω)2=bωˆj|bω|

If bω>0,T(0)=ˆj, and if bω<0,T(0)=ˆj
Answer
If bω>0,T(0)=ˆj, and if bω<0,T(0)=ˆj

If a>0,N(0)=ˆi, and if a<0,N(0)=ˆi

14) Given r(t)=acos(ωt)ˆi+bsin(ωt)ˆj, find the binormal vector B(0).

15) Given r(t)=2et,etcost,etsint, determine the unit tangent vector T(t).

Answer
T(t)=26,costsint6,cost+sint6=63,66(costsint),66(cost+sint)

16) Given r(t)=2et,etcost,etsint, find the unit tangent vector T(t) evaluated at t=0, T(0).

17) Given r(t)=2et,etcost,etsint, determine the unit normal vector N(t).

Answer
N(t)=0,22(sint+cost),22(costsint)

18) Given r(t)=2et,etcost,etsint, find the unit normal vector N(t) evaluated at t=0, N(0).

Answer
N(0)=0,22,22

19) Given r(t)=tˆi+t2ˆj+tˆk, find the unit tangent vector T(t). The graph is shown here:

CNX_Calc_Figure_13_03_207.jpg

Answer
T(t)=14t2+2<1,2t,1>

20) Find the unit tangent vector T(t) and unit normal vector N(t) at t=0 for the plane curve r(t)=t34t,5t22. The graph is shown here:

CNX_Calc_Figure_13_03_208.jpg

21) Find the unit tangent vector T(t) for r(t)=3tˆi+5t2ˆj+2tˆk.

Answer
T(t)=1100t2+13(3ˆi+10tˆj+2ˆk)

22) Find the principal normal vector to the curve r(t)=6cost,6sint at the point determined by t=π3.

23) Find T(t) for the curve r(t)=(t34t)ˆi+(5t22)ˆj.

Answer
T(t)=19t4+76t2+16((3t24)ˆi+10tˆj)

24) Find N(t) for the curve r(t)=(t34t)ˆi+(5t22)ˆj.

25) Find the unit tangent vector T(t) for r(t)=2sint,5t,2cost.

Answer
T(t)=22929cost,52929,22929sint

26) Find the unit normal vector N(t) for r(t)=2sint,5t,2cost.

Answer
N(t)=sint,0,cost

Arc Length Parameterizations

27) Find the arc-length function s(t) for the line segment given by r(t)=33t,4t. Then write the arc-length parameterization of r with s as the parameter.

Answer
Arc-length function: s(t)=5t; The arc-length parameterization of r(t): r(s)=(33s5)ˆi+4s5ˆj

28) Parameterize the helix r(t)=costˆi+sintˆj+tˆk using the arc-length parameter s, from t=0.

29) Parameterize the curve using the arc-length parameter s, at the point at which t=0 for r(t)=etsintˆi+etcostˆj

Answer
r(s)=(1+s2)sin(ln(1+s2))ˆi+(1+s2)cos(ln(1+s2))ˆj

Curvature and the Osculating Circle

30) Find the curvature of the curve r(t)=5costˆi+4sintˆj at t=π/3. (Note: The graph is an ellipse.)

CNX_Calc_Figure_13_03_209.jpg

31) Find the x-coordinate at which the curvature of the curve y=1/x is a maximum value.

Answer
The maximum value of the curvature occurs at x=1.

32) Find the curvature of the curve r(t)=5costˆi+5sintˆj. Does the curvature depend upon the parameter t?

33) Find the curvature κ for the curve y=x14x2 at the point x=2.

Answer
12

34) Find the curvature κ for the curve y=13x3 at the point x=1.

35) Find the curvature κ of the curve r(t)=tˆi+6t2ˆj+4tˆk. The graph is shown here:

CNX_Calc_Figure_13_03_210.jpg

Answer
κ49.477(17+144t2)3/2

36) Find the curvature of r(t)=2sint,5t,2cost.

37) Find the curvature of r(t)=2tˆi+etˆj+etˆk at point P(0,1,1).

Answer
122

38) At what point does the curve y=ex have maximum curvature?

39) What happens to the curvature as x for the curve y=ex?

Answer
The curvature approaches zero.

40) Find the point of maximum curvature on the curve y=lnx.

41) Find the equations of the normal plane and the osculating plane of the curve r(t)=2sin(3t),t,2cos(3t) at point (0,π,2).

Answer
y=6x+π and x+6y=6π

42) Find equations of the osculating circles of the ellipse 4y2+9x2=36 at the points (2,0) and (0,3).

43) Find the equation for the osculating plane at point t=π/4 on the curve r(t)=cos(2t)ˆi+sin(2t)ˆj+tˆk.

Answer
x+2z=π2

44) Find the radius of curvature of 6y=x3 at the point (2,43).

45) Find the curvature at each point (x,y) on the hyperbola r(t)=acosh(t),bsinh(t).

Answer
a4b4(b4x2+a4y2)3/2

46) Calculate the curvature of the circular helix r(t)=rsin(t)ˆi+rcos(t)ˆj+tˆk.

47) Find the radius of curvature of y=ln(x+1) at point (2,ln3).

Answer
10103

48) Find the radius of curvature of the hyperbola xy=1 at point (1,1).

A particle moves along the plane curve C described by r(t)=tˆi+t2ˆj. Use this parameterization to answer questions 49 - 51.

49) Find the length of the curve over the interval [0,2].

Answer
14[417+ln(4+17)] units 4.64678 units

50) Find the curvature of the plane curve at t=0,1,2.

51) Describe the curvature as t increases from t=0 to t=2.

Answer
The curvature is decreasing over this interval.

The surface of a large cup is formed by revolving the graph of the function y=0.25x1.6 from x=0 to x=5 about the y-axis (measured in centimeters).

52) [T] Use technology to graph the surface.

53) Find the curvature κ of the generating curve as a function of x.

Answer
κ=30x2/5(25+4x6/5)3/2

Note that initially your answer may be:
625x2/5(1+425x6/5)3/2

We can simplify it as follows:
625x2/5(1+425x6/5)3/2=625x2/5[125(25+4x6/5)]3/2=625x2/5(125)3/2[25+4x6/5]3/2=625125x2/5[25+4x6/5]3/2=30x2/5(25+4x6/5)3/2

54) [T] Use technology to graph the curvature function.

Contributors:

  • Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

  • Paul Seeburger (Monroe Community College) created question 6.

 


12.4E: Exercises for Section 12.4 is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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