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4.1E: Exercises for Section 12.1

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Introduction to Vector-Valued Functions

1) Give the component functions x=f(t) and y=g(t) for the vector-valued function r(t)=3sectˆi+2tantˆj.

Answer
Here we can say that f(t)=3sect,g(t)=2tant

so we have x(t)=3sect,y(t)=2tant.

2) Given r(t)=3sectˆi+2tantˆj, find the following values (if possible).

  1. r(π4)
  2. r(π)
  3. r(π2)

3) Sketch the curve of the vector-valued function r(t)=3sectˆi+2tantˆj and give the orientation of the curve. Sketch asymptotes as a guide to the graph.

Answer
Hyperbolic path along a horizontally oriented hyperbola.

Limits of Vector-Valued Functions

4) Evaluate limt0(etˆi+sinttˆj+etˆk)

5) Given the vector-valued function r(t)=cost,sint find the following values:

  1. limtπ4r(t)
  2. r(π3)
  3. Is r(t) continuous at t=π3?
  4. Graph r(t).
Answer

a. 22,22,
b. 12,32,
c. Yes, the limit as t approaches π3 is equal to r(π3),
d.

Counterclockwise oriented path on the unit circle.

6) Given the vector-valued function r(t)=t,t2+1, find the following values:

  1. limt3r(t)
  2. r(3)
  3. Is r(t) continuous at x=3?
  4. r(t+2)r(t)

7) Let r(t)=etˆi+sintˆj+lntˆk. Find the following values:

  1. r(π4)
  2. limtπ4r(t)
  3. Is r(t) continuous at t=π4?
Answer
a. ⟨eπ4,22,ln(π4)⟩;
b. ⟨eπ4,22,ln(π4)⟩;
c. Yes

For exercises 8 - 13, find the limit of the following vector-valued functions at the indicated value of t.

8) limt4t3,t2t4,tan(πt)

9) limtπ2r(t) for r(t)=etˆi+sintˆj+lntˆk

Answer
eπ2,1,ln(π2)

10) limte2t,2t+33t1,arctan(2t)

11) limte2tln(t),lntt2,ln(t2)

Answer
2e2ˆi+2e4ˆj+2ˆk

12) limtπ6cos2t,sin2t,1

13) limtr(t) for r(t)=2eti+etˆj+ln(t1)ˆk

Answer
The limit does not exist because the limit of ln(t1) as t approaches infinity does not exist.


Domain of a Vector-Valued Function

For problems 14 - 17, find the domain of the vector-valued functions.

14) Domain: r(t)=t2,t,sint

15) Domain: r(t)=t2,tant,lnt

Answer
Dr={t|t>0,t(2k+1)π2,wherekis any integer}

16) Domain: r(t)=t2,t3,32t+1

17) Domain: r(t)=csc(t),1t3,ln(t2)

Answer
Dr={t|t>3,tnπ,wherenis any integer}

18) a. Find the domain of r(t)=2etˆi+etˆj+ln(t1)ˆk.

b. For what values of t is r(t)=2etˆi+etˆj+ln(t1)ˆk continuous?

Answer
a. Dr:(1,)
b. All t such that t(1,)

19) Domain: r(t)=(arccost)ˆi+2t1ˆj+ln(t)ˆk

Answer
Dr:[12,1]

Visualizing Vector-Valued Functions

20) Describe the curve defined by the vector-valued function r(t)=(1+t)ˆi+(2+5t)ˆj+(1+6t)ˆk.

21) Let r(t)=cost,t,sint and use it to answer the following questions.

  1. For what values of t is r(t) continuous?
  2. Sketch the graph of r(t).
Answer
a. r is continuous for all real numbers, i.e., for tR.
b. Note that there should be a z on the vertical axis in the cross-section in image (a) below instead of the y.

Top image shows counterclockwise oriented path on the unit circle.   Bottom image shows corkscrew path with z-coordinate varying as the circular motion continues as in the image above.

22) Produce a careful sketch of the graph of r(t)=t2ˆi+tˆj.

In questions 23 - 25, use a graphing utility to sketch each of the vector-valued functions:

23) [T] r(t)=2cos2tˆi+(2t)ˆj

Answer
CNX_Calc_Figure_13_01_208.jpg

24) [T] r(t)=ecos(3t),esin(t)

25) [T] r(t)=2sin(2t),3+2cost

Answer
A figure eight oriented path.

Finding Equations in x and y for the Path Traced out by Vector-Valued Functions

For questions 26-33, eliminate the parameter t, write the equation in Cartesian coordinates, then sketch the graph of the vector-valued functions.

26) r(t)=2tˆi+t2ˆj
(Hint: Let x=2t and y=t2. Solve the first equation for t in terms of x and substitute this result into the second equation.)

27) r(t)=t3ˆi+2tˆj

Answer

y=23x, a variation of the cube-root function

Oriented path along the graph of y equals 2 times the cube-root of x.  Motion along the path is oriented from left-to-right.

28) r(t)=sintˆi+costˆj

29) r(t)=3costˆi+3sintˆj

Answer

x2+y2=9, a circle centered at (0,0) with radius 3, and a counterclockwise orientation

Counterclockwise motion along the circle of radius 3, centered at the origin.

30) r(t)=sint,4cost

31) r(t)=2sintˆi3costˆj

Answer

x24+y29=1, an ellipse centered at (0,0) with intercepts at x=±2 and y=±3, and a clockwise orientation

Ellipse with clockwise orientation passing thru (-2,0), (0, 3), (2, 0), (0, -3)

32) r(t)=tantˆi2sectˆj

33) r(t)=3sectˆi+4tantˆj

Answer

x29y216=1, a hyperbola centered at (0,0) with x-intercepts (3,0) and (3,0), with orientation shown

Oriented Hyperbola

Finding a Vector-Valued Function to Trace out the Graph of an Equation in x and y

For questions 34 - 40, find a vector-valued function that traces out the given curve in the indicated direction.

34) 4x2+9y2=36; clockwise and counterclockwise

35) y=x2; from left to right

Answer
r(t)=t,t2, where t increases

36) The line through P and Q where P is (1,4,2) and Q is (3,9,6)

37) The circle, x2+y2=36, oriented clockwise, with position (6,0) at time t=0.

Answer
r(t)=6costˆi+6sintˆj

38) The ellipse, x2+y236=1, oriented counterclockwise

Answer
r(t)=costˆi+6sintˆj

39) The hyperbola, y236x2=1, top piece is oriented from left-to-right

Answer
r(t)=tantˆi+6sectˆj

40) The hyperbola, x249y264=1, right piece is oriented from bottom-to-top

Answer
r(t)=7sectˆi+8tantˆj

Parameterizing a Piecewise Path

For questions 41 - 44, provide a parameterization for each piecewise path. Try to write a parameterization that starts with t=0 and progresses on through values of t as you move from one piece to another.

41)

Counterclockwise-oriented boundary of a closed region formed by y = x^4 and y equals the cube root of x. Clockwise-oriented boundary of a closed region formed by y = x^4 and y equals the cube root of x.

Answer
a. r1(t)=tˆi+t4ˆj for 0t1
r2(t)=tˆi+3tˆj for 1t0

So a piecewise parameterization of this path is:
r(t)={tˆi+t4ˆj,0t1(2t)ˆi+32tˆj,1<t2

b. r1(t)=tˆi+3tˆj for 0t1
r2(t)=tˆi+(t)4ˆj for 1t0

So a piecewise parameterization of this path is:
r(t)={tˆi+3tˆj,0t1(2t)ˆi+(2t)4ˆj,1<t2

42)

Counterclockwise-oriented boundary of a closed region formed by y = x^3 and y = 4x. Clockwise-oriented boundary of a closed region formed by y = x^3 and y = 4x.

43)

Counterclockwise-oriented boundary of a closed region formed by y = x^3 and y = 2 - x and the x-axis. Clockwise-oriented boundary of a closed region formed by y = x^3 and y = 2 - x and the x-axis.

Answer
a. r1(t)=tˆi+0ˆj for 0t2
r2(t)=tˆi+(2+t)ˆj for 2t1
r3(t)=tˆi+(t)3ˆj for 1t0

So a piecewise parameterization of this path is:
r(t)={tˆi,0t2(4t)ˆi+(t2)ˆj,2<t3(4t)ˆi+(4t)3ˆj,3<t4

b. r1(t)=tˆi+t3ˆj for 0t1
r2(t)=tˆi+(2t)ˆj for 1t2
r3(t)=tˆi+0ˆj for 2t0

So a piecewise parameterization of this path is:
r(t)={tˆi+t3ˆj,0t1tˆi+(2t)ˆj,1<t2(4t)ˆi,2<t4

44)

Counterclockwise-oriented boundary of a closed region formed by y = 1-x/2 and y = 3x/2 - 3 and y = 1 plus the square root of x. Clockwise-oriented boundary of a closed region formed by y = 1-x/2 and y = 3x/2 - 3 and y = 1 plus the square root of x.

Additional Vector-Valued Function Questions

For questions 45 - 48, consider the curve described by the vector-valued function r(t)=(50etcost)ˆi+(50etsint)ˆj+(55et)ˆk.

45) What is the initial point of the path corresponding to r(0)?

Answer
(50,0,0)

46) What is limtr(t)?

47) [T] Use technology to sketch the curve.

Answer
Partial path for r(t)=(50e^(−t) cos t)i+(50e^(−t) sin t)j+(5−5e^(−t))k.

48) Eliminate the parameter t to show that z=5r10 where r2=x2+y2.

49) [T] Let r(t)=costˆi+sintˆj+0.3sin(2t)ˆk. Use technology to graph the curve (called the roller-coaster curve) over the interval [0,2π). Choose at least two views to determine the peaks and valleys.

Answer
Two views of the path traced out by r(t)=(cos t)i + (sin t)j + (0.3 sin 2t)k.

50) [T] Use the result of the preceding problem to construct an equation of a roller coaster with a steep drop from the peak and steep incline from the “valley.” Then, use technology to graph the equation.

51) Use the results of the preceding two problems to construct an equation of a path of a roller coaster with more than two turning points (peaks and valleys).

Answer

One possibility is r(t)=costˆi+sintˆj+sin(4t)ˆk. By increasing the coefficient of t in the third component, the number of turning points will increase.

Path traced out by r(t)=(cos t)i + (sin t)j + (sin 4t)k.

52) Complete the following investigation.

  1. Graph the curve r(t)=(4+cos(18t))cos(t)ˆi+(4+cos(18t)sin(t))ˆj+0.3sin(18t)ˆk using two viewing angles of your choice to see the overall shape of the curve.
  2. Does the curve resemble a “slinky”?
  3. What changes to the equation should be made to increase the number of coils of the slinky?

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 problems 12, 14, 19, 22, 30-33, 37- 44.


This page titled 4.1E: Exercises for Section 12.1 is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

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