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6E: Chapter Exercises

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Chapter Review Exercises

Answers are provided to even-numbered problems.

Exercise 6E.1 True or false

True or False? Justify your answer with a proof or a counterexample.

1) The rectangular coordinates of the point (4,5π6) are (23,2).

2) The equations x=cosh(3t),y=2sinh(3t) represent a hyperbola.

3) The arc length of the spiral given by r=θ2 for 0θ3π is 94π3.

4) Given x=f(t) and y=g(t), if dxdy=dydx, then f(t)=g(t)+C, where C is a constant.

Answer

Solution 2: True, Solution 4: False. Imagine y=t+1,x=t+1.

Exercise 6E.2 sketch the parametric curve

For the following exercises, sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve.

5) x=1+t,y=t21,1t1

6) x=et,y=1e3t,0t1

7) x=sinθ,y=1cscθ,0θ2π

8) x=4cosϕ,y=1sinϕ,0ϕ2π

Answer

Solution 6: y=1x3

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Solution 8: x216+(y1)2=1

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Exercise 6E.3 sketch the polar curve

For the following exercises, sketch the polar curve and determine what type of symmetry exists, if any.

9) r=4sin(θ3)

10) r=5cos(5θ)

Answer

Solution 10: Symmetric about polar axis

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Exercise 6E.4 Polar Equation

For the following exercises, find the polar equation for the curve given as a Cartesian equation.

11) x+y=5

12) y2=4+x2

Answer

Solution 12: r2=4sin2θcos2θ

Exercise 6E.5 tangent line

For the following exercises, find the equation of the tangent line to the given curve. Graph both the function and its tangent line.

13) x=ln(t),y=t21,t=1

14) r=3+cos(2θ),θ=3π4

15) Find dydx,dxdy, and d2xdy2 of y=(2+et),x=1sin(t)

Answer

Solution 14: y=322+15(x+322)

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Exercise 6E.6 Area

For the following exercises, find the area of the region.

16) x=t2,y=ln(t),0te

17) r=1sinθ in the first quadrant

Answer

Solution 16: e22

Exercise 6E.7 Arc length

For the following exercises, find the arc length of the curve over the given interval.

18) x=3t+4,y=9t2,0t3

19) r=6cosθ,0θ2π. Check your answer by geometry.

Answer

Solution 18: 910

Exercise 6E.8 cartesian Equation

For the following exercises, find the Cartesian equation describing the given shapes.

20) A parabola with focus (2,5) and directrix x=6

21) An ellipse with a major axis length of 10 and foci at (7,2) and (1,2)

22) A hyperbola with vertices at (3,2) and (5,2) and foci at (2,6) and (2,4)

Answer

Solution 20: (y+5)2=8x+32, Solution 22: (y+1)216(x+2)29=1

Exercise 6E.9 Ecencentricity

For the following exercises, determine the eccentricity and identify the conic. Sketch the conic.

23) r=61+3cos(θ)

24) r=432cosθ

25) r=755cosθ

Answer

Solution 24: e=23, ellipse

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Exercise 6E.10 Application

26) Determine the Cartesian equation describing the orbit of Pluto, the most eccentric orbit around the Sun. The length of the major axis is 39.26 AU and minor axis is 38.07 AU. What is the eccentricity?

27) The C/1980 E1 comet was observed in 1980. Given an eccentricity of 1.057 and a perihelion (point of closest approach to the Sun) of 3.364 AU, find the Cartesian equations describing the comet’s trajectory. Are we guaranteed to see this comet again? (Hint: Consider the Sun at point (0,0).)

Answer

Solution 26: y219.032+x219.632=1,e=0.2447


This page titled 6E: Chapter Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform.

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