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 (2√3,−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 0≤θ≤3π 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
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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=t2−1,−1≤t≤1
6) x=et,y=1−e3t,0≤t≤1
7) x=sinθ,y=1−cscθ,0≤θ≤2π
8) x=4cosϕ,y=1−sinϕ,0≤ϕ≤2π
- Answer
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Solution 6: y=1−x3
Solution 8: x216+(y−1)2=1
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
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Solution 10: Symmetric about polar axis
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
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Solution 12: r2=4sin2θ−cos2θ
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=t2−1,t=1
14) r=3+cos(2θ),θ=3π4
15) Find dydx,dxdy, and d2xdy2 of y=(2+e−t),x=1−sin(t)
- Answer
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Solution 14: y=3√22+15(x+3√22)
Exercise 6E.6 Area
For the following exercises, find the area of the region.
16) x=t2,y=ln(t),0≤t≤e
17) r=1−sinθ in the first quadrant
- Answer
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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=9t−2,0≤t≤3
19) r=6cosθ,0≤θ≤2π. Check your answer by geometry.
- Answer
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Solution 18: 9√10
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
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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=43−2cosθ
25) r=75−5cosθ
- Answer
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Solution 24: e=23, ellipse
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
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Solution 26: y219.032+x219.632=1,e=0.2447