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Mathematics LibreTexts

11.6: Chapter 11 Review Exercises

  • Gilbert Strang & Edwin “Jed” Herman
  • OpenStax

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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.

Answer
True

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

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

Answer
False. Imagine y=t+1,x=t+1.

In exercises 5 -8, 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

Answer

y=1x3

Graph of a curve starting at (1, 0) and decreasing into the fourth quadrant.

7) x=sinθ,y=1cscθ,0θ2π

8) x=4cosϕ,y=1sinϕ,0ϕ2π

Answer

x216+(y1)2=1

Graph of an ellipse with center (0, 1), major axis horizontal and of length 8, and minor axis of length 2.

In exercises 9 - 10, sketch the polar curve and determine what type of symmetry exists, if any.

9) r=4sin(θ3)

10) r=5cos(5θ)

Answer

Symmetric about polar axis

Graph of a five-petaled rose with initial petal at θ = 0.

In exercises 11 - 12, find the polar equation for the curve given as a Cartesian equation.

11) x+y=5

12) y2=4+x2

Answer
r2=4sin2θcos2θ

In exercises 13 - 14, 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

Answer

y=322+15(x+322)

Graph of a peanut-shaped figure, with y intercepts at ±2 and x intercepts at ±4. The tangent line occurs in the second quadrant.

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

In exercises 16 -17, find the area of the region.

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

Answer
e22 units2

17) r=1sinθ in the first quadrant

In exercises 18 - 19, find the arc length of the curve over the given interval.

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

Answer
910 units

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

In exercises 20 - 22, find the Cartesian equation describing the given shapes.

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

Answer
(y+5)2=8x+32

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
(y+1)216(x+2)29=1

In exercises 23 - 25, determine the eccentricity and identify the conic. Sketch the conic.

23) r=61+3cosθ

24) r=432cosθ

Answer

e=23, ellipse

Graph of an ellipse with center near (1.5, 0), major axis nearly 5 and horizontal, and minor axis nearly 4.

25) r=755cosθ

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?

Answer
y219.032+x219.632=1,e=0.2447

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).)


This page titled 11.6: Chapter 11 Review Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin “Jed” Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform.

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