Skip to ContentGo to accessibility pageKeyboard shortcuts menu
Calculus Volume 2

# Review Exercises

Calculus Volume 2Review Exercises

Menu

### Review Exercises

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

322.

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

323.

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

324.

The arc length of the spiral given by $r=θ2r=θ2$ for $0≤θ≤3π0≤θ≤3π$ is $94π3.94π3.$

325.

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

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

326.

$x=1+t,x=1+t,$ $y=t2−1,y=t2−1,$ $−1≤t≤1−1≤t≤1$

327.

$x=et,x=et,$ $y=1−e3t,y=1−e3t,$ $0≤t≤10≤t≤1$

328.

$x=sinθ,x=sinθ,$ $y=1−cscθ,y=1−cscθ,$ $0≤θ≤2π0≤θ≤2π$

329.

$x=4cosϕ,x=4cosϕ,$ $y=1−sinϕ,y=1−sinϕ,$ $0≤ϕ≤2π0≤ϕ≤2π$

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

330.

$r = 4 sin ( θ 3 ) r = 4 sin ( θ 3 )$

331.

$r = 5 cos ( 5 θ ) r = 5 cos ( 5 θ )$

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

332.

$x + y = 5 x + y = 5$

333.

$y 2 = 4 + x 2 y 2 = 4 + x 2$

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

334.

$x=ln(t),x=ln(t),$ $y=t2−1,y=t2−1,$ $t=1t=1$

335.

$r=3+cos(2θ),r=3+cos(2θ),$ $θ=3π4θ=3π4$

336.

Find $dydx,dydx,$ $dxdy,dxdy,$ and $d2xdy2d2xdy2$ of $y=(2+e−t),y=(2+e−t),$ $x=1−sin(t)x=1−sin(t)$

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

337.

$x=t2,x=t2,$ $y=ln(t),y=ln(t),$ $0≤t≤e0≤t≤e$

338.

$r=1−sinθr=1−sinθ$ in the first quadrant

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

339.

$x=3t+4,x=3t+4,$ $y=9t−2,y=9t−2,$ $0≤t≤30≤t≤3$

340.

$r=6cosθ,r=6cosθ,$ $0≤θ≤2π.0≤θ≤2π.$ Check your answer by geometry.

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

341.

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

342.

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

343.

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

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

344.

$r = 6 1 + 3 cos ( θ ) r = 6 1 + 3 cos ( θ )$

345.

$r = 4 3 − 2 cos θ r = 4 3 − 2 cos θ$

346.

$r = 7 5 − 5 cos θ r = 7 5 − 5 cos θ$

347.

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?

348.

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 $(0,0).)(0,0).)$

Do you know how you learn best?
Kinetic by OpenStax offers access to innovative study tools designed to help you maximize your learning potential.
Order a print copy

As an Amazon Associate we earn from qualifying purchases.

Citation/Attribution

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License and you must attribute OpenStax.

Attribution information
• If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction
• If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
Access for free at https://openstax.org/books/calculus-volume-2/pages/1-introduction
Citation information

© Jun 15, 2022 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.