Calculus: Early Transcendentals (2nd Edition)
Calculus: Early Transcendentals (2nd Edition)
2nd Edition
ISBN: 9780321947345
Author: William L. Briggs, Lyle Cochran, Bernard Gillett
Publisher: PEARSON
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Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in ter…
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which …
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the …
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Chapter 10, Problem 28RE

Slopes of tangent lines

  1. a. Find all points where the following curves have vertical and horizontal tangent lines.
  2. b. Find the slope of the lines tangent to the curve at the origin (when relevant).
  3. c. Sketch the curve and all the tangent lines identified in parts (a) and (b).

28. r = 2 cos 2θ

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Chapter 10 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 10.1 - Explain how a pair of parametric equations...Ch. 10.1 - Prob. 2ECh. 10.1 - Prob. 3ECh. 10.1 - Give parametric equations that generate the line...Ch. 10.1 - Prob. 5ECh. 10.1 - Prob. 6ECh. 10.1 - Prob. 7ECh. 10.1 - Prob. 8ECh. 10.1 - Prob. 9ECh. 10.1 - Explain how to find points on the curve x = f(t),...
Ch. 10.1 - Prob. 11ECh. 10.1 - Prob. 12ECh. 10.1 - Prob. 13ECh. 10.1 - Prob. 14ECh. 10.1 - Prob. 15ECh. 10.1 - Prob. 16ECh. 10.1 - Prob. 17ECh. 10.1 - Prob. 18ECh. 10.1 - Prob. 19ECh. 10.1 - Prob. 20ECh. 10.1 - Prob. 21ECh. 10.1 - Prob. 22ECh. 10.1 - Prob. 23ECh. 10.1 - Prob. 24ECh. 10.1 - Prob. 25ECh. 10.1 - Prob. 26ECh. 10.1 - Parametric equations of circles Find parametric...Ch. 10.1 - Parametric equations of circles Find parametric...Ch. 10.1 - Parametric equations of circles Find parametric...Ch. 10.1 - Prob. 30ECh. 10.1 - Parametric equations of circles Find parametric...Ch. 10.1 - Prob. 32ECh. 10.1 - Prob. 33ECh. 10.1 - Prob. 34ECh. 10.1 - Prob. 35ECh. 10.1 - Prob. 36ECh. 10.1 - Parametric lines Find the slope of each line and a...Ch. 10.1 - Parametric lines Find the slope of each line and a...Ch. 10.1 - Parametric lines Find the slope of each line and a...Ch. 10.1 - Prob. 40ECh. 10.1 - Prob. 41ECh. 10.1 - Prob. 42ECh. 10.1 - Prob. 43ECh. 10.1 - Prob. 44ECh. 10.1 - Curves to parametric equations Give a set of...Ch. 10.1 - Curves to parametric equations Give a set of...Ch. 10.1 - Prob. 47ECh. 10.1 - Prob. 48ECh. 10.1 - More parametric curves Use a graphing utility to...Ch. 10.1 - More parametric curves Use a graphing utility to...Ch. 10.1 - More parametric curves Use a graphing utility to...Ch. 10.1 - More parametric curves Use a graphing utility to...Ch. 10.1 - More parametric curves Use a graphing utility to...Ch. 10.1 - More parametric curves Use a graphing utility to...Ch. 10.1 - Prob. 55ECh. 10.1 - Beautiful curves Consider the family of curves...Ch. 10.1 - Prob. 57ECh. 10.1 - Prob. 58ECh. 10.1 - Prob. 59ECh. 10.1 - Derivatives Consider the following parametric...Ch. 10.1 - Derivatives Consider the following parametric...Ch. 10.1 - Prob. 62ECh. 10.1 - Derivatives Consider the following parametric...Ch. 10.1 - Prob. 64ECh. 10.1 - Explain why or why not Determine whether the...Ch. 10.1 - Tangent lines Find an equation of the line tangent...Ch. 10.1 - Tangent lines Find an equation of the line tangent...Ch. 10.1 - Tangent lines Find an equation of the line tangent...Ch. 10.1 - Tangent lines Find an equation of the line tangent...Ch. 10.1 - Prob. 70ECh. 10.1 - Prob. 71ECh. 10.1 - Prob. 72ECh. 10.1 - Prob. 73ECh. 10.1 - Prob. 74ECh. 10.1 - Prob. 75ECh. 10.1 - Prob. 76ECh. 10.1 - Prob. 77ECh. 10.1 - Prob. 78ECh. 10.1 - Prob. 79ECh. 10.1 - Prob. 80ECh. 10.1 - Prob. 81ECh. 10.1 - Prob. 82ECh. 10.1 - Eliminating the parameter Eliminate the parameter...Ch. 10.1 - Eliminating the parameter Eliminate the parameter...Ch. 10.1 - Prob. 85ECh. 10.1 - Prob. 86ECh. 10.1 - Prob. 87ECh. 10.1 - Prob. 88ECh. 10.1 - Slopes of tangent lines Find all the points at...Ch. 10.1 - Slopes of tangent lines Find all the points at...Ch. 10.1 - Slopes of tangent lines Find all the points at...Ch. 10.1 - Slopes of tangent lines Find all the points at...Ch. 10.1 - Prob. 93ECh. 10.1 - Prob. 94ECh. 10.1 - Prob. 95ECh. 10.1 - Lissajous curves Consider the following Lissajous...Ch. 10.1 - Lam curves The Lam curve described by...Ch. 10.1 - Prob. 98ECh. 10.1 - Prob. 99ECh. 10.1 - Prob. 100ECh. 10.1 - Prob. 101ECh. 10.1 - Prob. 102ECh. 10.1 - Prob. 103ECh. 10.1 - Air drop A plane traveling horizontally at 80 m/s...Ch. 10.1 - Air dropinverse problem A plane traveling...Ch. 10.1 - Prob. 106ECh. 10.1 - Implicit function graph Explain and carry out a...Ch. 10.1 - Prob. 108ECh. 10.1 - Prob. 109ECh. 10.1 - Prob. 110ECh. 10.2 - Plot the points with polar coordinates (2,6) and...Ch. 10.2 - Prob. 2ECh. 10.2 - Prob. 3ECh. 10.2 - Prob. 4ECh. 10.2 - What is the polar equation of the vertical line x...Ch. 10.2 - What is the polar equation of the horizontal line...Ch. 10.2 - Prob. 7ECh. 10.2 - Prob. 8ECh. 10.2 - Graph the points with the following polar...Ch. 10.2 - Graph the points with the following polar...Ch. 10.2 - Prob. 11ECh. 10.2 - Prob. 12ECh. 10.2 - Prob. 13ECh. 10.2 - Points in polar coordinates Give two sets of polar...Ch. 10.2 - Converting coordinates Express the following polar...Ch. 10.2 - Converting coordinates Express the following polar...Ch. 10.2 - Converting coordinates Express the following polar...Ch. 10.2 - Converting coordinates Express the following polar...Ch. 10.2 - Converting coordinates Express the following polar...Ch. 10.2 - Converting coordinates Express the following polar...Ch. 10.2 - Converting coordinates Express the following...Ch. 10.2 - Converting coordinates Express the following...Ch. 10.2 - Converting coordinates Express the following...Ch. 10.2 - Converting coordinates Express the following...Ch. 10.2 - Converting coordinates Express the following...Ch. 10.2 - Converting coordinates Express the following...Ch. 10.2 - Prob. 27ECh. 10.2 - Prob. 28ECh. 10.2 - Prob. 29ECh. 10.2 - Prob. 30ECh. 10.2 - Prob. 31ECh. 10.2 - Prob. 32ECh. 10.2 - Prob. 33ECh. 10.2 - Prob. 34ECh. 10.2 - Prob. 35ECh. 10.2 - Prob. 36ECh. 10.2 - Prob. 37ECh. 10.2 - Prob. 38ECh. 10.2 - Prob. 39ECh. 10.2 - Prob. 40ECh. 10.2 - Graphing polar curves Graph the following...Ch. 10.2 - Graphing polar curves Graph the following...Ch. 10.2 - Prob. 43ECh. 10.2 - Prob. 44ECh. 10.2 - Graphing polar curves Graph the following...Ch. 10.2 - Graphing polar curves Graph the following...Ch. 10.2 - Graphing polar curves Graph the following...Ch. 10.2 - Graphing polar curves Graph the following...Ch. 10.2 - Prob. 49ECh. 10.2 - Prob. 50ECh. 10.2 - Prob. 51ECh. 10.2 - Prob. 52ECh. 10.2 - Using a graphing utility Use a graphing utility to...Ch. 10.2 - Using a graphing utility Use a graphing utility to...Ch. 10.2 - Prob. 55ECh. 10.2 - Using a graphing utility Use a graphing utility to...Ch. 10.2 - Using a graphing utility Use a graphing utility to...Ch. 10.2 - Using a graphing utility Use a graphing utility to...Ch. 10.2 - Prob. 59ECh. 10.2 - Prob. 60ECh. 10.2 - Prob. 61ECh. 10.2 - Cartesian-to-polar coordinates Convert the...Ch. 10.2 - Cartesian-to-polar coordinates Convert the...Ch. 10.2 - Cartesian-to-polar coordinates Convert the...Ch. 10.2 - Cartesian-to-polar coordinates Convert the...Ch. 10.2 - Prob. 66ECh. 10.2 - Prob. 67ECh. 10.2 - Prob. 68ECh. 10.2 - Prob. 69ECh. 10.2 - Prob. 70ECh. 10.2 - Prob. 71ECh. 10.2 - Prob. 72ECh. 10.2 - Prob. 73ECh. 10.2 - Prob. 74ECh. 10.2 - Circles in general Show that the polar equation...Ch. 10.2 - Prob. 76ECh. 10.2 - Prob. 77ECh. 10.2 - Prob. 78ECh. 10.2 - Prob. 79ECh. 10.2 - Prob. 80ECh. 10.2 - Prob. 81ECh. 10.2 - Equations of circles Find equations of the circles...Ch. 10.2 - Prob. 83ECh. 10.2 - Prob. 84ECh. 10.2 - Prob. 85ECh. 10.2 - Prob. 86ECh. 10.2 - Prob. 87ECh. 10.2 - Prob. 88ECh. 10.2 - Prob. 89ECh. 10.2 - Limiting limaon Consider the family of limaons r =...Ch. 10.2 - Prob. 91ECh. 10.2 - Prob. 92ECh. 10.2 - Prob. 93ECh. 10.2 - The lemniscate family Equations of the form r2 = a...Ch. 10.2 - The rose family Equations of the form r = a sin m...Ch. 10.2 - Prob. 96ECh. 10.2 - Prob. 97ECh. 10.2 - The rose family Equations of the form r = a sin m...Ch. 10.2 - Prob. 99ECh. 10.2 - Prob. 100ECh. 10.2 - Prob. 101ECh. 10.2 - Spirals Graph the following spirals. Indicate the...Ch. 10.2 - Prob. 103ECh. 10.2 - Prob. 104ECh. 10.2 - Prob. 105ECh. 10.2 - Prob. 106ECh. 10.2 - Enhanced butterfly curve The butterfly curve of...Ch. 10.2 - Prob. 108ECh. 10.2 - Prob. 109ECh. 10.2 - Prob. 110ECh. 10.2 - Prob. 111ECh. 10.2 - Cartesian lemniscate Find the equation in...Ch. 10.2 - Prob. 113ECh. 10.2 - Prob. 114ECh. 10.3 - Prob. 1ECh. 10.3 - Prob. 2ECh. 10.3 - Explain why the slope of the line tangent to the...Ch. 10.3 - What integral must be evaluated to find the area...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Slopes of tangent lines Find the slope of the line...Ch. 10.3 - Horizontal and vertical tangents Find the points...Ch. 10.3 - Horizontal and vertical tangents Find the points...Ch. 10.3 - Horizontal and vertical tangents Find the points...Ch. 10.3 - Prob. 18ECh. 10.3 - Prob. 19ECh. 10.3 - Prob. 20ECh. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Areas of regions Make a sketch of the region and...Ch. 10.3 - Prob. 37ECh. 10.3 - Prob. 38ECh. 10.3 - Prob. 39ECh. 10.3 - Prob. 40ECh. 10.3 - Prob. 41ECh. 10.3 - Prob. 42ECh. 10.3 - Prob. 43ECh. 10.3 - Prob. 44ECh. 10.3 - Prob. 45ECh. 10.3 - Multiple identities Explain why the point (1, 3/2)...Ch. 10.3 - Area of plane regions Find the areas of the...Ch. 10.3 - Area of plane regions Find the areas of the...Ch. 10.3 - Area of plane regions Find the areas of the...Ch. 10.3 - Area of plane regions Find the areas of the...Ch. 10.3 - Prob. 51ECh. 10.3 - Prob. 52ECh. 10.3 - Regions bounded by a spiral Let Rn be the region...Ch. 10.3 - Area of polar regions Find the area of the regions...Ch. 10.3 - Area of polar regions Find the area of the regions...Ch. 10.3 - Area of polar regions Find the area of the regions...Ch. 10.3 - Prob. 57ECh. 10.3 - Prob. 58ECh. 10.3 - Grazing goat problems Consider the following...Ch. 10.3 - Grazing goat problems Consider the following...Ch. 10.3 - Prob. 61ECh. 10.3 - Tangents and normals Let a polar curve be...Ch. 10.3 - Prob. 63ECh. 10.4 - Give the property that defines all parabolas.Ch. 10.4 - Prob. 2ECh. 10.4 - Give the property that defines all hyperbolas.Ch. 10.4 - Prob. 4ECh. 10.4 - Prob. 5ECh. 10.4 - What is the equation of the standard parabola with...Ch. 10.4 - Prob. 7ECh. 10.4 - Prob. 8ECh. 10.4 - Given vertices (a, 0) and eccentricity e, what are...Ch. 10.4 - Prob. 10ECh. 10.4 - What are the equations of the asymptotes of a...Ch. 10.4 - Prob. 12ECh. 10.4 - Graphing parabolas Sketch a graph of the following...Ch. 10.4 - Prob. 14ECh. 10.4 - Prob. 15ECh. 10.4 - Prob. 16ECh. 10.4 - Prob. 17ECh. 10.4 - Graphing parabolas Sketch a graph of the following...Ch. 10.4 - Prob. 19ECh. 10.4 - Equations of parabolas Find an equation of the...Ch. 10.4 - Equations of parabolas Find an equation of the...Ch. 10.4 - Prob. 22ECh. 10.4 - Prob. 23ECh. 10.4 - Equations of parabolas Find an equation of the...Ch. 10.4 - From graphs to equations Write an equation of the...Ch. 10.4 - From graphs to equations Write an equation of the...Ch. 10.4 - Prob. 27ECh. 10.4 - Prob. 28ECh. 10.4 - Prob. 29ECh. 10.4 - Prob. 30ECh. 10.4 - Prob. 31ECh. 10.4 - Prob. 32ECh. 10.4 - Equations of ellipses Find an equation of the...Ch. 10.4 - Equations of ellipses Find an equation of the...Ch. 10.4 - Equations of ellipses Find an equation of the...Ch. 10.4 - Prob. 36ECh. 10.4 - Prob. 37ECh. 10.4 - Prob. 38ECh. 10.4 - Prob. 39ECh. 10.4 - Prob. 40ECh. 10.4 - Prob. 41ECh. 10.4 - Prob. 42ECh. 10.4 - Prob. 43ECh. 10.4 - Prob. 44ECh. 10.4 - Equations of hyperbolas Find an equation of the...Ch. 10.4 - Equations of hyperbolas Find an equation of the...Ch. 10.4 - Equations of hyperbolas Find an equation of the...Ch. 10.4 - Prob. 48ECh. 10.4 - From graphs to equations Write an equation of the...Ch. 10.4 - From graphs to equations Write an equation of the...Ch. 10.4 - Eccentricity-directrix approach Find an equation...Ch. 10.4 - Eccentricity-directrix approach Find an equation...Ch. 10.4 - Eccentricity-directrix approach Find an equation...Ch. 10.4 - Eccentricity-directrix approach Find an equation...Ch. 10.4 - Prob. 55ECh. 10.4 - Prob. 56ECh. 10.4 - Prob. 57ECh. 10.4 - Prob. 58ECh. 10.4 - Prob. 59ECh. 10.4 - Prob. 60ECh. 10.4 - Tracing hyperbolas and parabolas Graph the...Ch. 10.4 - Tracing hyperbolas and parabolas Graph the...Ch. 10.4 - Tracing hyperbolas and parabolas Graph the...Ch. 10.4 - Tracing hyperbolas and parabolas Graph the...Ch. 10.4 - Prob. 65ECh. 10.4 - Hyperbolas with a graphing utility Use a graphing...Ch. 10.4 - Prob. 67ECh. 10.4 - Prob. 68ECh. 10.4 - Tangent lines Find an equation of the tine tangent...Ch. 10.4 - Tangent lines Find an equation of the tine tangent...Ch. 10.4 - Tangent lines Find an equation of the tine tangent...Ch. 10.4 - Prob. 72ECh. 10.4 - Prob. 73ECh. 10.4 - Prob. 74ECh. 10.4 - Prob. 75ECh. 10.4 - The ellipse and the parabola Let R be the region...Ch. 10.4 - Tangent lines for an ellipse Show that an equation...Ch. 10.4 - Prob. 78ECh. 10.4 - Volume of an ellipsoid Suppose that the ellipse...Ch. 10.4 - Area of a sector of a hyperbola Consider the...Ch. 10.4 - Volume of a hyperbolic cap Consider the region R...Ch. 10.4 - Prob. 82ECh. 10.4 - Prob. 83ECh. 10.4 - Golden Gate Bridge Completed in 1937, San...Ch. 10.4 - Prob. 85ECh. 10.4 - Prob. 86ECh. 10.4 - Prob. 87ECh. 10.4 - Prob. 88ECh. 10.4 - Shared asymptotes Suppose that two hyperbolas with...Ch. 10.4 - Focal chords A focal chord of a conic section is a...Ch. 10.4 - Focal chords A focal chord of a conic section is a...Ch. 10.4 - Focal chords A focal chord of a conic section is a...Ch. 10.4 - Prob. 93ECh. 10.4 - Prob. 94ECh. 10.4 - Confocal ellipse and hyperbola Show that an...Ch. 10.4 - Approach to asymptotes Show that the vertical...Ch. 10.4 - Prob. 97ECh. 10.4 - Prob. 98ECh. 10.4 - Prob. 99ECh. 10 - Explain why or why not Determine whether the...Ch. 10 - Prob. 2RECh. 10 - Prob. 3RECh. 10 - Prob. 4RECh. 10 - Prob. 5RECh. 10 - Prob. 6RECh. 10 - Prob. 7RECh. 10 - Prob. 8RECh. 10 - Eliminating the parameter Eliminate the parameter...Ch. 10 - Prob. 10RECh. 10 - Parametric description Write parametric equations...Ch. 10 - Parametric description Write parametric equations...Ch. 10 - Prob. 13RECh. 10 - Prob. 14RECh. 10 - Parametric description Write parametric equations...Ch. 10 - Parametric description Write parametric equations...Ch. 10 - Tangent lines Find an equation of the line tangent...Ch. 10 - Prob. 18RECh. 10 - Prob. 19RECh. 10 - Sets in polar coordinates Sketch the following...Ch. 10 - Prob. 21RECh. 10 - Prob. 22RECh. 10 - Polar conversion Write the equation...Ch. 10 - Polar conversion Consider the equation r = 4/(sin ...Ch. 10 - Prob. 25RECh. 10 - Prob. 26RECh. 10 - Prob. 27RECh. 10 - Slopes of tangent lines a. Find all points where...Ch. 10 - Slopes of tangent lines a. Find all points where...Ch. 10 - Slopes of tangent lines a. Find all points where...Ch. 10 - Prob. 31RECh. 10 - The region enclosed by all the leaves of the rose...Ch. 10 - The region enclosed by the limaon r = 3 cosCh. 10 - The region inside the limaon r = 2 + cos and...Ch. 10 - Prob. 35RECh. 10 - Prob. 36RECh. 10 - The area that is inside the cardioid r = 1 + cos ...Ch. 10 - Prob. 38RECh. 10 - Prob. 39RECh. 10 - Prob. 40RECh. 10 - Conic sections a. Determine whether the following...Ch. 10 - Prob. 42RECh. 10 - Prob. 43RECh. 10 - Prob. 44RECh. 10 - Tangent lines Find an equation of the line tangent...Ch. 10 - Prob. 46RECh. 10 - Tangent lines Find an equation of the line tangent...Ch. 10 - Tangent lines Find an equation of the line tangent...Ch. 10 - Prob. 49RECh. 10 - Prob. 50RECh. 10 - Prob. 51RECh. 10 - Prob. 52RECh. 10 - Prob. 53RECh. 10 - Prob. 54RECh. 10 - Eccentricity-directrix approach Find an equation...Ch. 10 - Prob. 56RECh. 10 - Prob. 57RECh. 10 - Prob. 58RECh. 10 - Prob. 59RECh. 10 - Prob. 60RECh. 10 - Prob. 61RECh. 10 - Prob. 62RECh. 10 - Prob. 63RECh. 10 - Prob. 64RECh. 10 - Prob. 65RECh. 10 - Prob. 66RECh. 10 - Prob. 67RECh. 10 - Prob. 68RECh. 10 - Prob. 69RECh. 10 - Prob. 70RECh. 10 - Prob. 71RE
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