Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
7th Edition
ISBN: 9781337552516
Author: Ron Larson, Bruce H. Edwards
Publisher: Cengage Learning
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Chapter 12.4, Problem 11E
To determine

To calculate: The unit tangent vector and a set of parametric equations for the tangent line to the space curve r(t)=costi+3sintj+(3t−4)k at point P (1,0,−4).

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Chapter 12.4, Problem 10E
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(a) Find a direction vector for the first line, which is given in parametric form. (b) Find parametric equations for the second line in terms of the parameter t. (c) Show that the two lines intersect at a single point by finding the values of s and t that result in the same point. Then find the point of intersection.
5: A potato cannon launches a potato straight up from a height of 5 feet with an initial velocity of 128 ft/sec. The horizontal distance from the cannon launch and the vertical height from the ground, in feet, are modeled by a set of parametric equations. . Write a set of parametric equations that will model the height of the potato as a function of time, t. . Find the average rate of change in horizontal position over the first two seconds. Show your work and units. What is the average rate of change in vertical motion over the first two seconds? Show your work and units.
x(t) = acos((a + b)t)       y(t) = acos((a − b)t) Use the GeoGebra tool to graph the parametric equations on the domain [-π, 0] and include the orientation for the following cases: a = 2 and b = 1 a = 3 and b = 2 a = 4 and b = 3 a = 5 and b = 4 Discuss the role of a and b in the parametric equations shown above. Discuss why the domain is given as [-π, 0]. What would change if the domain were given as [0, π] instead?

Chapter 12 Solutions

Calculus: Early Transcendental Functions

Ch. 12.1 - CONCEPT CHECK Vector-Valued Function Describe how...Ch. 12.1 - Prob. 2ECh. 12.1 - Finding the domain In exercises 3-10 find the...Ch. 12.1 - Prob. 4ECh. 12.1 - Finding the domain In exercises 3-10 find the...Ch. 12.1 - Finding the domain In exercises 3-10 find the...Ch. 12.1 - Finding the Domain In Exercises 3-10, find the...Ch. 12.1 - Finding the Domain In Exercises 3-10, find the...Ch. 12.1 - Prob. 9ECh. 12.1 - Finding the domain In exercises 3-10 find the...
Ch. 12.1 - Evaluating a function In Exercises 11 and 12...Ch. 12.1 - Evaluating a function In Exercises 11 and 12...Ch. 12.1 - Writing a Vector-Valued Function In Exercises...Ch. 12.1 - Writing a Vector-Valued Function In Exercises...Ch. 12.1 - Prob. 15ECh. 12.1 - Prob. 16ECh. 12.1 - Prob. 17ECh. 12.1 - Prob. 18ECh. 12.1 - Matching In Exercises 19-22. match the equation...Ch. 12.1 - Prob. 20ECh. 12.1 - Prob. 21ECh. 12.1 - Matching In Exercises 19-22, match the equation...Ch. 12.1 - Prob. 23ECh. 12.1 - Prob. 24ECh. 12.1 - Prob. 25ECh. 12.1 - Prob. 26ECh. 12.1 - Prob. 27ECh. 12.1 - Prob. 28ECh. 12.1 - Prob. 29ECh. 12.1 - Prob. 30ECh. 12.1 - Prob. 31ECh. 12.1 - Prob. 32ECh. 12.1 - Prob. 33ECh. 12.1 - Prob. 34ECh. 12.1 - Prob. 35ECh. 12.1 - Prob. 36ECh. 12.1 - Prob. 37ECh. 12.1 - Prob. 38ECh. 12.1 - Prob. 39ECh. 12.1 - Prob. 40ECh. 12.1 - Transformation of a vector valued valued in...Ch. 12.1 - Transformations of Vector-Valued Functions In...Ch. 12.1 - Prob. 43ECh. 12.1 - Prob. 44ECh. 12.1 - Prob. 45ECh. 12.1 - Prob. 46ECh. 12.1 - Prob. 47ECh. 12.1 - Prob. 48ECh. 12.1 - Prob. 49ECh. 12.1 - Prob. 50ECh. 12.1 - Prob. 51ECh. 12.1 - Prob. 52ECh. 12.1 - Prob. 53ECh. 12.1 - Prob. 54ECh. 12.1 - Prob. 55ECh. 12.1 - Prob. 56ECh. 12.1 - Prob. 57ECh. 12.1 - Prob. 58ECh. 12.1 - Prob. 59ECh. 12.1 - Prob. 60ECh. 12.1 - Prob. 61ECh. 12.1 - Prob. 62ECh. 12.1 - Prob. 63ECh. 12.1 - Prob. 64ECh. 12.1 - Finding a Limit In Exercises 65-70, find the limit...Ch. 12.1 - Prob. 66ECh. 12.1 - Prob. 67ECh. 12.1 - Prob. 68ECh. 12.1 - Finding a Limit In Exercises 65-70, find the limit...Ch. 12.1 - Finding a Limit In Exercises 65-70, find the limit...Ch. 12.1 - Continuity of a Vector-Valued Function In...Ch. 12.1 - Prob. 72ECh. 12.1 - Prob. 73ECh. 12.1 - Prob. 74ECh. 12.1 - Prob. 75ECh. 12.1 - Prob. 76ECh. 12.1 - Prob. 77ECh. 12.1 - Prob. 78ECh. 12.1 - Prob. 79ECh. 12.1 - Prob. 80ECh. 12.1 - Prob. 81ECh. 12.1 - HOW DO YOU SEE IT? The four figures below are...Ch. 12.1 - Proof Let r(t) and u(t) be vector-valued functions...Ch. 12.1 - Proof Let r(t) and u(t) be vector-valued functions...Ch. 12.1 - Proof Prove that if r is a vector-valued function...Ch. 12.1 - Prob. 86ECh. 12.1 - Prob. 87ECh. 12.1 - Prob. 88ECh. 12.1 - Prob. 89ECh. 12.1 - Prob. 90ECh. 12.2 - Prob. 1ECh. 12.2 - Prob. 2ECh. 12.2 - Prob. 3ECh. 12.2 - Prob. 4ECh. 12.2 - Prob. 5ECh. 12.2 - Prob. 6ECh. 12.2 - Prob. 7ECh. 12.2 - Prob. 8ECh. 12.2 - Prob. 9ECh. 12.2 - Prob. 10ECh. 12.2 - Prob. 11ECh. 12.2 - Prob. 12ECh. 12.2 - Prob. 13ECh. 12.2 - Prob. 14ECh. 12.2 - Prob. 15ECh. 12.2 - Prob. 16ECh. 12.2 - Prob. 17ECh. 12.2 - Prob. 18ECh. 12.2 - Prob. 19ECh. 12.2 - Prob. 20ECh. 12.2 - Prob. 21ECh. 12.2 - Prob. 22ECh. 12.2 - Prob. 23ECh. 12.2 - Prob. 24ECh. 12.2 - Prob. 25ECh. 12.2 - Prob. 26ECh. 12.2 - Finding Intervals on Which a Curve Is Smooth In...Ch. 12.2 - Finding Intervals on Which a Curve Is Smooth In...Ch. 12.2 - Finding Intervals on Which a Curve Is Smooth In...Ch. 12.2 - Finding Intervals on Which a Curve Is Smooth In...Ch. 12.2 - Finding Intervals on Which a Curve Is Smooth In...Ch. 12.2 - Prob. 32ECh. 12.2 - Prob. 33ECh. 12.2 - Prob. 34ECh. 12.2 - Prob. 35ECh. 12.2 - Prob. 36ECh. 12.2 - Using Two Methods In Exercises 37 and 38, Find (a)...Ch. 12.2 - Prob. 38ECh. 12.2 - Prob. 39ECh. 12.2 - Prob. 40ECh. 12.2 - Finding an Indefinite Integral In Exercises 39-46,...Ch. 12.2 - Prob. 42ECh. 12.2 - Prob. 43ECh. 12.2 - Prob. 44ECh. 12.2 - Prob. 45ECh. 12.2 - Prob. 46ECh. 12.2 - Prob. 47ECh. 12.2 - Prob. 48ECh. 12.2 - Prob. 49ECh. 12.2 - Prob. 50ECh. 12.2 - Prob. 51ECh. 12.2 - Prob. 52ECh. 12.2 - Finding an Antiderivative In Exercises 53-58, find...Ch. 12.2 - Prob. 54ECh. 12.2 - Prob. 55ECh. 12.2 - Prob. 56ECh. 12.2 - Finding an Antiderivative In Exercises 53-58, find...Ch. 12.2 - Prob. 58ECh. 12.2 - Prob. 59ECh. 12.2 - Prob. 60ECh. 12.2 - Prob. 61ECh. 12.2 - Prob. 62ECh. 12.2 - Prob. 63ECh. 12.2 - Prob. 64ECh. 12.2 - Prob. 65ECh. 12.2 - Prob. 66ECh. 12.2 - Prob. 67ECh. 12.2 - Prob. 68ECh. 12.2 - Particle Motion A particle moves in the xy-plane...Ch. 12.2 - Prob. 70ECh. 12.2 - Prob. 71ECh. 12.2 - Prob. 72ECh. 12.2 - Prob. 73ECh. 12.2 - Prob. 74ECh. 12.2 - Prob. 75ECh. 12.2 - Prob. 76ECh. 12.3 - CONCEPT CHECK Velocity Vector An object moves...Ch. 12.3 - Prob. 2ECh. 12.3 - Prob. 3ECh. 12.3 - Prob. 4ECh. 12.3 - Prob. 5ECh. 12.3 - Prob. 6ECh. 12.3 - Prob. 7ECh. 12.3 - Prob. 8ECh. 12.3 - Prob. 9ECh. 12.3 - Prob. 10ECh. 12.3 - Prob. 11ECh. 12.3 - Prob. 12ECh. 12.3 - Prob. 13ECh. 12.3 - Prob. 14ECh. 12.3 - Prob. 15ECh. 12.3 - Prob. 16ECh. 12.3 - Prob. 17ECh. 12.3 - Prob. 18ECh. 12.3 - Prob. 19ECh. 12.3 - Prob. 20ECh. 12.3 - Prob. 21ECh. 12.3 - Prob. 22ECh. 12.3 - Prob. 23ECh. 12.3 - Prob. 24ECh. 12.3 - Finding a Position Vector by Integration In...Ch. 12.3 - Prob. 26ECh. 12.3 - Prob. 27ECh. 12.3 - Prob. 28ECh. 12.3 - Prob. 29ECh. 12.3 - Prob. 30ECh. 12.3 - Prob. 31ECh. 12.3 - Prob. 32ECh. 12.3 - Prob. 33ECh. 12.3 - Prob. 34ECh. 12.3 - Projectile Motion In Exercises 27-40, use the...Ch. 12.3 - A bomber is flying horizontally at an altitude of...Ch. 12.3 - Prob. 37ECh. 12.3 - Prob. 38ECh. 12.3 - Prob. 39ECh. 12.3 - Prob. 40ECh. 12.3 - Prob. 41ECh. 12.3 - Prob. 42ECh. 12.3 - Shot-Put Throw The path of a shot thrown at an...Ch. 12.3 - Shot-Put Throw A shot is thrown from a height of...Ch. 12.3 - Prob. 45ECh. 12.3 - Prob. 46ECh. 12.3 - Prob. 47ECh. 12.3 - Prob. 48ECh. 12.3 - Prob. 49ECh. 12.3 - Prob. 50ECh. 12.3 - Prob. 51ECh. 12.3 - Circular Motion In Exercises 51 and 52, use the...Ch. 12.3 - Prob. 53ECh. 12.3 - Prob. 54ECh. 12.3 - Prob. 55ECh. 12.3 - Prob. 56ECh. 12.3 - Prob. 57ECh. 12.3 - HOW DO YOU SEE IT? The graph shows the path of a...Ch. 12.3 - Proof Prove that when an object is traveling at a...Ch. 12.3 - Prob. 60ECh. 12.3 - Prob. 61ECh. 12.3 - Prob. 62ECh. 12.3 - Prob. 63ECh. 12.4 - Prob. 1ECh. 12.4 - Prob. 2ECh. 12.4 - Prob. 3ECh. 12.4 - Prob. 4ECh. 12.4 - Prob. 5ECh. 12.4 - Prob. 6ECh. 12.4 - Prob. 7ECh. 12.4 - Prob. 8ECh. 12.4 - Prob. 9ECh. 12.4 - Prob. 10ECh. 12.4 - Prob. 11ECh. 12.4 - Prob. 12ECh. 12.4 - Prob. 13ECh. 12.4 - Prob. 14ECh. 12.4 - Finding the Principal Unit Normal Vector In...Ch. 12.4 - Finding the Principal Unit Normal Vector In...Ch. 12.4 - Finding the Principal Unit Normal Vector In...Ch. 12.4 - Finding the Principal Unit Normal Vector In...Ch. 12.4 - Prob. 19ECh. 12.4 - Prob. 20ECh. 12.4 - Prob. 21ECh. 12.4 - Prob. 22ECh. 12.4 - Prob. 23ECh. 12.4 - Prob. 24ECh. 12.4 - Finding Tangential and Normal Components of...Ch. 12.4 - Prob. 26ECh. 12.4 - Prob. 27ECh. 12.4 - Prob. 28ECh. 12.4 - Prob. 29ECh. 12.4 - Prob. 30ECh. 12.4 - Prob. 31ECh. 12.4 - Prob. 32ECh. 12.4 - Prob. 33ECh. 12.4 - Prob. 34ECh. 12.4 - Prob. 35ECh. 12.4 - Prob. 36ECh. 12.4 - Prob. 37ECh. 12.4 - Prob. 38ECh. 12.4 - Finding Tangential and Normal Components of...Ch. 12.4 - Prob. 40ECh. 12.4 - Prob. 41ECh. 12.4 - Prob. 42ECh. 12.4 - Finding Vectors An object moves along the path...Ch. 12.4 - Prob. 44ECh. 12.4 - Prob. 45ECh. 12.4 - Prob. 46ECh. 12.4 - Prob. 47ECh. 12.4 - Prob. 48ECh. 12.4 - Prob. 49ECh. 12.4 - Prob. 50ECh. 12.4 - Prob. 51ECh. 12.4 - Prob. 52ECh. 12.4 - Prob. 53ECh. 12.4 - Prob. 54ECh. 12.4 - Prob. 55ECh. 12.4 - Prob. 56ECh. 12.4 - Projectile Motion Find the tangential and normal...Ch. 12.4 - Prob. 58ECh. 12.4 - Prob. 59ECh. 12.4 - Prob. 60ECh. 12.4 - Air Traffic Control Because of a storm, ground...Ch. 12.4 - Projectile Motion A plane flying at an altitude of...Ch. 12.4 - Prob. 63ECh. 12.4 - Prob. 64ECh. 12.4 - Prob. 65ECh. 12.4 - Prob. 66ECh. 12.4 - Prob. 67ECh. 12.4 - Prob. 68ECh. 12.4 - Prob. 69ECh. 12.4 - Prob. 70ECh. 12.4 - Prob. 71ECh. 12.4 - Prob. 72ECh. 12.4 - Proof Prove that the sector T(t) is 0 for an...Ch. 12.4 - Prob. 74ECh. 12.4 - Prob. 75ECh. 12.4 - Prob. 76ECh. 12.5 - Curvature Consider points P and Q on a curve What...Ch. 12.5 - Prob. 2ECh. 12.5 - Prob. 3ECh. 12.5 - Prob. 4ECh. 12.5 - Prob. 5ECh. 12.5 - Prob. 6ECh. 12.5 - Prob. 7ECh. 12.5 - Prob. 8ECh. 12.5 - Prob. 9ECh. 12.5 - Prob. 10ECh. 12.5 - Prob. 11ECh. 12.5 - Prob. 12ECh. 12.5 - Prob. 13ECh. 12.5 - Prob. 14ECh. 12.5 - Prob. 15ECh. 12.5 - Prob. 16ECh. 12.5 - Investigation Consider the graph of the...Ch. 12.5 - Prob. 18ECh. 12.5 - Prob. 19ECh. 12.5 - Prob. 20ECh. 12.5 - Finding Curvature In Exercises 19-22, find the...Ch. 12.5 - Prob. 22ECh. 12.5 - Prob. 23ECh. 12.5 - Prob. 24ECh. 12.5 - Prob. 25ECh. 12.5 - Prob. 26ECh. 12.5 - Prob. 27ECh. 12.5 - Prob. 28ECh. 12.5 - Prob. 29ECh. 12.5 - Prob. 30ECh. 12.5 - Prob. 31ECh. 12.5 - Prob. 32ECh. 12.5 - Finding Curvature In Exercises 29-36, find the...Ch. 12.5 - Prob. 34ECh. 12.5 - Prob. 35ECh. 12.5 - Prob. 36ECh. 12.5 - Prob. 37ECh. 12.5 - Prob. 38ECh. 12.5 - Prob. 39ECh. 12.5 - Finding Curvature In Exercises 37-40, find the...Ch. 12.5 - Prob. 41ECh. 12.5 - Prob. 42ECh. 12.5 - Prob. 43ECh. 12.5 - Prob. 44ECh. 12.5 - Prob. 45ECh. 12.5 - Prob. 46ECh. 12.5 - Prob. 47ECh. 12.5 - Prob. 48ECh. 12.5 - Prob. 49ECh. 12.5 - Maximum Curvature In Exercises 49-54, (a) find the...Ch. 12.5 - Prob. 51ECh. 12.5 - Prob. 52ECh. 12.5 - Maximum Curvature In Exercises 49-54, (a) find the...Ch. 12.5 - Prob. 54ECh. 12.5 - Prob. 55ECh. 12.5 - Prob. 56ECh. 12.5 - Prob. 57ECh. 12.5 - Prob. 58ECh. 12.5 - Prob. 59ECh. 12.5 - Prob. 60ECh. 12.5 - Prob. 61ECh. 12.5 - Prob. 62ECh. 12.5 - Prob. 63ECh. 12.5 - Prob. 64ECh. 12.5 - Prob. 65ECh. 12.5 - The smaller the curvature of a bend in a road, the...Ch. 12.5 - Prob. 67ECh. 12.5 - Prob. 68ECh. 12.5 - Prob. 69ECh. 12.5 - Prob. 70ECh. 12.5 - Prob. 71ECh. 12.5 - Prob. 72ECh. 12.5 - Prob. 73ECh. 12.5 - Prob. 74ECh. 12.5 - Prob. 75ECh. 12.5 - Prob. 76ECh. 12.5 - Prob. 77ECh. 12.5 - Prob. 78ECh. 12.5 - Prob. 79ECh. 12.5 - Prob. 80ECh. 12.5 - Prob. 81ECh. 12.5 - Prob. 82ECh. 12.5 - True or False? In Exercises 83-86, determine...Ch. 12.5 - Prob. 84ECh. 12.5 - Prob. 85ECh. 12.5 - Prob. 86ECh. 12.5 - Prob. 87ECh. 12.5 - Prob. 88ECh. 12.5 - Prob. 89ECh. 12.5 - Prob. 90ECh. 12.5 - Prob. 91ECh. 12.5 - Prob. 92ECh. 12.5 - Prob. 93ECh. 12.5 - Prob. 94ECh. 12 - Domain and Continuity In Exercises 1-4, (a) And...Ch. 12 - Prob. 2RECh. 12 - Prob. 3RECh. 12 - Prob. 4RECh. 12 - Evaluating a Function In Exercises 5 and 6....Ch. 12 - Prob. 6RECh. 12 - Prob. 7RECh. 12 - Prob. 8RECh. 12 - Prob. 9RECh. 12 - Prob. 10RECh. 12 - Prob. 11RECh. 12 - Prob. 12RECh. 12 - Prob. 13RECh. 12 - Prob. 14RECh. 12 - Prob. 15RECh. 12 - Prob. 16RECh. 12 - Prob. 17RECh. 12 - Prob. 18RECh. 12 - Prob. 19RECh. 12 - Prob. 20RECh. 12 - Prob. 21RECh. 12 - Prob. 22RECh. 12 - Prob. 23RECh. 12 - Prob. 24RECh. 12 - Prob. 25RECh. 12 - Prob. 26RECh. 12 - Prob. 27RECh. 12 - Prob. 28RECh. 12 - Prob. 29RECh. 12 - Prob. 30RECh. 12 - Prob. 31RECh. 12 - Prob. 32RECh. 12 - Prob. 33RECh. 12 - Prob. 34RECh. 12 - Prob. 35RECh. 12 - Prob. 36RECh. 12 - Prob. 37RECh. 12 - Prob. 38RECh. 12 - Prob. 39RECh. 12 - Prob. 40RECh. 12 - Prob. 41RECh. 12 - Prob. 42RECh. 12 - Prob. 43RECh. 12 - Prob. 44RECh. 12 - Prob. 45RECh. 12 - Prob. 46RECh. 12 - Prob. 47RECh. 12 - Prob. 48RECh. 12 - Prob. 49RECh. 12 - Prob. 50RECh. 12 - Prob. 51RECh. 12 - Prob. 52RECh. 12 - Prob. 53RECh. 12 - Finding Tangential and Normal Components of...Ch. 12 - Prob. 55RECh. 12 - Prob. 56RECh. 12 - Prob. 57RECh. 12 - Prob. 58RECh. 12 - Prob. 59RECh. 12 - Prob. 60RECh. 12 - Prob. 61RECh. 12 - Prob. 62RECh. 12 - Prob. 63RECh. 12 - Prob. 64RECh. 12 - Prob. 65RECh. 12 - Finding Curvature In Exercises 63-66, find the...Ch. 12 - Prob. 67RECh. 12 - Prob. 68RECh. 12 - Finding Curvature in Rectangular Coordinates In...Ch. 12 - Finding Curvature in Rectangular Coordinates In...Ch. 12 - Finding Curvature in Rectangular Coordinates In...Ch. 12 - Prob. 72RECh. 12 - Prob. 73RECh. 12 - Cornu Spiral The cornu spiral is given by...Ch. 12 - Prob. 2PSCh. 12 - Prob. 3PSCh. 12 - Projectile Motion Repeat Exercise 3 for the case...Ch. 12 - Prob. 5PSCh. 12 - Cardioid Consider the cardioid r=1cos,02 as shown...Ch. 12 - Prob. 7PSCh. 12 - Prob. 8PSCh. 12 - Prob. 9PSCh. 12 - Prob. 10PSCh. 12 - Prob. 11PSCh. 12 - Exit Ramp A highway has an exit ramp that begins...Ch. 12 - Prob. 13PSCh. 12 - Ferris Wheel You want to toss an object to a...
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