Calculus: Early Transcendental Functions
7th Edition
ISBN: 9781337552516
Author: Ron Larson, Bruce H. Edwards
Publisher: Cengage Learning
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Chapter 12.2, Problem 6E
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Motion around a circle of radius a is described by the 2D vector-valued function r(t) = ⟨a cos(t), a sin(t)⟩. Find the derivative r′ (t) and the unit tangent vector T(t), and verify that the tangent vector to r(t) is always perpendicular to r(t).
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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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- Header & Footer Text Consider the vector-valued function = ( 4 cos(t) + 2 sin(t), -cos(t) + V5 4 sin(t), V105 f(t)%3D 20 V105 VI05 sin(e) (a) Compute the first derivative and the nmagnitude of the first derivative. (Hint: Make sure to simplify your answer for the magnitude as much as possible! There will be magic cancellations.) (b) Are ^r(t) and ^r0°(t) perpendicular for all t? (Hint: You do not need to compute the second derivative. Use the formula (1*OF)' = (*"(t) · (1) = 26"(1) · i'(1) and your answer from part a.) (c) Use your answer from part b. to compute the principal normal vector in termns of the second derivative r00(E). (Hint: Your answer should involve the symbol r 0t). Do not compute the second derivative).arrow_forwardDetermine if the derivative of the vector-valued function exists at the specified point. (Your instructors prefer angle bracket notation for vectors. If the derivative exists at the specified point, enter its value. If the derivative does not exist, enter DNE.) FC) = (coste), tan(t), 3t cos(t) at t, =arrow_forwardThe question belowarrow_forward
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