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Finding Velocity and Acceleration Vectors in Space In Exercises 11-20, the position vector r describes the path of an object moving in space. (a) Find the velocity vector, speed, and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given value of t. r ( t ) = 〈 ln t , 1 t 2 t 4 〉 t = 3

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Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337552516

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Chapter
Section
BuyFindarrow_forward

Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781337552516
Chapter 12.3, Problem 20E
Textbook Problem
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Finding Velocity and Acceleration Vectors in Space In Exercises 11-20, the position vector r describes the path of an object moving in space.

(a) Find the velocity vector, speed, and acceleration vector of the object.

(b) Evaluate the velocity vector and acceleration vector of the object at the given value of t.

r ( t ) = ln t , 1 t 2 t 4 t = 3

(a)

To determine

To Calculate: The velocity vector, speed, and acceleration vector of the object.

Given r(t)=lnt,1t2,t4t=3

Explanation of Solution

Calculation:

Here

r(t)=lnt.i^+1t2.j^+t4.k^

The velocity vector is

v(t)=ddt(r'(t))=1t.i^2t3.j^+4t3.k^The speed of the object is r'(t)=1t2+4t6+16t6

(b)

To determine

The velocity vector, speed, and acceleration vector of the object.

Given r(t)=lnt,1t2,t4t=3

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Chapter 12 Solutions

Calculus: Early Transcendental Functions
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What...Ch. 12.5 - Curvature Given a twice-differentiable function...Ch. 12.5 - Investigation Consider the function f(x)=x4x2. (a)...Ch. 12.5 - Motion of a Particle A particle moves along the...Ch. 12.5 - Investigation Find all a and b such that the two...Ch. 12.5 - HOW DO YOU SEE IT? Using the graph of the ellipse,...Ch. 12.5 - Sphere and Paraboloid A sphere of radius 4 is...Ch. 12.5 - The smaller the curvature of a bend in a road, the...Ch. 12.5 - Center of Curvature Let C be a curve given by...Ch. 12.5 - Center of Curvature Use the result of Exercise 67...Ch. 12.5 - Curvature A curve C is given by the polar equation...Ch. 12.5 - Curvature Use the result of Exercise 69 to find...Ch. 12.5 - Curvature Given the polar curve r=ea,a0, use the...Ch. 12.5 - Curvature at the Pole Show. that the formula for...Ch. 12.5 - Curvature at the Pole In Exercises 73 and 74, use...Ch. 12.5 - Curvature at the Pole In Exercises 73 and 74, use...Ch. 12.5 - Proof For a smooth curve given by the parametric...Ch. 12.5 - Horizontal Asymptotes Use the result of Exercise...Ch. 12.5 - Curvature of a Cycloid Use the result of Exercise...Ch. 12.5 - Tangential and Normal Components of Acceleration...Ch. 12.5 - Frictional Force A 5500-pound vehicle is driven at...Ch. 12.5 - Frictional Force A 6400-pound vehicle is driven at...Ch. 12.5 - Curvature Verify that the curvature at any point...Ch. 12.5 - Formulas for Curvature Use the definition of...Ch. 12.5 - True or False? 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Cardioid Consider the cardioid r=1cos,02 as shown...Ch. 12 - Proof If r(t) is a nonzero differentiable function...Ch. 12 - Satellite A communications satellite moves in a...Ch. 12 - Binormal Vector In Exercises 9-11, use the...Ch. 12 - Binormal Vector In Exercises 9-11, use the...Ch. 12 - Binormal Vector In Exercises 9-11, use the...Ch. 12 - Exit Ramp A highway has an exit ramp that begins...Ch. 12 - Arc Length and Curvature Consider the...Ch. 12 - Ferris Wheel You want to toss an object to a...

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