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Calculus: Early Transcendental Fun...

6th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781285774770

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BuyFindarrow_forward

Calculus: Early Transcendental Fun...

6th Edition
Ron Larson + 1 other
Publisher: Cengage Learning
ISBN: 9781285774770
Chapter 12.2, Problem 48E
Textbook Problem
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Finding an Indefinite Integral In Exercises 43–50, find the indefinite integral.

( e t i + sin t j + cos t k ) d t

To determine

To calculate: The indefinite integral [eti+sintj+costk]dt.

Explanation of Solution

Given:

The vector valued function:

r(t)=eti+sintj+costk.

The integral of a vector valued function r(t)=f(t)i+g(t)j+h(t)k is given by:

r(t)dt=[f(t)dt]i+[g(t)dt]j+[h(t)dt]k

Calculation:

The vector valued function is:

r(t)=eti+sintj+costk

The integral of a vector valued function r(t)=f(t)i+g(t)j+h(t)k is given by:

r(t)dt=[f(t)dt]i+[g(t)dt]j+[h(t)dt]k

Here,

f(t)=et

Integrating both sides, we get,

f(t)dt=etdt=et+c1

and

g(t)=sint

Integrating both sides, we get,

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Chapter 12 Solutions

Calculus: Early Transcendental Functions (MindTap Course List)
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Ch. 12.1 - Evaluating a Function In Exercises 912. evaluate...Ch. 12.1 - Evaluating a FunctionIn Exercises 912. evaluate...Ch. 12.1 - Writing a Vector-Valued Function In Exercises...Ch. 12.1 - Writing a Vector-Valued Function In Exercises...Ch. 12.1 - Writing a Vector-Valued Function In Exercises...Ch. 12.1 - Writing a Vector-Valued Function In Exercises...Ch. 12.1 - Think About It In Exercises 17 and 18, find r( t)...Ch. 12.1 - Think About It In Exercises 17 and IS, find...Ch. 12.1 - Matching In Exercises 19-22. match the equation...Ch. 12.1 - Matching In Exercises 19-22. match the equation...Ch. 12.1 - Matching In Exercises 19-22. match the equation...Ch. 12.1 - Matching In Exercises 19-22, match the equation...Ch. 12.1 - Think about it In Exercises 23-30 sketch the plane...Ch. 12.1 - Think about it In Exercises 23-30 sketch the plane...Ch. 12.1 - Think about it In Exercises 23-30 sketch the plane...Ch. 12.1 - Think about it In Exercises 23-30 sketch the plane...Ch. 12.1 - Think about it In Exercises 23-30 sketch the plane...Ch. 12.1 - Think about it In Exercises 23-30 sketch the plane...Ch. 12.1 - Think about it In Exercises 23-30 sketch the plane...Ch. 12.1 - Think about it In Exercises 23-30 sketch the plane...Ch. 12.1 - Sketching a space curve In Exercises 31-38 sketch...Ch. 12.1 - Sketching a space curve In Exercises 31-38 sketch...Ch. 12.1 - Sketching a space curve In Exercises 31-38 sketch...Ch. 12.1 - Sketching a space curve In Exercises 31-38 sketch...Ch. 12.1 - Sketching a space curve In Exercises 31-38 sketch...Ch. 12.1 - Sketching a space curve In Exercises 31-38 sketch...Ch. 12.1 - Sketching a space curve In Exercises 31-38 sketch...Ch. 12.1 - Sketching a space curve In Exercises 31-38 sketch...Ch. 12.1 - Identifying Common Curve In Exercises 39 and 40...Ch. 12.1 - Identifying Common Curve In Exercises 39 and 40...Ch. 12.1 - Identifying a Common Curve In Exercises 3942, use...Ch. 12.1 - Identifying a Common Curve In Exercises 3942, use...Ch. 12.1 - Transformation of a vector valued valued in...Ch. 12.1 - Transformations of Vector-Valued Functions In...Ch. 12.1 - Representing a graph by vector valued function In...Ch. 12.1 - Representing a graph by vector valued function In...Ch. 12.1 - Representing a graph by vector valued function In...Ch. 12.1 - Representing a graph by vector valued function In...Ch. 12.1 - Representing a graph by vector valued function In...Ch. 12.1 - Representing a graph by vector valued function In...Ch. 12.1 - Representing a graph by vector valued function In...Ch. 12.1 - Representing a graph by vector valued function In...Ch. 12.1 - Representing a Graph by Vector-Valued FunctionIn...Ch. 12.1 - Representing a Graph by Vector-Valued FunctionIn...Ch. 12.1 - Representing a Graph by Vector-Valued FunctionIn...Ch. 12.1 - Representing a Graph by Vector-Valued FunctionIn...Ch. 12.1 - Representing a Graph by Vector-Valued FunctionIn...Ch. 12.1 - Representing a Graph by Vector-Valued FunctionIn...Ch. 12.1 - Representing a Graph by Vector-Valued FunctionIn...Ch. 12.1 - Representing a Graph by Vector-Valued Function In...Ch. 12.1 - Sketching a Curve Show that the vector-valued...Ch. 12.1 - Sketching a Curve Show that the vector-valued...Ch. 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 - 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 - Finding a Limit In Exercises 65-70, find the limit...Ch. 12.1 - Finding a Limit In Exercises 6368, find the limit...Ch. 12.1 - Continuity of a Vector-Valued Function In...Ch. 12.1 - Continuity of a Vector-Valued Function In...Ch. 12.1 - Continuity of a Vector-Valued Function In...Ch. 12.1 - Continuity of a Vector-Valued Function In...Ch. 12.1 - Continuity of a Vector-Valued Function In...Ch. 12.1 - Continuity of a Vector-Valued Function In...Ch. 12.1 - Writing a Transformation In Exercises 7578....Ch. 12.1 - Writing a Transformation In Exercises 7578,...Ch. 12.1 - Writing a Transformation In Exercises 7578,...Ch. 12.1 - Writing a Transformation In Exercises 7578,...Ch. 12.1 - Continuity of a Vector-Valued Function State the...Ch. 12.1 - Comparing Functions Which of the following...Ch. 12.1 - Playground Slide The outer edge of a playground...Ch. 12.1 - HOW DO YOU SEE IT? 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(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 75 and 76, 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? In Exercises 83-86, determine...Ch. 12.5 - True or False? In Exercises 8588, determine...Ch. 12.5 - True or False? In Exercises 8588, determine...Ch. 12.5 - True or False? In Exercises 83-86, determine...Ch. 12.5 - Keplers Laws In Exercises 87-94, you are asked to...Ch. 12.5 - Keplers Laws In Exercises 87-94, you are asked to...Ch. 12.5 - Keplers Laws In Exercises 87-94, you are asked to...Ch. 12.5 - Show that rGMLrr=e is a constant vector.Ch. 12.5 - Prove Keplers First Law: Each planet moves in an...Ch. 12.5 - Assume that the elliptical orbit r=ed1+ecos is in...Ch. 12.5 - Prove Keplers Second Law: Each ray from the sun to...Ch. 12.5 - Prove Keplers Third Law: The square of the period...Ch. 12 - Domain and Continuity In Exercises 1-4, (a) And...Ch. 12 - Domain and Continuity In Exercises 1-4, (a) And...Ch. 12 - Domain and Continuity In Exercises 14, (a) find...Ch. 12 - Domain and Continuity In Exercises 1-4, (a) And...Ch. 12 - Evaluating a Function In Exercises 5 and 6....Ch. 12 - Evaluating a Function In Exercises 5 and 6....Ch. 12 - Writing a Vector-Valued Function In Exercises 7...Ch. 12 - Writing a Vector-Valued Function In Exercises 7...Ch. 12 - Sketching a Curve In Exercises 9-12, sketch the...Ch. 12 - Sketching a Curve In Exercises 9-12, sketch the...Ch. 12 - Sketching a Curve In Exercises 9-12, sketch the...Ch. 12 - Sketching a Curve In Exercises 9-12, sketch the...Ch. 12 - Representing a Graph by a Vector-Valued Function...Ch. 12 - Representing a Graph by a Vector-Valued Function...Ch. 12 - Representing a Graph by a Vector-Valued Function...Ch. 12 - Representing a Graph by a Vector-Valued Function...Ch. 12 - Finding a Limit In Exercises 17 and 18, find the...Ch. 12 - Finding a Limit In Exercises 17 and 18, find the...Ch. 12 - Higher-Order Differentiation In Exercises 19 and...Ch. 12 - Higher-Order Differentiation In Exercises 19 and...Ch. 12 - Higher-Order Differentiation In Exercises 21 and...Ch. 12 - Higher-Order Differentiation In Exercises 21 and...Ch. 12 - Using Properties of the Derivative In Exercises 25...Ch. 12 - Using Properties of the Derivative In Exercises 25...Ch. 12 - Finding an Indefinite Integral In Exercises 27-30,...Ch. 12 - Finding an Indefinite Integral In Exercises 2528,...Ch. 12 - Finding an Indefinite Integral In Exercises 27-30,...Ch. 12 - Finding an Indefinite Integral In Exercises 2528,...Ch. 12 - Evaluating a Definite Integral In Exercises 31-34,...Ch. 12 - Evaluating a Definite Integral In Exercises 2932,...Ch. 12 - Evaluating a Definite Integral In Exercises 31-34,...Ch. 12 - Evaluating a Definite Integral In Exercises 31-34,...Ch. 12 - Finding an Antiderivative In Exercises 35 and 36,...Ch. 12 - Finding an Antiderivative In Exercises 35 and 36,...Ch. 12 - Finding Velocity and Acceleration Vectors in Space...Ch. 12 - Finding Velocity and Acceleration Vectors in Space...Ch. 12 - Finding Velocity and Acceleration Vectors in Space...Ch. 12 - Finding Velocity and Acceleration Vectors in Space...Ch. 12 - Projectile Motion In Exercises 41 and 42, use the...Ch. 12 - Projectile Motion In Exercises 3942, use the model...Ch. 12 - Projectile Motion In Exercises 3942, use the model...Ch. 12 - Projectile Motion In Exercises 3942, use the model...Ch. 12 - Finding the Unit Tangent Vector In Exercises 43...Ch. 12 - Finding the Unit Tangent Vector In Exercises 43...Ch. 12 - Finding a Tangent Line In Exercises 45 and 46,...Ch. 12 - Finding a Tangent Line In Exercises 45 and 46,...Ch. 12 - Finding the Principal Unit Normal Vector In...Ch. 12 - Finding the Principal Unit Normal Vector In...Ch. 12 - Finding the Principal Unit Normal Vector In...Ch. 12 - Finding the Principal Unit Normal Vector In...Ch. 12 - Finding Tangential and Normal Components of...Ch. 12 - Finding Tangential and Normal Components of...Ch. 12 - r(t)=2ti3tj,[0,5] Finding the Arc Length of a...Ch. 12 - Finding the Arc Length of a Plane Curve In...Ch. 12 - Finding the Arc Length of a Plane Curve In...Ch. 12 - Finding the Arc Length of a Plane Curve In...Ch. 12 - Finding the Arc Length of a Curve in Space In...Ch. 12 - Finding the Arc Length of a Curve in Space In...Ch. 12 - Finding the Arc Length of a Curve in Space In...Ch. 12 - Finding the Arc Length of a Curve in Space In...Ch. 12 - Finding Curvature In Exercises 63-66, find the...Ch. 12 - Finding Curvature In Exercises 63-66, find the...Ch. 12 - Finding Curvature In Exercises 63-66, find the...Ch. 12 - Finding Curvature In Exercises 63-66, find the...Ch. 12 - Finding Curvature In Exercises 67 and 68, find the...Ch. 12 - Finding Curvature In Exercises 67 and 68, find the...Ch. 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 - Finding Curvature in Rectangular Coordinates In...Ch. 12 - Frictional Force A 7200-pound vehicle is driven at...Ch. 12 - Cornu Spiral The cornu spiral is given by...Ch. 12 - Radius of Curvature Let T be the tangent line at...Ch. 12 - Projectile Motion A bomber is flying horizontally...Ch. 12 - Projectile Motion Repeat Exercise 3 for the case...Ch. 12 - Cycloid Consider one arch of the cycloid as shown...Ch. 12 - 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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