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Math
CalculusMultivariable Calculus8th Edition

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Chapter 13.2, Problem 20E
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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

Solutions

Chapter
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Problem 1E
Problem 2E
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Problem 7E
Problem 8E
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Problem 11E
Problem 12E
Problem 13E
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Problem 17E
Problem 18E
Problem 19E
Problem 20E
Problem 21E
Problem 22E
Problem 23E
Problem 24E
Problem 25E
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Problem 28E
Problem 29E
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Problem 31E
Problem 32E
Problem 33E
Problem 34E
Problem 35E
Problem 36E
Problem 37E
Problem 38E
Problem 39E
Problem 40E
Problem 41E
Problem 42E
Problem 43E
Problem 44E
Problem 45E
Problem 46E
Problem 47E
Problem 48E
Problem 49E
Problem 50E
Problem 51E
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Problem 53E
Problem 54E
Problem 55E
Problem 56E
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Problem 58E
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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find the unit tangent vector T(t) at the point with the given value of the parameter t.

20. r(t) = sin2t i + cos2t j + tan2t k, t = π/4

To determine

To find: The unit tangent vector T(t) for the vector r(t)=sin2t i+cos2t j+tan2t k at t=π4.

Explanation

Formula used:

Write the expression to find unit tangent vector T(t) for the vector r(t) at a finite value of scalar parameter t.

T(t)=r′(t)|r′(t)| (1)

Here,

r′(t) is the tangent vector, which is the derivative of vector r(t).

Find the tangent vector r′(t) by differentiating each component of the vector r(t) as follows.

ddt[r(t)]=ddt(sin2t i+cos2t j+tan2t k)r′(t)=ddt(sin2t)i+ddt(cos2t)j+ddt(tan2t)k

Use the following formula to compute the expression.

ddtcost=−sintddtsint=costddtsin2t=2sint(ddtsint)ddtcos2t=2cost(ddtcost)

ddttan2t=2tant(ddttant)ddttant=sec2t

Compute the expression r′(t)=ddt(sin2t)i+ddt(cos2t)j+ddt(tan2t)k by using the formulae as follows.

r′(t)=2sintddt(sint)i+2costddt(cost)j+2tantddt(tant)k=2sint(cost)i+2cost(−sint)j+2tant(sec2t)k=(2sintcost)i−(2sintcost)j+(2tantsec2t)k

Calculation of tangent vector r′(t) at t=π4:

Substitute π4 for t in the expression r′(t)=(2sintcost)i−(2sintcost)j+(2tantsec2t)k

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Sect-13.1 P-1ESect-13.1 P-2ESect-13.1 P-3ESect-13.1 P-4ESect-13.1 P-5ESect-13.1 P-6ESect-13.1 P-7ESect-13.1 P-8ESect-13.1 P-9ESect-13.1 P-10E
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