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CalculusCalculus10th Edition

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Chapter 10.3, Problem 15E
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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Finding Equations of Tangent Lines In Exercises 19-22, find an equation of the tangent line to the curve at each given point.

x = 2 cot θ , y = 2 sin 2 θ , ( − 2 3 , 3 2 ) , ( 0 , 2 ) , ( 2 3 , 1 2 )

To determine

To Calculate: The equation of tangent line to the provided curve x=2cosθ, y=2sin2θ at the points (−23,32),(23,12),(0,2).

Explanation

Given:

The parametric equation,

x=2cotθy=2sin2θ

And, the points (−23,32),(23,12),(0,2).

Formula used:

Point-slope form,

(y−y1)=m(x−x1).

Calculation:

Consider the parametric equations,

x=2cotθy=2sin2θ

Differentiate x=2cotθ with respect to θ, to get,

dxdθ=−2cosec2θ ...... (1)

Differentiate y=2sin2θ with respect to θ, to get,

dydθ=4sinθcosθ ...... (2)

Divide equation (2) by (1), to get,

dydx=dydθdxdθ=4sinθcosθ−2cosec2θ=−2sin3θcosθ ….. (3)

Find value of θ at (−23,32).

Substitute x=−23 in the parametric equation x=2cotθ.

That is.,

−23=2cotθcotθ=−13 …… (4)

And, substitute y=32 in y=2sin2θ.

That is.,

32=2sin2θsinθ=32 …… (5)

From equation (4) and (5),

θ=2π3

So, slope of equation at θ=2π3 is,

Slope=(dydx)θ=2π3=−2sin32π3cos2π3=−2(32)3(−12)=(338)

Now, use point slope form, (y−y1)=m(x−x1).

Therefore,

Equation of tangent line at (−23,32) is,

(y−32)=338(x+23)2y−3=334(x+23)8y−12=33x+633x−8y+18=0

Therefore, equation of tangent line at (−23,32) is 33x−8y+18=0

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Trigonometry (MindTap Course List)

Note: Exercises preceded by an asterisk are of a more challenging nature. In Exercises 29 to 36, Sketch and des...

Elementary Geometry For College Students, 7e

Solve the following exercises by using formulas. Ordinary annuity Annuity Payment Time Nominal Interest Future ...

Contemporary Mathematics for Business & Consumers

Use De Moivres Theorem to find the value of each of the following. Leave your answers in standard form a+bi. (3...

Elements Of Modern Algebra

Finding an Indefinite Integral In Exercises 15-36, find the indefinite integral and check the result by differe...

Calculus: Early Transcendental Functions (MindTap Course List)

Finding a Trigonometric Integral In Exercises 1726, find the trigonometric integral. 11sind

Calculus of a Single Variable

In Exercises 11-20, find the accumulated amount A if the principal P is invested at the interest rate of r/year...

Finite Mathematics for the Managerial, Life, and Social Sciences

In Exercises 5 to 10, classify each statement as simple, conditional, a conjunction, or a disjunction. If Alice...

Elementary Geometry for College Students

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