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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find the slope of the tangent line to the given curve at the point corresponding to the specified value of the parameter.

24. r = 3 + cos 3θ; θ = π/2

To determine

To find: The slope of the tangent line to the given curve at the point corresponding to r=3+cos3θ and θ=π2 .

Explanation

Given:

The parametric equation for the variable r is as follows.

r=3+cos3θ (1)

The parametric equation for the variable θ is as follows.

θ=π2 (2)

Differentiating (1) with respect to θ we get,

drdθ=3sin3θ (3)

The Cartesian equation of variable x is as follows.

x=rcosθ (4)

The Cartesian equation of variable y is as follows.

y=rsinθ (5)

Differentiating (4) with respect to θ we get,

dxdθ=drdθcosθrsinθ (6)

Differentiating (5) with respect to θ we get,

dydθ=drdθsinθ+rcosθ (7)

Dividing equation (7) by (6) we get,

dydx=drdθsinθ+rcosθdrdθcosθrsinθ (8)

Substitute equation (1) and (3) in (9) we get,

dydx=(3sin3θ)sinθ+(3+cos3θ)cosθ(3sin3θ)cosθ(3+cos3θ)sinθ (9)

Substitute equation (2) in (9) we get,

dydx=(3sin3θ)

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