   Chapter 10, Problem 23RE

Chapter
Section
Textbook Problem

# Find the slope of the tangent line to the given curve at the point corresponding to the specified value of the parameter.23. r = e−θ; θ = π

To determine

To find: The slope of the tangent line to the given curve at the point corresponding to r=eθ and θ=π .

Explanation

Given:

The parametric equation for the variable r is as follows.

r=eθ (1)

The parametric equation for the variable y is as follows.

θ=π (2)

Calculation:

The Cartesian equation of variable x is as follows

x=rcosθ (3)

The Cartesian equation of variable y is as follows

y=rsinθ (4)

Differentiate equation (1) with respect to θ .

drdθ=eθ (5)

Differentiate equation (3) with respect to θ .

dxdθ=drdθcosθrsinθ (6)

Differentiate equation (3) with respect to θ we get

dydθ=drdθsinθ+rcosθ (7)

Divide the equation (7) by (6).

dydx=drdθsinθ+rcosθdrdθcosθrsinθ (8)

Substitute the terms from equation (1) and (5) in (8)

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