   Chapter 10.3, Problem 9E

Chapter
Section
Textbook Problem

# Finding Slope and Concavity In Exercises 9-18, find dy/dx and d2y/dx2, and find the slope and concavity (if possible) at the given value of the parameter.Parametric Equations Parameter x = 4 cos θ ,   y   =   4   sin   θ θ = π 4

To determine

To calculate:

The value of dydx, d2ydx2 slope and concavity of parametric equations x=4sinθ,y=4cosθ, at

parameter θ=π4.

Explanation

Given:

The parametric equations,

x=4sinθy=4cosθ

And, parameter θ=π4.

Calculation:

Consider the equations,

x=4sinθy=4cosθ

Differentiate x=4sinθ with respect to θ, to get,

dxdθ=4cosθ …..(1)

Differentiate y=4cosθ with respect to θ, to get,

dydθ=4sinθ ……(2)

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

dy/dθdx/dθ=dydx=4cosθ4sinθ=cotθ …..(3)

Now, in order to find slope, substitute θ=π4 in equation (3).

That is.,

(dydx)θ=π4=cotπ4=1

Therefore,

At θ=π4, slope is 1

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