   Chapter 10, Problem 44RE

Chapter
Section
Textbook Problem

Finding Slope and Concavity In Exercises 39-46, find d y / d x and d 2 y / d x 2 , and find the slope and concavity (if possible) at the given value of the parameter.Parametric Equations Parameter x = 5 + cos θ ,   y = 3 + 4 sin θ θ = π 6

To determine

To calculate: The slope and concavity for equation x=5+cosθ, y=3+4sinθ at the given value of the parameter θ=π6 and also calculate dydx, d2ydx2.

Explanation

Given:

The provided parametric equations are x=5+cosθ, y=3+4sinθ and the value of parameter is θ=π6

Formula Used:

The formula for derivative is:

dydx=dydθdxdθ

Calculation:

The provided parametric equations are x=5+cosθ and y=3+4sinθ.

Since,

dydx=dydθdxdθ

So,

dydθ=ddθ(3+4sinθ)=ddθ(3)+ddt(4sinθ)=4cosθ

Then

dydθ=4cosθ

Likely

dxdθ=ddθ(5+cosθ)=ddθ(5)+ddθ(cosθ)=sinθ

Therefore,

dxdθ=sinθ

Hence,

dydx=dydθdxdθ=4cosθsinθ=4cotθ

Now,

d2ydx2=ddθ(dydx</

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