Chapter 10, Problem 44RE

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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+cosÎ¸, y=3+4sinÎ¸ and the value of parameter is Î¸=Ï€6

Formula Used:

The formula for derivative is:

dydx=dydÎ¸dxdÎ¸

Calculation:

The provided parametric equations are x=5+cosÎ¸ and y=3+4sinÎ¸.

Since,

dydx=dydÎ¸dxdÎ¸

So,

dydÎ¸=ddÎ¸(3+4sinÎ¸)=ddÎ¸(3)+ddt(4sinÎ¸)=4cosÎ¸

Then

dydÎ¸=4cosÎ¸

Likely

dxdÎ¸=ddÎ¸(5+cosÎ¸)=ddÎ¸(5)+ddÎ¸(cosÎ¸)=âˆ’sinÎ¸

Therefore,

dxdÎ¸=âˆ’sinÎ¸

Hence,

dydx=dydÎ¸dxdÎ¸=4cosÎ¸âˆ’sinÎ¸=âˆ’4cotÎ¸

Now,

d2ydx2=ddÎ¸(dydx</

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