   Chapter 10.3, Problem 10E

Chapter
Section
Textbook Problem

# Finding Slope and Concavity In Exercises 9-18, find d y / d x and d 2 y / d x 2 , and find the slope b and concavity (if possible) at the given value of the parameter. Parametric Equations Parameter x = cos θ ,   y = 3 sin θ θ = 0

To determine

To calculate: The value of dydx and d2ydx2 for x=cosθ,y=3sinθ at θ=0 and also find the slope and concavity.

Explanation

Given:

The parametric equation, x=cosθ,y=3sinθ at θ=0.

Formula used:

For implicit differentiation of the form, x=f(t) and y=g(t), then,

dydx=dydtdxdt.

Calculation:

Here, x=cosθ,y=3sinθ.

Now, differentiate x and y with respect to θ,

dxdθ=ddθ(cosθ)=sinθ

And,

dydθ=ddθ(3sinθ)=3cosθ

Therefore,

dydx=dydθdxdθ=3cosθsinθ=3cotθ

Again, differentiate dydx with respect to x,

d2ydx2=ddx(dydx)=ddx(3cotθ)=ddθ(3cotθ)dθdx=3csc2θ

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