   Chapter 10.3, Problem 13E

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 and concavity (if possible) at the given value of the parameter.Parametric Equations Parameter x = cos 3 θ , y = sin 3 θ θ = π 4

To determine

To calculate: The value of dydx, d2ydx2, and slope and concavity of parametric equations x=cos3θ,y=sin3θ, at parameter θ=π4.

Explanation

Given:

The parametric equations,

x=cos3θy=sin3θ

And, parameter θ=π4.

Formula used:

If a smooth curve is given by the parametric equations x(t) and y(t), then the slope of the curve is,

dydx=dy/dtdx/dt, dxdt0

Calculation:

Consider the equations,

x=cos3θy=sin3θ

Differentiate x=cos3θ with respect to t, to get,

dxdθ=3cos2θsinθ …..(1)

Differentiate y=sin3θ with respect to t, to get,

dydθ=3sin2θcosθ ……(2)

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

dydx=dydθdxdθ=3sin2θcosθ3cos2θsinθ=tanθ …..(3)

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

That is.,

(dydx)θ=π4=tan(π4)=1

Therefore, at θ=π4, slope is 1

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