Chapter 10.3, Problem 15E

### 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 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 = 2 + sec θ , y = 1 + 2 tan θ θ = − π 3

To determine

To calculate: The value of dydx, d2ydx2 slope and concavity of parametric equations x=2+secθ,y=1+2tanθ, At parameter θ=π3.

Explanation

Given:

The parametric equations are,

x=2+secÎ¸y=1+2tanÎ¸

And, parameter is Î¸=âˆ’Ï€3.

Formula used:

Differentiation of some standard functions are

d(constant)dx=0, d(secÎ¸)dÎ¸=secÎ¸tanÎ¸, d(tanÎ¸)dÎ¸=2sec2Î¸

Chain rule:

dudv=dudxÃ—dxdy............Ã—dzdv

Calculation:

Consider the equations,

x=2+secÎ¸y=1+2tanÎ¸

Differentiate the equation x=2+secÎ¸ with respect to Î¸, to obtain,

dxdÎ¸=secÎ¸tanÎ¸ â€¦.. (1)

Differentiate the equation y=1+2tanÎ¸ with respect to Î¸, to obtain,

dydÎ¸=2sec2Î¸ â€¦â€¦ (2)

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

dydx=2sec2Î¸secÎ¸tanÎ¸=2cosecÎ¸ â€¦.. (3)

For the slope.

Put Î¸=âˆ’Ï€3 in equation (3).

That is,

(dydx)Î¸=âˆ’Ï€3=2cosec(âˆ’Ï€3)=âˆ’433

Therefore,

At Î¸=âˆ’Ï€3, slope is âˆ’433

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