Chapter 10.3, Problem 16E

### 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 = t , y = t − 1 t = 5

To determine

To calculate: The value of dydx, d2ydx2, and slope and concavity of parametric equations x=t,y=t1, at Parameter t=5.

Explanation

Given:

The parametric equations,

x=ty=tâˆ’1

And, parameter t=5.

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, dxdtâ‰ 0

Calculation:

Consider the parametric equations,

x=ty=tâˆ’1

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

dxdt=12t â€¦.. (1)

Differentiate y=tâˆ’1 with respect to t, to get,

dydt=12tâˆ’1 â€¦â€¦ (2)

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

dydx=dy/dtdx/dt=12tâˆ’112t=ttâˆ’1 â€¦.. (3)

Now, in order to find slope at t=5, substitute t=5 in equation (3).

That is.,

(dydx)t=5=55âˆ’1=54=1.11

Therefore, at t=5, slope is 1.11

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