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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Finding Slope and Concavity In Exercises 39–46, find dy/dx and d2y/dx2, and find the slope and concavity (if possible) at the given value of the parameter.

Parametric Equations Parameter

x = 1 t ,   t =t 2

t=–2

To determine

To calculate: The value of dydx,d2ydx2, slope and concavity for the parametric equation x=1t,y=2t+3 at t=1.

Explanation

Given:

The parametric equation is x=1t,y=2t+3 and the condition is t=1.

Formula used:

For concavity, if d2ydx2>0, then graph is concave upward and if d2ydx2<0, then graph is concave downward.

Parametric form of first derivative is,

dydx=(dydt)(dxdt)

Parametric form of second derivative is,

d2ydx2=ddt(dydx)dxdt

Calculation:

Consider the parametric equation x=1t,y=2t+3 and the condition t=1.

Now for the value of dydx, divide numerator and denominator by dt as both x and y are functions of t,

dydx=(dydt)(dxdt)

Differentiate x=1t and y=2t+3 with respect to t.

dydx=dydtdxdt=2+01t2=21t2=2t2

For the value at t=1, substitute 1 for t in dydx.

dydx|t=1=2(1)2=2(1)=2

Thus, the value of dydx at t=1 is 2.

Also dydx represent the slope for any graph, so the slope is 2.

Now for the value of d2ydx2, again differentiate dydx with respect to t.

d2ydx2=ddt(dydx)dxdt=(ddt(2t2)(1t2))=(4t(1t2))=4t3

For the value at t=1, substitute 1 for t in d2ydx2.

d2ydx2|t=1=4(1)3=4(1)=4

Thus, the value of d2ydx2 at t=1 is 4.

Recall that, for concavity, if d2ydx2>0, then graph is concave upward and if d2ydx2<0, then graph is concave downward

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