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Chapter 3: Q3 TF. (page 301)

Parametric curves: Imagine the curve traced in the xy-plane by

the coordinates (x, y) = (3z + 1, z2鈭� 4) as z varies, where the

parameter z is a function of time t.

If the parameter z moves at 5 units per second,

find the instantaneous rate of change of the x- and

y-coordinates as the curve passes through the

point (7, 0).

Short Answer

Expert verified

dxdt=15units/sec

dydt=20 units/sec

Step by step solution

01

Step 1. Given Information 

(x,y)=(3z+,z2-4)

anddzdt=5units/sec

02

Step 2. Calculation 

Here, (x,y)=(3z+1,z2-4)

Therefore,x=3z+1&y=z2-4

Differentiating the above two equations with respect to time both sides we get,

dxdt=3dzdt&dydt=2zdzdt

Since dzdt=5&(x,y)=(7,0)

Therefore, y=0

or, z2-4=0

Therefore, z=2. Since for z=-2, xnot equals to 7.

Then dxdt=3.5

=15

and dydt=2.2.5

=20

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