Let f(x, y) = 5x² - sin(7x - 6y). Now, assume that x and y are functions of u and v, given by:

x(u, v) = 6u - 4v², y(u, v) = 3u² - v.

Consider the composite function f(x(u, v), y(u, v)). What is the value of this function at (u, v) = (-1, 1)?

The value of the composite function f ( x ( u , v ) , y ( u , v )) at the point ( u , v ) = ( − 1 , 1 ) is approximately 500.9903 . This is calculated by first determining x and y using the given functions and then substituting these into f ( x , y ) . The final result is derived from evaluating the sine function involved in the formula.
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Answered by Anonymous | 2025-07-04

To solve the problem, we need to find the value of the composite function f ( x ( u , v ) , y ( u , v )) at ( u , v ) = ( − 1 , 1 ) .
First, recall the given function: f ( x , y ) = 5 x 2 − sin ( 7 x − 6 y )
Additionally, we have: x ( u , v ) = 6 u − 4 v 2 y ( u , v ) = 3 u 2 − v
Now, let's find the specific values for x and y when u = − 1 and v = 1 :

Calculate x ( − 1 , 1 ) : x ( − 1 , 1 ) = 6 ( − 1 ) − 4 ( 1 ) 2 = − 6 − 4 = − 10

Calculate y ( − 1 , 1 ) : y ( − 1 , 1 ) = 3 ( − 1 ) 2 − 1 = 3 ( 1 ) − 1 = 2


Now, we have x = − 10 and y = 2 . Substitute these values into the function f ( x , y ) :

Evaluate f ( − 10 , 2 ) : f ( − 10 , 2 ) = 5 ( − 10 ) 2 − sin ( 7 ( − 10 ) − 6 ( 2 )) Simplify further: = 5 ( 100 ) − sin ( − 70 − 12 ) = 500 − sin ( − 82 )

Simplify the sine term: sin ( − 82 ) = − sin ( 82 ) because sine is an odd function. Therefore, we have: 500 + sin ( 82 )


Without a calculator, the exact value of sin ( 82 ) in terms of simple numbers cannot be further simplified. Thus, the value of the function is:
f ( − 10 , 2 ) ≈ 500 + sin ( 82 )
In summary, the value of the composite function f ( x ( u , v ) , y ( u , v )) at ( u , v ) = ( − 1 , 1 ) is 500 + sin ( 82 ) .

Answered by SophiaElizab | 2025-07-07