Question

(a) If z = f(x, y) = x^2 + 6xy - y^2, find the differential dz. (b) If x changes from 2 to 2.07 and y changes from 3 to 2.93, compare the values of ?z and dz. SOLUTION (a) The definition of the differential gives dz = ?z/?x dx + ?z/?y dy = ( )dx + ( )dy. (b) Putting x = 2, dx = ?x = 0.07, y = 3, and dy = ?y = -0.07, we get dz = [ (2) + 6(3)](0.07) + [6(2) - 2(3)]( ) = . The increment of z is as follows. (Round your final answer to two decimal places.) ?z = f(2.07, 2.93) - f(2, 3) = [( )^2 + 6(2.07)(2.93) - (2.93)^2] - [2^2 + 6(2)(3) - ^2] = .

          (a) If z = f(x, y) = x^2 + 6xy - y^2, find the differential dz.

(b) If x changes from 2 to 2.07 and y changes from 3 to 2.93, compare the values of ?z and dz.

SOLUTION
(a) The definition of the differential gives
dz = ?z/?x dx + ?z/?y dy
= ( )dx + ( )dy.

(b) Putting x = 2, dx = ?x = 0.07, y = 3, and dy = ?y = -0.07, we get
dz = [ (2) + 6(3)](0.07) + [6(2) - 2(3)]( )
= .

The increment of z is as follows. (Round your final answer to two decimal places.)
?z = f(2.07, 2.93) - f(2, 3)
= [( )^2 + 6(2.07)(2.93) - (2.93)^2] - [2^2 + 6(2)(3) - ^2]
= .

(a) If z = f(x, y) = x^2 + 6xy - y^2, find the differential dz.

(b) If x changes from 2 to 2.07 and y changes from 3 to 2.93, compare the values of ?z and dz.

SOLUTION
(a) The definition of the differential gives
dz = ?z/?x dx + ?z/?y dy
= ( )dx + ( )dy.

(b) Putting x = 2, dx = ?x = 0.07, y = 3, and dy = ?y = -0.07, we get
dz = [ (2) + 6(3)](0.07) + [6(2) - 2(3)]( )
= .

The increment of z is as follows. (Round your final answer to two decimal places.)
?z = f(2.07, 2.93) - f(2, 3)
= [( )^2 + 6(2.07)(2.93) - (2.93)^2] - [2^2 + 6(2)(3) - ^2]
= .

Added by Matthew H.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

04/13/2023

Step 1

For x: $$ \frac{\partial f}{\partial x} = 2x + 6y $$ For Y: $$ \frac{\partial f}{\partial Y} = 6x + 2Y $$ Now, we can find the differential dz using the definition: $$ dz = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial Y} dy $$ Substitute the Show more…

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(a) If z = f(x, y) = x^2 + 6xy - y^2, find the differential dz. (b) If x changes from 2 to 2.07 and y changes from 3 to 2.93, compare the values of Δz and dz. SOLUTION (a) The definition of the differential gives dz = ∂z/∂x dx + ∂z/∂y dy = ( )dx + ( )dy. (b) Putting x = 2, dx = Δx = 0.07, y = 3, and dy = Δy = -0.07, we get dz = [ (2) + 6(3)](0.07) + [6(2) - 2(3)]( ) = . The increment of z is as follows. (Round your final answer to two decimal places.) Δz = f(2.07, 2.93) - f(2, 3) = [( )^2 + 6(2.07)(2.93) - (2.93)^2] - [2^2 + 6(2)(3) - ^2] = .