The length and width of a rectangle are measured as 42 cm and 28 cm, respectively, with an error in measurement of at most 0.1 cm in each. Use differentials to estimate the maximum error in the calculated area of the rectangle.

For a differentiable function of two variables, [tex]z=f(x,y)[/tex], we define the differentials [tex]dx[/tex] and [tex]dy[/tex] to be independent variables; that is, they can be given any values. Then the differential [tex]dz[/tex], also called the total differential, is defined by

[tex]dz=f_x(x,y)dx+f_y(x,y)dy=\frac{\partial z}{\partial x}dx+\frac{\partial z}{\partial y}dy[/tex]

The area [tex]A[/tex] of a rectangle with length [tex]L[/tex] and width [tex]W[/tex] is

[tex]A=L\cdot W[/tex]

So, the differential of [tex]A[/tex] is

[tex]dA=\frac{\partial A}{\partial L}dL+\frac{\partial A}{\partial W}dW=WdL+LdW[/tex]

To find the largest error in the area we take the largest error in the measurement of [tex]W[/tex] and of [tex]L[/tex]. Therefore we take [tex]dW=0.1[/tex] and [tex]dL=0.1[/tex] along with [tex]L=42[/tex], [tex]W=28[/tex]. This gives

[tex]dA=(28)\cdot(0.1)+(42)\cdot(0.1)\\\\dA=2.8+4.2\\\\dA=7[/tex]

Thus the maximum error in the calculated area of the rectangle is [tex]7\:cm^{2}[/tex].