Exercise 1082Calculus › Multivariable calculus › Partial derivativesL1Added 3 Nov 2026

The differential

#differential#linearization#approximation

Statement

Let $f(x,y)=x^3+y^3$ . Compute the differential at $(1,2)$ and use it to approximate $f(1.01,1.99)$ .

Hint

$df=f_x(a,b)\,dx+f_y(a,b)\,dy$ with $dx=0.01$ , $dy=-0.01$ .

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Partial derivatives at $(1,2)$ . $f_x=3x^2=3$ , $f_y=3y^2=12$ .

Differential. $df=3\,dx+12\,dy$ .

Approximation. $f(1,2)=9$ , so $f(1.01,1.99)\approx9+3(0.01)+12(-0.01)=9-0.09=8.91$ .

Exact value. $1.01^3+1.99^3\approx1.0303+7.8806=8.9109$ , very close.

Conclusion. Linear approximation gives $\approx8.91$ .