Estimate $\sqrt{(10.1)^2 − (2.9)^3 − (3.05)^2}$ via differentials

Question

This is a question from my Math for Economics assignment.

If it was a single variable question, like $\sqrt{(2.9)^3}$, I would use tangent line equation $y-y_1=m(x-x_1)$ where $x$ is $2.9$, $x1$ is $3$, $y_1$ is $\sqrt{27}$, and $m$ is $\frac{1}{2\sqrt{27}}$. Then I would solve for $y$ and find the approximation.

But in a case with multiple variables I think chain rule should be somehow implemented but I don't know how considering that I can work with only 1 variable at a time using the tangent line equation.

Any help is greatly appreciated, thank you for your responses.

Answer 1

If $F(M)=F(x,y,z)=\sqrt{x^2-y^3-z^2}$ and $M_0=(10, 3,3)$ then $$F(M_0+H)=F(M_0)+\langle F'(M_0),H\rangle + \|H\|\epsilon (H)$$ where $\epsilon (H)\to_{H\to 0} 0$ and where $\langle M',M\rangle=x'x+y'y+z'z.$ Here $H=(0.1,-0.1,0.05),$ $\|H\|^2= (0.1)^2+(0.1)^2+(0.05)^2$ and $F'(M)=\frac{1}{2F(M)}(2x,-3y^2,-2z).$ The desired approximation is $F(M_0)+\langle F'(M_0),H\rangle.$

Answer 2

Here is an equivalent solution (with final result) in notation more familiar to a first course in multivariable calculus.

Let $f(x,y,z)=\sqrt{x^2-y^3-z^2}$ and $(x_0,y_0,z_0)=(10, 3,3)$.

Then we can form the total differential (where we could use $\Delta x$ instead of $dx$ etc.):

$dz=\frac{\partial f}{\partial x} dx+\frac{\partial f}{\partial y} dy+\frac{\partial f}{\partial z} dz$

Writing $f'(x,y,z) =\frac1{2f}(2x,-3y^2,-2z)$ which does borrow notation from above solution;

setting $(dx,dy,dz)=(.1,-.1,.05)$

evaluating $f'(10,3,3) =\frac1{2f(10,3,3)}(20,-27,-6)$

and finally substituting all in the above expression for $dz$ results in $8.275$ as an approximation to the original value, which is $8.266$.