03 Partial Derivatives and Differentiation Free Online FlashCards

What does a partial derivative measure?

A partial derivative measures change in one coordinate direction while the other variables are held constant.

Find $f_x$ for $f(x,y)=x^2y+e^{xy}$.

Holding y constant gives $f_x=2xy+ye^{xy}$.

Find $f_y$ for $f(x,y)=x^2y+e^{xy}$.

Holding x constant gives $f_y=x^2+xe^{xy}$.

What is the equality of mixed partials?

If the mixed partial derivatives are continuous in a neighborhood of a point, then $f_{xy}=f_{yx}$ there.

Give a sufficient condition for differentiability.

Continuous first partial derivatives on an open neighborhood of a point are sufficient to guarantee differentiability there.

What does differentiability mean geometrically?

Differentiability means the function has a linear first-order approximation with an error that is small relative to the distance from the base point.

Find the tangent plane for $f=x^2+xy+y^2$ at $(1,2)$.

At $(1,2)$, $f=7$, $f_x=4$, and $f_y=5$, so the tangent plane is $z=7+4(x-1)+5(y-2)=4x+5y-7$.

What is the linearization of $f(x,y)$ at $(a,b)$?

The linearization at $(a,b)$ is $L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)$.

Approximate $\sqrt{(3.02)^2+(3.99)^2}$ by linearization.

Using the linearization at $(3,4)$, $f(3.02,3.99)\approx 5+\frac{3}{5}(0.02)+\frac{4}{5}(-0.01)=5.004$.

What is the differential of $z=f(x,y)$?

For $z=f(x,y)$, the differential is $df=f_x\,dx+f_y\,dy$. It gives the linear approximation to a small change in $f$.

Estimate the volume error for the given box measurements.

For $V=lwh$, $dV=wh\,dl+lh\,dw+lw\,dh=0.16+0.20+0.40=0.76$ cubic units.

State the chain rule along a one-parameter path.

If $z=f(x,y)$, $x=x(t)$, and $y=y(t)$, then $\frac{dz}{dt}=f_x\frac{dx}{dt}+f_y\frac{dy}{dt}$.