03 Partial Derivatives and Differentiation Online Quiz Questions

When computing the partial derivative $f_x$ of a function $f(x,y)$, which procedure is correct?

The existence of both first partial derivatives at a single point is, by itself, sufficient to guarantee that a function is differentiable at that point.

What is the standard name for the result stating that $f_{xy}=f_{yx}$ when the mixed partial derivatives are continuous in a neighborhood?

Let $f(x,y)=x^2y+e^{xy}$. Which expression is $f_x(x,y)$?

Which quantities are needed to construct the tangent plane to $z=f(x,y)$ at $(a,b,f(a,b))$? Select all correct choices.

For a differentiable function $z=f(x,y)$, the tangent plane, viewed as a function of $x$ and $y$, is the linearization at the base point.

A rectangular box has $l=10$, $w=4$, and $h=2$, with measurement errors $dl=0.02$, $dw=0.01$, and $dh=0.01$. Using differentials, what is the approximate volume error in cubic units?

For $f(x,y)=x^2+xy+y^2$, what is the tangent plane at the point $(1,2,7)$?

Suppose $z=f(x,y)$, where $x=x(u,v)$ and $y=y(u,v)$. Which chain-rule formulas are correct? Select all correct choices.

Use linearization at $(3,4)$ to approximate $f(3.02,3.99)$ for $f(x,y)=\sqrt{x^2+y^2}$. Show the linearization and the numerical estimate.

For a level surface $F(x,y,z)=c$, what is the geometric role of $\nabla F$ at a point on the surface?

Let $f(x,y)=x^2y+e^{xy}$. What is the value of $f_y(1,0)$?

True or false: If both first partial derivatives of a function exist at a single point, then the function must be differentiable at that point.

For $f(x,y)=x^2+xy+y^2$, which equation is the tangent plane to the graph at $(1,2,7)$?

Let $z=x^2+y^2$, where $x=u+v$ and $y=u-v$. What is the value of $z_u$ when $u=3$?

Which condition justifies concluding that $f_{xy}=f_{yx}$ near a point?

Use the linearization of $f(x,y)=\sqrt{x^2+y^2}$ at $(3,4)$ to approximate $f(3.02,3.99)$. Enter the approximation rounded to three decimal places.

A rectangular box has $l=10$, $w=4$, and $h=2$. If $dl=0.02$, $dw=0.01$, and $dh=0.01$, which value is the differential estimate for the volume error?