Brain Gains (Rapid Recall, Jivin' Generation)

  1. Consider the function $$x^3 - y^3 - 12 x + 3 y - 4.$$
    • Compute the directional derivative of the function at the point $(1,2)$ in the direction $(3,4)$.
    • Give an equation of the tangent plane to the function at the point $(1,2, f(1,2))$.
    • Give an equation of the tangent line to the level curve that passes through the function at the point $(1,2)$.
    • Locate all critical points of the function.
    • One critical point is $(2,1)$. Use the eigenvalues of $D^2f(2,1)$ to classify this critical point as a maximum, minimum, or saddle point. We'll use software to examine the other 3 points.
  2. Consider the function $w(x,y)$, where $x$ and $y$ both depend on $r$ and $h$. Compute $\frac{\partial w}{\partial r}$.
  3. Let $f(x,y)=x^2+y$, and $g(x,y)=4x+3y$. Find the locations of all local maxima and minima of $f$ on the curve $g(x,y) = 12$. In other words, solve the system $\vec \nabla f = \lambda \vec \nabla g$ and $g(x,y)=12$.

Solution

We have $\vec \nabla f = (2x,1)$ and $\vec \nabla g = (4,3)$. The equation $\vec \nabla f = \lambda \vec \nabla g$ gives us $2x=\lambda\cdot 4$ and $1 = \lambda 3$. The second equation tells us $\lambda =1/3$, and the first equation tells us $x=\lambda\cdot 2=2/3$. Substitution into $4x+3y=12$ tells us $y=(12-8/3)/3$.

Group Problems

  1. For the function $f(x,y)=x^2+4xy+3y^2-10x-18y$, verify that the first derivative $Df(x,y)$ and second derivative $D^2f(x,y)$ are $$Df(x,y) = \begin{bmatrix}2x+4y-10&4x+6y-18\end{bmatrix}\quad\text{and}\quad D^2f(x,y) = \begin{bmatrix}\begin{matrix}2\\4\end{matrix}&\begin{matrix}4\\6\end{matrix}\end{bmatrix}. $$
    • Solve $Df(x,y)=\begin{bmatrix}0&0\end{bmatrix}$, to find the critical points of this function. [Check: $(x,y)=(3,1)$.]
    • Find the eigenvalues of $D^2f(3,1)$. [Check: $\lambda = 4\pm\sqrt{20} = 4\pm 2\sqrt{5}$.]
    • Does the function $f$ have a local max, local min, or saddle at $(3,1)$? Explain.
  2. Consider the function $f(x,y,z) = 3xy+z^2$. We'll be analyzing the level surface that passes through the point $P=(1,-3,2)$.
    • Compute the differential $df$, and then evaluate the differential at $P$.
    • For a level surface, the output remains constant (so $df=0$). If we let $(x,y,z)$ be a point on the surface really close to $P$, then we have $dx=x-1$, $dy=y-(-3)$ and $dz = z-?$. Plug this information into the differential at $P$ to obtain an equation of the tangent plane.
    • Give an equation of the tangent plane to the level surface of $f$ that passes through $(1,2,-3)$.
    • Give an equation of the tangent plane to the level surface of $f$ that passes through $(a,b,c)$.
    • What relationship exists between the gradient of $f$ at $P$ and the tangent plane through $P$?
  3. Suppose a plane passes through the point $(a,b,c)$ and has normal vector $(A,B,C)$. Give an equation of that plane.
  4. Give an equation of the tangent plane to $xy+z^2=7$ at the point $P=(-3,-2,1)$.
  5. Give an equation of the tangent plane to $z=f(x,y)=xy^2$ at the point $P=(4,-1,f(4,-1))$.
  6. Suppose $h$ is a function of $p$ and $q$, where $p$ and $q$ are both functions of $x$, $y$, and $z$. Compute the partial derivatives of $h$ with respect to $x$, $y$, and $z$.