Grad School Comment

NOTE: Graduate school should be free! (Via Tuition waivers and Teaching/Research assistantships, for most people in STEM this is true.)

Brain Gains (Rapid Recall, Jivin' Generation)

  • Consider the function $f(x,y) = x^2y+7y$. We'll focus on what happens at the point $(1,2)$. Note that $f(1,2) = 16$.
    • Compute the differential $df(1,2)$ and gradient $\vec \nabla f(1,2)$.
    • Let $(x,y,z)$ be a point on the tangent plane to $f$ at $(1,2)$. The differentials $dx = x-1$, $dy = y-2$, and $dz = z-16$ represent changes in $x$, $y$, and $z$ on the tangent plane. Give an equation of the tangent plane to $f$ at $(1,2,16)$.
    • Give an equation of the tangent line to the level curve of $f$ at $(1,2)$ (what does $dz$ equal on a level curve).
    • Compute the directional derivative of $f$ at $(1,2)$ in the direction $(3,4)$.
  • Let $f(x,y) = x^2\sin(y)+y^3$.
    • Compute $f_x$ and then compute $\dfrac{\partial}{\partial x}(f_x)$ and $\dfrac{\partial}{\partial y}\left(\dfrac{\partial f}{\partial x}\right)$
    • Compute $f_y$ and then compute $\dfrac{\partial}{\partial x}(f_y)$ and $\dfrac{\partial^2f}{\partial y^2}$
  • Find a number $c$ so that the vectors $(1+c,2)$ and $(4,6)$ lie on the same line (are parallel or antiparallel).

Group Problems

  1. Let $f(x,y) = 9-x^2-y^2$ with $\vec r(t) = (2\cos t,3\sin t)$.
    1. State $f(\vec r(t))$ and then compute $\frac{df}{dt}$.
    2. Compute the differential $df$ in terms of $x,y,dx,dy$, and then compute the differentials $dx$ and $dy$ in terms of $t$ and $dt$.
    3. Use substitution from your previous computations to obtain $df$ in terms of $t$ and $dt$. Then state $df/dt$.
  2. Give an equation of the tangent plane to $f(x,y) = 9-x^2-y^2$ at the point $(2,-3)$. [Find the differential, and then substitute $dx = x-2$, $dy = y-?$, $dz = ?$.]
  3. Give an equation of the tangent line to the level curve of $f(x,y) = 9-x^2-y^2$ at the point $(2,-3)$. [Your answer will be very similar to the one above. What changes?]
  4. Let $g(x,y)=x\cos(xy)$.
    1. Compute $g_x$ and $g_y$, and then state the first derivative $Dg(x,y)$.
    2. Compute the second partials $\dfrac{\partial}{\partial x}\left(\dfrac{\partial f}{\partial x}\right)$, $\dfrac{\partial}{\partial y}\left(\dfrac{\partial f}{\partial x}\right)$, $\dfrac{\partial}{\partial x}\left(\dfrac{\partial f}{\partial y}\right)$, and $\dfrac{\partial}{\partial y}\left(\dfrac{\partial f}{\partial y}\right)$.
    3. State the second derivative $D^2g(x,y)$ (it should be a 2 by 2 matrix).
  5. Find the directional derivative of $f(x,y)=xy^2$ at $P=(4,-1)$ in the direction $(-3,4)$. [Check: $D_{(-3,4)}f(4,-1) = \vec\nabla f(4,-1)\cdot \frac{(-3,4)}{5}=(-8,16)\cdot \frac{(-3,4)}{5}=88/5$.]
  6. Give an equation of the tangent plane to $xy+z^2=7$ at the point $P=(-3,-2,1)$. [Check: $(-2)(x-(-3))+(-3)(y-(-2))+2(1)(z-1)=0$. ]
  7. Give an equation of the tangent plane to $z=f(x,y)=xy^2$ at the point $P=(4,-1,f(4,-1))$. [Check: $z-4 = (-1)^2(x-4)+2(4)(-1)(y-(-1))$.]
  8. Consider the function $f(x,y,z) = 3xy+z^2$. We'll be analyzing the surface at the point $P=(1,-3,2)$.
    • If $dx=0.1$, $dy=0.2$ and $dz=0.3$, then what is $df$ at $P$.
    • Find the directional derivative of $f$ at $P$ in the direction $(1,-2,2)$.
    • Give an equation of the tangent plane to the level surface of $f$ that passes through $P$.
    • Give an equation of the tangent plane to the level surface of $f$ that passes through $(a,b,c)$.