Rapid Recall
- Use the level curves of $f$ shown on the front board. At each labeled point, draw the gradient.
Solution
- See the board.
- State $a$ and $b$ so that $\vec u=(-8,4,5)$ and $\vec v=(4,a,b)$ lie along the same line.
Solution
- $(4,-2,-\frac{5}{2})$.
- Give an equation of the tangent line to $f(x) = x^2$ at $x=3$. Give your answer using point-slope form.
Solution
- The equation is $y-9=6(x-3)$. This is the same as the differential notation $$\underbrace{dy}_{y-9}=\underbrace{f'(3)}_{6}\underbrace{dx}_{x-3},$$ where $dy=y-9$ (from 9 to $y$) and $dx=x-3$ (from 3 to $x$).
Group problems
- Consider the function $f(x,y)=e^x\sin y$ and the path $\vec r(t) = (t^2,t^3)$.
- Compute $f(\vec r(t))$ and then compute $\frac{df}{dt}$.
- Find $df$ in terms of $dx$ and $dy$. Then find $dx$ and $dy$ in terms of $dt$.
- Use substitution to find $df$ in terms of $dt$. Then state $\frac{df}{dt}$.
- In you previous work, label each of $f_x$, $f_y$, $\frac{dx}{dt}$ and $\frac{dy}{dt}$.
- Consider the function $f(x,y,z) = 4x^2+4y^2+z^2$. We'll be analyzing the surface at the point $P=(1/2,0,\sqrt{3})$.
- Compute $f(1/2,0,\sqrt{3})$. Then draw the level surface that passes through this point. So draw the ellipsoid $4=4x^2+4y^2+z^2$.
- Compute the gradient $\vec\nabla f(x,y,z)$, and then give $\vec\nabla f(P)$.
- Compute the differential $df$, and then 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 $dx=x-1/2$, $dy=y-0$ and $dz = z-?$. Plug this information into the differential to obtain an equation of the tangent plane to the surface.
- 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 $(a,b,c)$.
- We will find the points on the curve $g(x,y)=xy^2=16$ that minimize the function $f(x,y)=x^2+y^2$.
- Compute $\vec \nabla f$ and $\vec \nabla g$.
- To find the points where $\vec \nabla f$ and $\vec \nabla g$ are either parallel or anti-parallel, we need to solve $\vec \nabla f=\lambda\vec \nabla g$ together with $g(x,y) = 16$. Write the three equations that results from this.
- Solve the system above (you should get $x=2$ and $y=\pm \sqrt{8}=\pm 2\sqrt{2}$).
|
Sun |
Mon |
Tue |
Wed |
Thu |
Fri |
Sat |