Rapid Recall
- A rover moves in a straight line to a new spot that is 3 m north and 4 m west of its current location. The rover's height drops 2 m along the way. What is the average slope of the hill in the direction the rover moved?
- A rover is located on a hill whose elevation is given by $z=f(x,y) = 3xy+y^2$. Compute the differential $dz$ and write it in the form $dz=(?_1)dx+(?_2)dy.$ In your solution, circle $f_x$ and put a box around $\ds\frac{\partial f}{\partial y}$.
- A rectangle with width $x$ and height $y$ has area given by $A=xy$. If $x$ increases by $dx$ and $y$ increases by $dy$, then use differentials to estimate the increase in area of the box. (I'll draw a picture on the chalkboard that visually shows the answer.)
Group problems
- A rover is located on a hill whose elevation is given by $z=f(x,y) = 3x^2+2xy+4y^2$.
- Compute the differential $dz$ in terms of $x$, $y$, $dx$, and $dy$, and write it in the form $dz=(?_1)dx+(?_2)dy.$
- Write your answer above as the dot product of two vectors, i.e., $dz = (??, ??)\cdot(dx, dy).$
What does the vector $(dx, dy)$ represent physically? - State $\dfrac{\partial f}{\partial x}$ and $f_y$. Then give $\vec \nabla f$ (same as $Df$ since $f$ is scalar valued).
- State the differential at the point $P=(1,1)$ (the spot where the rover currently resides).
- What is the slope of the hill at $P=(x,y)=(1,1)$ in the direction $(dx,dy)=(1,0)$?
- What is the slope of the hill at $P=(1,1)$ in the direction $(0,1)$?
- What is the slope of the hill at $P=(1,1)$ in the direction $(3,4)$?
- The sides of a rectangle $x=3$ ft and $y=2$ ft, with tolerances $dx = .1$ ft and $dy = 0.05$ ft. Use differentials to estimate the tolerance on the area $A=xy$ that results from the given tolerances on $x$ and $y$.
- Let $g(x,y) =x^2y$.
- Give $g_x$ and $\dfrac{\partial g}{\partial y}$. Then state $\vec \nabla g$.
- Find the directional derivative (slope) of $g$ at $P=(3,1)$ in the direction $(-3,2)$.
- Find the directional derivative of $g$ at $P=(3,1)$ in the direction $(2,-5)$.
- The sides of a box are supposed to be $x=3$ ft by $y=2$ ft by $z=1$ ft, with tolerances $dx = .1$ ft by $dy = 0.05$ ft by $dz=0.02$ ft. Use differentials to estimate the tolerance on surface area $A=2xy+2yz+2xz$ that results from the given tolerances.
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