Application of Partial Derivatives Four organisms meet at $(1,2)$ in a field where the temperature is described by $T(x,y)=100-y^2-3x^3$ with $x$ and $y$ given in m and $T$ in °C. I have to tell them where to go best depending on their preferences:


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*A thermophilic organism wants to turn to higher temperatures

*A phychorophilic organism wants to turn to lower temperatures

*An extotherm organism wants to maintain its body temperature

*An organism (with unknown preference) goes to direction (4,1). What is the slope that this organism experiences? It goes into that direction for 2m. What is the temperature at this destination.



My thoughts


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*Here I simply calculated  the gradient vector, because it is pointing towards the greatest increase. $\nabla f=<-9x^2, -2y>$

*Here the organism should go to the opposite direction of the gradient vector. How do I express this mathematically?

*I don't know how to approach this problem.

*What do I do here? Do I calculate the tangent plane equation at (4,1)?

 A: The gradient is better $\nabla f=-9x^2\hat i-2y\hat j$
The best for each case is
$\mathbf v_1=\nabla f=-9x^2\hat i-2y\hat j$ maximum increasing temperature.
Maximum decreasing temperature $\mathbf v_2=-\nabla f=9x^2\hat i+2y\hat j$
Moving with no increasing or decreasing of temperature. Look for a vector such that $\nabla f·\mathbf v_3=0$ With two solutions $\mathbf v_{3l}=2y\hat i-9x^2\hat j$ and  $\mathbf v_{3r}=-2y\hat i+9x^2\hat j$
Moving in the direction of $\mathbf v_4=4\hat i+\hat j$ we need an unitary vector in that direction and compute the dot product with the gradient to get the ratio of temperature rising wrt distance,
$\hat{\mathbf v}_4=\dfrac{\sqrt{5}}{5}(4\hat i+\hat j)$
$I=\nabla f·\mathbf{\hat v}_4=\dfrac{\sqrt{5}}{5}(-36x^2-2y)$
For the increment when moving the organism 2 meters in this direction, I am not sure if you need an approximation, and in this case you simply multiply $I$ by 2, or you need the exact increment, and now the calculations are $T_e=T(1+4,2+1)-T(1,2)$
In fact, all the way we are working with the plane tangent to the plot of the function in $\mathbb R^3$. In this case, the plane has the equation $-9x^2X-2yY-Z=0$
