# Problems 1-3 (10 pts. each)

The malthusian model or "exponential growth" model for a population of size $N$ purpose that at time $t$ is proportional to which of the following? Circle the correct answer: a. The carrying capacity of the population's environment at time $t$. b. The initial population, $N_0$. c. The population at time $t$, $N(t)$. d. The exponential constant $e$A certain fish population grows according to the differential equation
Where $N$ is the size of the population and $t$ is the time in weeks. Assuming the initial population of fish was not zero, which of the following statements is true?

- The fish population will die out within a week.
- The population of fish will approach 4000 and then remain relatively stable. -The population of fish will slowly approach 4000 and then die out.
- The population of fish will grow at a rate of 20 fish a week every week until there are 4000 fish.

3. A team of star scientists is observing a red supergiant. They have a rough model of how the star's volume changes according to its current volume:
The left-right axis is volume $V$, and the up-down axis is The model is given by for some function $g$, and the graph above is $V$ versus $g(V)$.

The star is currently at the indicated equilibrium $V_0.g(V_0)=0$ and $g$ is positive near this equilibrium both to the left and to the right.

Which of the following statements describes the predected behavior of the star under this model? (Circle all that apply.)

The star is currently at the indicated equilibrium $V_0.g(V_0)=0$ and $g$ is positive near this equilibrium both to the left and to the right.

Which of the following statements describes the predected behavior of the star under this model? (Circle all that apply.)

- a. If the volume of the star decreases at all from $V_0$, then it will return to equilibrium $V_0$.
- b. The equilibrium of $V_0$ is stable. if the volume is perturbed slightly in either direction, it will tend toward $V_0$.
- c. The equilibrium at $V_0$ is unstable. If the volume is perturbed slightly in either direction, it will tend toward $V_0$.
- d. $g$ is not differentiable at $V_0$ so it is impossible to determine the star's behavior.
- e. If the volume of the star increase at all from $V_0$, it will continue to grow toward $V_1$.
- f. The equilibrium at $V_1$ is stable.
- g. The equilibrium at $V_1$ is unstable.

4. Let x and y be nonzero in $R^n$. What is the formula for the angle between these vectors?

5. What is the distance between the two points (2,-1,3) and (7,4,3) in 3-space?

6. Louis is trying to find a function $f(x,y)$ whose graph $z=f(x,y) is the plane containing the point (-2,0,1) and which is prependicular to the vector [1,3,-1]' the' indicates a column vector).

Help Louis out by finding the values of $a,b,$and $c$

$f(x,y)=ax+by+c$.

where $a,b,c$ are constants.Help Louis out by finding the values of $a,b,$and $c$

7. Find the solution to the following differential equation which passes through the point (0,2):

8. Find the equilibria of the following differential equation and determine whether they are stable or unstable:

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