1. ## Another calculus problems.

1. Find the limit, if it exists, if it does not exist, explain why

lim n to infinity (n!)^(1/n)

Recall the n factorial is defined to be the product of all the integers between 1 and

n inclusive.

2. You are driving through the remote Ontario wilderness, and notice that you are low on fuel. According to your map, which has a grid measuring a 1km scale drawn on it
(ie, the distance from one grid line to the next represents 1km in the real world), you are at the point (0,1) and the road you are on appears to be described by the curve

y= arcsin(x)+sqr root of (1-x^2)

You have enough gas to drive for exactle half a km along this road, and there is a gas station at the end of the road, at the point (1, 1/2 pi) Do you make it to the gas station? The following approximation may be useful: 1/(sqr root of 2) approx equal to 0.707...

3. Given the equation dp/dt=kP-m and an initial population P(0) equal to P0. Solve the initial value problem for the population

2. 3. $\displaystyle \int\frac{dp}{kp-m}=\int dt\Rightarrow\frac{ln(kp-m)}{k}=t+c\Rightarrow ln(kp-m)=kt+c_2\Rightarrow$

$\displaystyle e^{ln(kp-m)}=c_3e^{kt}\Rightarrow kp-m=c_3e^{kt}\Rightarrow p(t)=\frac{c_3e^{kt}+m}{k}$

Solve for P(0)