Here is some help with the first three problems. Mine in red.
1) The farmer plans to fence a regular pasture adjacent to a river. The pasture must contain 2,000,000 square meters in order to provide enough grass for the herd. What dimensions would require the least amount of fencing if no fencing is needed along the river?
Draw a diagram. Called the sides x, x, and y. The length is then 2x+y (excluding the river side). Restriction xy = 2,000,000. Solve for x from the restriction, then plug into the perimeter function:
Use calculus to optimize this function. Find the first derivative, then critical numbers, then show that you have minimized the perimeter.
2) A population of 500 bacteria is introduced into a culture and grows in number according to the equation p(t)=500 (1+4t/50+t^2) where t is measured in hours. Find the rate at which the population is growing at t=2.
Find the first derivative using the quotient rule, then plug in t=2. The value you get is a rate measured by the units bacteria/hour.
3) find the horizontal and vertical asymptote of y=2x/x-3
Horizontal: The degrees of the numerator and denominator are equal, therefore the horizontal asymptote is the ratio of the leading coefficient
Vertical: x values that make ONLY the denominator 0, so it's
Good luck on the final!