Problem 1 A population of dinosaurs produces offspring which numbers
P ( t ) = − 0 .
3 t 2 + 0 .
1 t + 10 for times t in the interval [ − 4 , 0]. At the same time, dinosaurs die from various causes, deaths numbering D ( t ) = 0 .
2 t
2
+ 0 .
1 t in the same timespan.
a)The net increase of the population is offspring - death numbers. Find the rate of change of this net increase at t = − 4.
b) Find the equation of the tangent line of P ( t ) at t = − 4 and sketch it, together with the graph of P , over the interval [ − 4 , 0]. Your graph should include the values of P at − 4 , − 2 , 0.
Problem 2 Consider the function
2 t − 3 f ( t ) = t 2 + 2 t − 4 a) Find the derivative of f using quick rules.
b) Find all relative maxima and minima of f .
c) Tell where the function f is increasing and where it is decreasing.
d) Find the maximum and minimum value of f on the interval [ − 3 , 0].
e) Find the equation of the tangent line to the graph of f at t = 2.
Problem 3 A farmer wants to fence in a rectangular field using strong fence on two opposite sides, costing $ 5 per foot, and cheap fence on the two other sides, costing $ 4 per foot.
a) What is the minimum cost to fence in 4000 square feet in this manner?
b) What if the farmer can use an existing 60-foot wall instead of strong fence?
Problem 4 Find the following derivatives, using Quick Rules. No need to simplify.
y =
√
3 x
2 − x + 3 y = x 2 − 4 y = ( x
2 − 1)
2
4 x + 2 y = ( x
3
+ 2 x − 1)
100

Problem 5 A cylindrical barrel is to be built from sheet metal. Assume that the surface area of the barrel is
A = 2 πr
2
+ 2 πhr if h is its height and r the radius of the base. The volume is V = πr
2 h . If you want to build a barrel with a volume of 432 π cubic feet, what are the dimensions that use the least amount of material?