Texts:
Problem 1a: Establish, using the definition of the derivative, that the derivative of f(x) is f'(x) = -if'(c) if c is a constant.
b) The volume V in liters of 3 g of CO at 27Ā°C is given in terms of its pressure p in atmospheres by the formula.
What is the rate of change of V with respect to p when p = 2 atm?
c) Plot V(p) and V'(p) on the same graph. Be sure to include all appropriate axis labels, arrows, and scale markings.
Problem 2: The Robbins cinquefoil (Potentilla robbinsiana) is a small, extremely rare, yellow-flowered alpine plant found exclusively in New Hampshire's White Mountains. It was on the list of endangered species from the 1950s until 2002, when recovery efforts became successful.
a) At the time the Robbins cinquefoil was taken off the endangered species list, there were about 14,000 plants in a single group on Mount Washington. If we assume that the growth of the plant population is exponential and that the population of this group will double every six years, then a formula for the number of plants at time t, where t represents the number of years after June 2002, is N(t) = 14,000 * 2^(t/6).
How many plants do we expect to be in this group exactly eighteen years after June 2002?
b) When do we expect 170,000 plants to be in this group? Please give a year and a month.
c) How fast is the number of plants expected to grow eighteen years after June 2002, and how fast is it expected to grow twenty years after June 2002? Please give your answer in units of plants per year.
Second quadrant where x < -8. You will need to use implicit differentiation.