Section 4.4 Additional Exponential Models

Curve Sketching

Determine where the function is increasing and decreasing and where its graph is concave up and down. Sketch the graph of the function showing as many key features as possible (high and low points, points of inflection)

#18 y = x – ln x (x > 0)

#8

#12

#10 (Use Calculator)

Applications

Learning Curve

Learn fast early, then level of to a learning “capacity”

#32 ADVERTISING

When professors select texts for their courses, they usually choose from among the books on their shelves. For this reason, most publishers send complimentary copies of new texts to professors teaching related courses. The mathematics editor at a major publishing house estimates that if x thousand complimentary copies are distributed, the first-year sales of a certain new mathematics text will be approximately thousand copies.

a. Sketch this sales function.

b. How many copies can the editor expect to sell in the first year if no complimentary copies are sent out?

c. How many copies can the editor expect to sell in the first year if 10,000 complementary copies are sent out?

d. If the editor’s estimate is correct, what is the most optimistic projection for the first-year sales of the text?

Logistics Curve

rises like exponential function first, then turns over (inflection point ) and flattens out.

#28 POPULATION GROWTH

It is estimated that t years from now, the population of a certain country will be million.

a. Sketch the graph of P(t).

b. What is the current population?

c. What will the population be 50 years from now?

d. What will happen to the population in the long run?

MARGINAL ANALYSIS

A business estimates that when x thousand people are employed, its profit will be P(x) million dollars, where for x > 0. What level of employment maximizes profit? What is the maximum profit?