5th SaturdaySections 3.4 – 3.6; 4.2 – 4.4, 4.6

Find the derivative of the function:

Example 1

Find the derivative of the function:

Example 2

Find the derivative of the function:

Example 3

The number of deer, N, on a wildlife preserve x years after the herd was first introduced is modeled by the formula:

(a) Find the rate of change model for this function.

(b) Find and interpret .

Example 4

The activity of a certain chemical reaction (measured in U per 100μL) x minutes into an experiment is given by:

(a) Find and interpret .

(b) Find and interpret the percent rate of change at x = 8.

Example 5

Find the derivative of each of the following functions.

(a)

(b)

(c)

Example 6

Suppose the price (in $) of a certain item in month x where x = 1 is January, x = 2 is February and so on is modeled by

Suppose the number of this item sold in month x is modeled by the function

Find the rate of change in the revenue from sales of this item in February and also in August.

Example 7

The number of students (in millions) enrolled full time in American public colleges and universities is modeled by .

The tuition (including fees) paid by such students is modeled by the function thousand dollars per student.

In both functions, x = 1 represents Fall of 2000, x = 2 represents Fall of 2001, and so on.

Find the rate of change in total tuition in American public colleges and universities in the Fall of 2010.

Example 8

Locate all extrema points for the function graphed below.

Example 9

Find all extrema points of the function

on the interval [-6, 6].

Example 10

The percentage of people aged 15 or older in the U.S. who are sleeping x hours after 9:00PM is modeled by:

Find the absolute maximum and minimum values of S between 9:00PM and 8:00AM. What time do these occur?

Example 11

The population of Kentucky from 1981 to 1991 can be modeled by thousand people x years

after 1980.

When was the population the highest and when was it the lowest during this span? What were the highest and lowest populations?

Example 12

State all inflection points of the function .

Identify each as a location of least/most rapid increase or decrease.

Example 13

The population of Kentucky from 1981 to 1991 can be modeled by thousand people x years

after 1980.

When was the population decreasing most rapidly? What were the rate of change in the population and the population in that year?

Example 14

A rancher has 200 feet of wire fence. He wishes to create a rectangular corral into which he will build a 6-foot wide wooden gate. What dimensions will result in the corral with the largest area?

Example 15

A company needs to make a rectangular storage container (no lid) that has a volume of 10m3. The length of the base needs to be twice its width. Material for the sides costs $6 per square meter while material for the bottom of the container costs $10 per square meter. Find the cost for the cheapest container that the company can make?

Example 16

An advertising company uses posters that have total area of 180in2. One of their clients wants to have a poster that has 1-inch margins at the bottom and on both sides while having a 2-inch margin at the top. What are the dimensions of the poster from this company having the largest printed area?

Example 17