section 2.1 the derivative calculus for business, economics, the social and life sciences copyright...

Section 2.1

The Derivative

Calculus for Business, Economics, the Social and Life Sciences

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Learning Objectives

Examine slopes of tangent lines and rates of change

Define the derivative, and study its basic properties

Compute and interpret a variety of derivatives using the definition

Study the relationship between differentiability and continuity

Average Rate of Change

The average rate of change of function f on the interval [a,b] is given by

Note that this is the “slope” of a function between two points.

Quick example:Let , then the average rate of change of f between x=-1 and x=2 is

Note that because f is linear, this is just the slope of the line, which equals -2.

EXAMPLE 1 Average Rate of Change in Cost

The cost to produce k quarts of kettle corn is dollars. What is the average rate of change in cost when changing from producing 10 to 20 quarts? Include units.


SOLUTION

The average rate of change from k = 10 to k = 20 is

Then we have

and


SOLUTION

So the average rate is

The units on the average are in units of the output (cost in dollars) divided by input (quarts produced). Thus the average rate of change is 0.2 dollars per quart.

EXAMPLE 2 Average Rate of Change on a Small Interval

Compute the average rate of change of ...(a)… on the interval [1,2].(b)... on the interval [1,1.1].(c)… “at” x = 1.


SOLUTION

(a)… on the interval [1,2].


SOLUTION

(b) ... on the interval [1,1.1].


SOLUTION

(c) … “at” x = 1.

Here we first compute the average rate of change for an arbitrary distance, h, from x = 1:


SOLUTION

If we observe the trend in this expression as h approaches zero, we should get a sense for the slope “at” the value x = 1:

The Derivative

The derivative of function f is given by

At a point c, is the slope of the tangent line to f at c.

To compute the derivative of a function is to differentiate.

Quick example:Let , the derivative of f is

Like the average rate of change computed earlier, this is merely the slope of the line.

EXAMPLE 3 Instantaneous Change in Cost

The cost in dollars to produce k quarts of kettle corn is

Compute the derivative of the cost function. At what instantaneous rate is cost changing when 10 quarts are produced?


SOLUTION

We compute the derivative by evaluating the limit


SOLUTION


SOLUTION

Thus

Then we find the instantaneous rate of change at ,that is, the derivative at 10:

When producing 10 quarts of kettle corn, cost is increasing at a rate of 0.3 dollars per additional quart produced.

EXAMPLE 4 Change in U.S. Demand for Rice

The demand for rice in the U.S. in 2009 approximately followed where p is

the price per ton and D is the demand in millions of tons of rice.(a)Find and interpret (b)Find the equation of the tangent line to D at p = 500.


SOLUTION

(a)Using the definition of the derivative, we get

We can remove the fractions by multiplying by


SOLUTION

The fraction is still undefined at h=0, so we apply an algebra trick, multiplying by the conjugate


SOLUTION


SOLUTION

so The units on demand are millions of tons of rice, and the units on price are dollars per ton.

Because the derivative is negative, at a unit price of $500 per ton, demand is decreasing by about 4,470 tons per $1 increase in unit price.


SOLUTION

(b) Find the equation of the tangent line to D at p = 500. The tangent line to a function f is defined to be the line passing through the point and having slope equal to the derivative at that point.


SOLUTION First, we find the value of D at p=500:

So we know that the tangent line passes through the point


SOLUTION Next, we use the derivative of D for the slope of the tangent line:

Finally, we use the point-slope formula and simplify:


SOLUTION

The graph of along with its tangent line

at p = 500.

Continuity vs. DifferentiabilityA function f is differentiable at if

is defined.

If a function is differentiable at a point, then it is continuous at that point.

Note that being continuous at a point does not guarantee that the function is differentiable there.

Quick example:Let , and recall that

and so the derivative of f is

Note that the derivative cannot exist at because the derivatives on each side disagree.

section 2.1 the derivative calculus for business, economics, the social and life sciences copyright...

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