<< Chapter < Page Chapter >> Page >
In this section, you will:
  • Find the average rate of change of a function.
  • Use a graph to determine where a function is increasing, decreasing, or constant.
  • Use a graph to locate local maxima and local minima.
  • Use a graph to locate the absolute maximum and absolute minimum.

Gasoline costs have experienced some wild fluctuations over the last several decades. [link] http://www.eia.gov/totalenergy/data/annual/showtext.cfm?t=ptb0524. Accessed 3/5/2014. lists the average cost, in dollars, of a gallon of gasoline for the years 2005–2012. The cost of gasoline can be considered as a function of year.

y 2005 2006 2007 2008 2009 2010 2011 2012
C ( y ) 2.31 2.62 2.84 3.30 2.41 2.84 3.58 3.68

If we were interested only in how the gasoline prices changed between 2005 and 2012, we could compute that the cost per gallon had increased from $2.31 to $3.68, an increase of $1.37. While this is interesting, it might be more useful to look at how much the price changed per year . In this section, we will investigate changes such as these.

Finding the average rate of change of a function

The price change per year is a rate of change    because it describes how an output quantity changes relative to the change in the input quantity. We can see that the price of gasoline in [link] did not change by the same amount each year, so the rate of change was not constant. If we use only the beginning and ending data, we would be finding the average rate of change    over the specified period of time. To find the average rate of change, we divide the change in the output value by the change in the input value.

Average rate of change = Change in output Change in input = Δ y Δ x = y 2 y 1 x 2 x 1 = f ( x 2 ) f ( x 1 ) x 2 x 1

The Greek letter Δ (delta) signifies the change in a quantity; we read the ratio as “delta- y over delta- x ” or “the change in y divided by the change in x . ” Occasionally we write Δ f instead of Δ y , which still represents the change in the function’s output value resulting from a change to its input value. It does not mean we are changing the function into some other function.

In our example, the gasoline price increased by $1.37 from 2005 to 2012. Over 7 years, the average rate of change was

Δ y Δ x = $ 1.37 7 years 0.196  dollars per year

On average, the price of gas increased by about 19.6¢ each year.

Other examples of rates of change include:

  • A population of rats increasing by 40 rats per week
  • A car traveling 68 miles per hour (distance traveled changes by 68 miles each hour as time passes)
  • A car driving 27 miles per gallon (distance traveled changes by 27 miles for each gallon)
  • The current through an electrical circuit increasing by 0.125 amperes for every volt of increased voltage
  • The amount of money in a college account decreasing by $4,000 per quarter

Rate of change

A rate of change describes how an output quantity changes relative to the change in the input quantity. The units on a rate of change are “output units per input units.”

The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.

Δ y Δ x = f ( x 2 ) f ( x 1 ) x 2 x 1

Questions & Answers

what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Find that number sum and product of all the divisors of 360
jancy Reply
exponential series
prime number?
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
Practice Key Terms 9

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?