rate of change

(noun)

Ratio between two related quantities that are changing.

Related Terms

Examples of rate of change in the following topics:

  • Rates of Change

    • Linear equations often include a rate of change.  
    • If two points in time and the total distance traveled is known the rate of change, also known as slope, can be determined.  
    • The rate of change is the speed of his run; distance over time.
    • Our speed (rate of change) is simply the slope of the line connecting the two points.
    • Anything that involves a constant rate of change can be nicely represented with a line with the slope.

  • Increasing, Decreasing, and Constant Functions

    • As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways.
    • The average rate of change of an increasing function is positive, and the average rate of change of a decreasing function is negative.
    • The figure below shows examples of increasing and decreasing intervals on a function.
    • Look at the graph from left to right on the $x$-axis; the first part of the curve is decreasing from infinity to the $x$-value of $-1$ and then the curve increases.  
    • Identify whether a function is increasing, decreasing, constant, or none of these

  • Linear Mathematical Models

    • A linear model includes the rate of change $(m)$ and the initial amount, the y-intercept $b$.
    • The total cost is equal to the rate per mile times the number of miles driven plus the cost for the flat fee:
    • Second, in order to write the equations representing each train's total distance in terms of time, calculate the rate of change for each train.
    • Since train A is traveling towards train B, which has a greater $y$ value, train A's rate of change must be positive and equal to its speed of $50$.
    • Train B is traveling towards A, which has a lesser $y$ value, giving B a negative rate of change: $-80$.

  • Population Growth

    • The rate $r$ by which the population is growing is itself a function of four variables.
    • ${\Delta}P$ denotes the change in population.
    • $PGR$ is the rate of change in population over a certain span of time: $t_2-t_1$.
    • A growth rate of $0$ means stagnation in population size.
    • Projections for high (red), medium (orange) and low (green) rates of change are represented accounting for the splitting off of the curve after 2016.

  • Exponential Decay

    • Consider the decrease of a population that occurs at a rate proportional to its value.
    • This rate at which the population is decreasing remains constant but as the population is continually decreasing the overall decline becomes less and less steep.
    • Exponential rate of change can be modeled algebraically by the following formula:
    • The decay constant is indeed a constant, but the form of the equation (the negative exponent of e) results in an ever-changing rate of decline.
    • Given a sample of carbon in an ancient, preserved piece of flesh, the age of the sample can be determined based on the percentage of radioactive carbon-13 remaining. 1.1% of carbon is C-13 and it decays to carbon-12.  

  • Solving Problems with Logarithmic Graphs

    • Some functions with rapidly changing shape are best plotted on a scale that increases exponentially, such as a logarithmic graph.
    • This means that for small changes in the independent variable there are very large changes in the dependent variable.
    • The fact that the rate is ever-increasing (and steeply so) means that changing scale (scaling the axes by $5$, $10$ or even $100$) is of little help in making the graph easier to interpret.
    • $T$ would generate the expected curve, but the scale would be such that minute changes go unnoticed and the large scale effects of the relationship dominate the graph: It is so big that the "interesting areas" won't fit on the paper on a readable scale.
    • Taking the logarithm of each side of the equations yields: $logj=log{(\sigma\tau ) }^4 $.

  • Inverse Variation

    • In the case of inverse variation, the increase of one variable leads to the decrease of another.
    • If the driver shifts into neutral gear, the car's speed will decrease at a constant rate as time increases, eventually coming to a stop.
    • Revisiting the example of the decelerating car, let's say it starts at 50 miles per hour and slows at a constant rate.
    • Note that realistically, other factors (e.g., friction), will influence the rate of deceleration.
    • Notice what happens when you change the variants.

  • Graphs of Exponential Functions, Base e

    • The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e., percentage increase or decrease) in the dependent variable.
    • If the change is positive, this is called exponential growth and if it is negative, it is called exponential decay.
    • For example, because a radioactive substance decays at a rate proportional to the amount of the substance present, the amount of the substance present at a given time can be modeled with an exponential function.
    • Also, because the the growth rate of a population of bacteria in a petri dish is proportional to its size, the number of bacteria in the dish at a given time can be modeled by an exponential function such as $y=Ae^{kt}$ where $A$ is the number of bacteria present initially (at time $t=0$) and $k$ is a constant called the growth constant.
    • The graph of $y=e^x$ (black) lies between that of $y=2^x$ (blue) and $y=3^x$ (green).

  • Formulas and Problem-Solving

    • For example, one can use a linear equation to determine the amount of interest accrued on a home equity line of credit after a given amount of time.
    • Consider the hypothetical situation in which you need money to make home improvements and can open a $20,000 credit line at an interest rate of 2.5% per year.
    • Where I is interest, p is the principal amount loaned ($20,000), r is the interest rate (2%, or 0.02) per year, and T is the number of years elapsed (15 months will be 1.25 years).
    • Graph of gratuity as a function of the price of the bill, y=0.18x, where gratuity of 18%.
    • How would the equation change if you wanted to tip 20%?

  • Problem-Solving

    • According to the United States Census Bureau, over the last 100 years (1910 to 2010), the population of the United States of America is exponentially increasing at an average rate of one and a half percent a year (1.5%).
    • GDP per capita has grown at an exponential rate of approximately two percent per year for two centuries.
    • Compound interest at a constant interest rate provides exponential growth of the capital.
    • In general, a real world application that can be modeled with an exponential function is one that will continuously increase at an increasing rate based on the current value.
    • How would you change the graph based on your example?