Rates of Change and Limits

    In the simplest words, calculus is the study of how functions change; and the instigating idea is the limit. We will first study how a function changes, say from to , and we do so by utilizing the concept of the limit of a function. As our first step in the study of the calculus, we motivate how limits arise by studying slopes of secant lines  (as Fermat did).    

(1) Definition (Average Rate of Change) Suppose is a function of say When a change in the variable is made from to there is a corresponding change to the namely The average rate of change of with respect to is

        Average rate of change

and is also known as the difference quotient of the function

    The quantity is called the absolute rate of change and sometimes can be much harder to determine than We will study techniques that allow us to estimate the value of the absolute rate of change.

(2) Example (Average Rate of Change)  Let Find the average rate of change from to

    Solution. The average rate of change of from to is given by,



which is also the slope of the secant line through and Here is the graph of and the secant line through these two points.

(3) Example (Average Velocity)  If a billiard is dropped from a height of 500 feet, its height at time is given by the position function where is measured in feet and is measured in seconds. Find the average velocity over the intervals and

    Solution. For the interval the object falls from a height of feet to a height of feet. The average velocity is
    
feet/second.

For the interval the object falls from a height of feet to a height of feet. The average velocity is
    
feet/second.

Note that the average velocities are negative indicating that the object is moving downward.

(4) Definition (Limit of a Function) Suppose that the domain of f contains points x arbitrarily close to c but different from c. Then means that the functional values can be made arbitrarily close to a unique number by choosing sufficiently close to (but not equal to ).

(5) Example (Finding a Limit by a Table) Find the limit of as approaches using a table of functional values for and

    Solution. We compute,  



Thus as x approaches from the left we estimate that approaches ; and as x approaches from the right we estimate that approaches Therefore, we estimate

(6) Proposition (Two-Sided Limits) The two-sided limit exists if and only if the one-sided limits and both exist and, in this case, and so,

(7) Definition (Limit of a Function) Suppose that the domain of f contains points x arbitrarily close to c but different from c. Then means, for all there exists , such that



for any in the domain of

(8) Example (Necessity of a Formal Definition) We will use a guessing method to show why the formal definition of a limit is a necessity. Use tables of values to find the limit,

    Solution. As before, we construct a table of values.



From the table it appears that   However, if we persevere with smaller values of the next table



suggests



In fact, which is easily proven once the formal limit definition is used to prove some interesting limit rules and continuity is discussed.

(9) Example (Limits with Piecewise Functions) Determine the values of the following limits based upon the given graph of the function

    (a)     (b)     (c)
    

(10) Exercise (Limits with Piecewise Functions) Determine the values of the following limits based upon the given graph of the function

    (a)     (b)     (c)
    
    (d)     (e)     (f)
    

(11) Example (Limits with Piecewise Functions) Determine the values of the following limits based upon the given graph of the function

    (a)     (b)     (c)
    
    (d)     (e)     (f)

    (g)     (h)     (i)

    

(12) Example (Limits with Piecewise Functions) Determine the values of the following limits based upon the given graph of the function

    (a)     (b)     (c)
    
    (d)     (e)     (f)

    (g)     (h)     (i)

(13) Example (Limits with Piecewise Functions) Determine the values of the following limits based upon the given graph of the function

    (a)     (b)     (c)
    
    (d)     (e)     (f)

    (g)     (h)     (i)

(14) Definition (Instantaneous Rate of Change) As the average rate of change approaches the instantaneous rate for change; that is,

    Instantaneous Rate of Change

and is also known as the derivative of at

(15) Example (Instantaneous Rate of Change)  Let Find the instantaneous rate of change at

    Solution. Using the definition of the instantaneous rate of change,















the instantaneous rate for change of at is 2.

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