Question

x−cf(x)−f(c)​ is known as a difference quotient. We will eventually use difference quotients to help us define and find the instantaneous rate of change of the function. That is, the rate of change of a function f at one specific domain value of the function, namely x=c 1. Is it possible to construct a difference quotient when x=c ? Explain your answer. Exploration We will examine a tire on a car being punctured by a sharp object. As the air escapes from the tire, the distance d, in inches, between the rin of the tire and the street is a function of time t in seconds, where t≥0. From data collected, it was determined that this situation can be modeled by the exponential equation d(t)=6(1.490 −t ) Notice that this equation suggests that, at t=0, the rim was 6 inches above the ground. Our goal is to estimate the rate of change of d(t) at precisely the time t=3 seconds. This will be accomplished by making use of the average rate of change and difference quotient concepts that were previously defined. 2. On graph paper, make a table of values and plot the graph of d(t) for t≥0. [Hint: You may want to use a graphing calculator or a software program to compute the values.] 3. Find the average rate of change of d(t) between the values (a) t=1sec. and t=3sec. Also find the average rate of change of d(t) between the values (b) t=2sec. and t=3 sec. and between (c) t=2.5sec. and t=3sec, and between (d) t=2.9sec. and t=3sec. 4. What do the average rates of change found in Exercise 3 represent with respect to the rim of the tire and its proximity to the street. 5. We want to find out how fast the rim of the tire is approaching the street at the instant t=3sec. Discuss how using the average rate of change is an approximation but not exact at the instant t=3sec. The Connection to Limits 6. Notice that your answers for each interval supplied above seem to be approaching what is known as a limit or limiting value. Can you estimate this value? 7. What is the significance of the numerical sign of the limiting value in relation to the motion of the rim of the tire?

Answer 1

The difference quotient cannot be constructed when x=c because it results in division by zero.

Step-by-step explanation:

The difference quotient x−cf(x)−f(c) is commonly used to find the instantaneous rate of change of a function at a specific value of x=c. However, it is not possible to construct a difference quotient when x=c because it results in division by zero. When finding the instantaneous rate of change, we need to approach the specific value of x, but not actually use that value.