Mat 210 - Rates of Change Practice



[pic]Mat210 - More Practice with Rates of Change

Print Out and Practice Exercises

The "Average Rate of Change" is the slope of a function over an ______________. The line that connects the two points is called a ____________________line.

 

"Instantaneous Rate of Change" is the slope of a function at a________________. The line that passes through this point (and just "nicks" the graph) is called a ________________ line. For a tangent line to exist at a point, the function's graph must be smooth and continuous.

 

The Instantaneous Rate of Change of a Function is called a D____________________. The key difference between the average rate of change and the instantaneous rate of change, is that of a rate over some interval, which is just an average, and the exact rate at any instant desired. This is why the slope of the tangent line, or d___________________, touches the graph only at one place (instant.)

 

The units of a derivative, which are the units of slope, are always the: [pic]

 

Problem 1: Consider the given table, which is a city's population as a function of years.

|Population Data Table |

|Year |

Money spent on advertising in thousands of dollars: |50 |100 |150 |200 |250 |300 |350 | |profit (in $millions) |2.4 |5.1 |6.4 |6.5 |5.8 |4.3 |1.9 | | 

• Determine the rate of change in the company's profits as they increase advertising spending from $150,000 to $300,000.

• Find a model for this set of data.

• Determine the average rate of change in the company's profits as they increase spending from $75,000 to $220,000.

• Approximate the instantaneous rate of change in the company's profits if they spend $175,000. What does this mean in the context of the problem?

• Approximate the instantaneous rate of change in the company's profits if they spend $300,000. What does this mean in the context of the problem? How does this compare to part d?

• Using your model, determine the optimal amount that the company should spend on advertising. Explain your methods. What would be the company's profits? Explain the significance of the tangent line at this point.

• Give the units associated with the derivative. Explain.

 

 

Problem 3: The percentage of high school seniors who have used a graphing calculator can be modeled by the logistic ("S shaped") function:

 

[pic]

 

Where t is the number of years since 1986 and p is percent.

 

• On your own sheet of graph paper, sketch a graph of this function over the interval 0 £ t £ 14. According to the model, what percent of high school seniors have used a graphing calculator as of 2004?

• Determine the average rate of change of calculator use between the years 1993 and 1999.

• During what one-year period (e.g. 1986-1987, etc.) did calculator usage increase the fastest? By how fast?

• Is there any time during 1986-2000 when calculator usage is decreasing? Explain.

• On your graph, carefully determine the point at which the rate of change in calculator usage was growing the fastest. Sketch in a tangent line at this point, and carefully determine its slope.

• What do you notice about this tangent line compared to other possible tangent lines on this graph? Is there a relationship between this point and concavity? Explain.

• Give the units associated with the derivative. Explain.

 

 

Problem 4: (Curve sketching) The population of a city is modeled by a function P(t), where t is time in years since 1980, and P is the population. The following information is given:

 

• P (0) = 25,000

• P (17) = 45,000

• [pic]is negative for t > 12, and positive for t < 12.

 

• From this data, sketch a possible graph of P(t). Use 0 £ t £ 20 as your interval.

• What has occurred in the city's growth rate in 1992?

• The information given is minimal, but some assumptions can be made. Consider the following assumptions and explain if they are valid assumptions (supported by the information) or not valid.

• The population in 1994 was greater than the population in 1997.

• The population in 1991 was greater than the population in 1997.

• The average rate of change in the city's population from 1992 to 2000 was decreasing.

• In 1985, the city was experiencing growth.

• It is possible to determine the population of the city in 1992.

• There were more people in the city in 2000 than in 1980.

d) Give the units associated with the derivative. Explain.

 

Problem 5: (Understanding Information given by the First and Second Derivatives) To find where the rate of change of a function is zero, we look for x values where the slope of the t____________________ is zero. This is because the slope of a h____________________ line is zero. The procedure is this: first we find the d____________________ of the function. We set the d____________________ equal to z_________ and solve for the i__________ v___________, which is frequently x.

We use this procedure also to find relative minimum and maximum function values. The x values where the slope of the tangent line is zero or does not exist are called c____________ v________________. After finding the c_____________ v_________________, we can determine whether the y value at each is is a relative m________________ or m__________________ by using either the f____________________ d____________________ t______________ or the s_____________ d_____________ t__________________.

To find where the rate of change (r.o.c.) is a maximum or minimum, we find the s__________________ d___________________ and set it equal to zero, since the s___________________ d______________________ is the slope graph for the f____________________ d___________________. The geatest or least (slowest) rate of change occurs at the i____________________ p_____________________.

 

We also use the second derivative to determine the c_______________________ or type of curvature of the graph in a region. If the second derivative is n__________________ at a point, then the graph is c_____________________ d_________________. If the second derivative is p______________________, then the graph is c__________________ u_______.

If a graph is concave down, then at a critical point we have a relative m___________________. If the graph is concave up, then we have a relative m______________________ at the critical point.

[pic]

last update: 11/20/04 sw

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download