Welcome to Mrs. Young's calculus class blog! Each week, I will start a new post. Students, you can write questions for me or chat with each other about how to solve a particular problem. As part of your class participation grade each week, every student must comment at least once to my post or another student's comment. I look forward to spending this year with you. Enjoy!

Sunday, September 16, 2012

September 17th - 21st

This week we are going to finish chapter 2! On Monday we will finish lesson 2.4 and will learn how to find the slope of a line with only point. This will let us find the slope of the tangent line to a curve at a particular point, which will help us find instantaneous rate of change. The chapter 2 test will be Thursday, 9/20. Make sure you are studying a little bit each day! For this week, please watch the video to the right. This gives a re-cap of lesson 2.4 - how to find the slope of a tangent line using the slope of secant lines. For your post, I'd like you to tell me in your own words what the difference is between average rate of change and instantaneous rate of change and how that relates to secants lines and tangent lines. Then, I'd like you to give me a real life example of average rate versus instantaneous rate of change.
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17 comments:

  1. BenIvanSeptember 20, 2012 at 10:53 AM

    The average rate of change can best be described as the average amount that something changes over a period of time. It is the secant line which is the average slope between two points. The instantaneous rate of change is the slope of the tangent line. This line only goes through one point, which will give you a certain value at one individual time. It is the limit as the two points of the secant line become one point. An example is the speed of a car. At any one time you can look at the speedometer and see how fast you are going (Instantaneous), but if you go 20 miles in one hour, that's average.

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    1. Mrs. YoungSeptember 20, 2012 at 4:50 PM

      Ben, average rate of change in your example would be more like if you're average speed in one hour is 60 mph. That's not exactly how fast you may be going at a specific second in that hour. You could be going faster or slower than that, but your different speeds might average out to 60.

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  2. Daniel BurketSeptember 20, 2012 at 2:11 PM

    The average rate of change is the slope of the line between two points. The instanteous rate of change is the slope of a line that only goes through one point in a function. Secant lines (lines that go through two points, e.g. in a circle) are used to find the average rate of change. Tangent lines (lines that only go through one point) are used to find the instanteous rate of change.

    In a track meet, someone may average a 5:30 pace (minutes per mile) over a two mile race. This would be the average rate of change. The instanteous rate of change would be the runner's pace at a specific point in the race. At this point, the runner could be running a 5:10 pace.

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  3. UnknownSeptember 20, 2012 at 3:47 PM

    Average rate is the rate of changing of time. It is slope of a secant line between two points on a curve. Instantaneous rate is the rate of changing of one moment. It is slope of a tangent line of a point which secant line's two point become same. When you decide that the price of manufacturing of some items, you can use average rate of changing for finding average price between the number of items. And when you want to know the price for one item, you can use instantaneous rate.

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    1. Mrs. YoungSeptember 20, 2012 at 4:54 PM

      Bonnie, the price of an item is not a rate. You could talk about the number of items manufactured. For instance, the average number of items manufactured over a certain amount of days could be 2,000 per day, but an instantaneous rate would be how many were manufactured on a specific day. Perhaps one day, only 1,800 were manufactured and another day 2,200 were manufactured.

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    2. UnknownSeptember 20, 2012 at 5:15 PM

      Thank you !!:)

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  4. rachel schillingSeptember 20, 2012 at 3:54 PM

    Average rate of change is finding the slope using two points on a curve, where instantaneous rate of change is finding the slope of a single point. The line that is formed from the two points on a curve is called the secant line and the line that touches the curve at only one point is the tangent line. As the horizontal distance between the two points of the secant line reaches zero, it becomes the tangent line. If the number of grass blades was recorded each day for a year, the average rate would be the average number of grass blades per day and the instantaneous rate of change would be how many blades of grass grew on a particular day.

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  5. UnknownSeptember 20, 2012 at 4:33 PM

    Average rate of change is change over a period of time, however instataneous rate of change is change in a moments time. The tangent line only goes through one sat of values, wheres the secant line consists two points. Therefore the tangent line is used to figure out the instanteous rate of change, and the secant like is used to find the average rate of change.
    If i drive from Ohio to west virginia, a distance of 350 miles, and i leave ohio at 12pm, and get to west virginia at 5pm, a total of 5 hours. instantaneous rate of change can tell me at what time during the trip what speed i was traveling at. Average rate of change however is taking total miles and diving it by the number of hours the trip toiokj me, which would give me a average rate of change of 70mph.

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  6. jacob eversoleSeptember 21, 2012 at 6:19 AM

    The average rate of change is the change of slope between two points and the instantaneous rate of change is the slope of just one point. Secant lines are used to find the average rate of change and tangent lines are used to find the instantaneous rate of change. For example, if you wanted to find out how steep a ski mountain is you would use average rate of change, but if you want to find out how steep the edge of a jump is on the mountain compared to the ground, you would use the tangent line.

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    1. Mrs. YoungSeptember 21, 2012 at 6:22 AM

      I'm not sure I quite understand your example. Remember that we're talking about rates, which are changes in quantities over time. You could talk about the speed of a skier down a hill and his average speed from 2 minutes to 10 minutes versus his instantaneous speed at 5 minutes itself.

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  7. Isaac BarringerSeptember 21, 2012 at 6:21 AM

    The average rate of change is just the slope of a function. The instantaneous rate of change is the change over a specific period of time. When finding the average rate of change is you use what is called the secant line which goes through two point over the curved line. When finding the instantaneous rate of change, you use a tangent line which is only one point on the curved line. A real world example for average rate of change is if you calculate the rain fall over a certain period of time. But for instantaneous rate of change, you calculate how much rain fell at a certain point in time.

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  8. UnknownSeptember 21, 2012 at 6:43 AM

    the average rate of change is an amount of something over an amount of time, which is also expressed by the slope of the secant line. The instantaneous rate of change is the amount of something at a specific point in time. Which is also known as the tangent line. The tangent line only touches one point of the graph, which gives the instantaneous rate at that given moment. A real world example would be the electrical gauge out side of your house which tells you the instantaneous rate at the given time. And at the end of the month your bill tells you the average rate of change of your electricity.

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  9. AnonymousSeptember 21, 2012 at 8:46 AM

    Average rate of change between two points on a function (slope) and is represented by a secant line. Instantaneous rate of change is the change at a specific point on the function and is represented by a tangent line. As the horizontal distance between the two points on the secant line become 0 that is when you get your tangent line. A real world example would be if a 10 gallon bucket had a leak and continued to leak until it was empty 5 min later you could calculate the average rate at which the bucket emptied. You could also figure out the instantaneous rate by figuring out how fast the bucket was leaking at exactly 2 min.

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  10. Grace HillSeptember 21, 2012 at 9:56 AM

    The average rate of change in a function is the slope of a line between two points. The instantaneous rate of change in a function is the slope in a singular point. Secant lines go through two points on a circle and are used to find the average rate of change while tangent lines only go through one point and are used to find the instantaneous rate of change. If I am traveling to New York, it takes about ten hours to reach my destination. I would say my average rate is 65 miles per hour over a period of ten hours. But for the instantaneous rate, I could potentially look at my speed after 4.6 hours of travel, and find that I had been driving 68 miles per hour at that moment.

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  11. SarahDunn2013September 21, 2012 at 10:16 AM

    Average rate of change is the slope of a straight line connecting two points of a graph. It does not necessarily follow the curve of the graph but represents the total change in the y-value over the total change in the x-value. Instantaneous rate of change, however, is the rate at which the curve is changing at a certain point on the graph. It is more useful when looking at the graphs of a curve rather than a straight line. The line used to find an average rate of change is a secant line because it crosses the graph at two points. When the secant line is “backed-up” along the function until it reaches the tangent line, or the line which only touches a single point on the graph, this is where the instantaneous rate of change can be found.
    When traveling somewhere and you arrive at your destination, you might say we traveled 400 miles in 6 hours, therefore we were traveling around 67 mi/hr. This is an example of an average rate of change. Because it is not possible for a constant speed to have been maintained the throughout the trip, a graphical representation would not appear as a straight line. This is where an instantaneous rate might help you find how fast you were traveling at a particular point in the journey. You could figure how fast you were moving at half way through the trip, etc.

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  12. Carly CannonSeptember 21, 2012 at 10:34 AM

    The average rate of change is the change between two points on a function; otherwise known as slope. On a function the secant line is the average rate of change. Instantaneous rate of change is the rate of change at a single point on the function. The tangent line on a function is the instantaneous rate of change. A real world example would be the amount of cars sold over a year (average rate of change)and then amount of cars sold on a single day (instantaneous rate of change).

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