mhf4ub_unit_4_lesson_17.pdf - 17 MHF4U-B Properties of the...

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17 MHF4U-B Properties of the Tangent to a Curve
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Lesson 17, page 1 Advanced Functions MHF4U-B Copyright © 2012 The Ontario Educational Communications Authority. All rights reserved. ilc.org Introduction A person drives to work every day and finds she travels about 50 km in about an hour. This means that her average speed is about 50 km/h on a normal day. During the drive, however, there are times when the car has stopped, is accelerating, decelerating, or travelling at a constant speed either greater or less than 50 km/h. There are in fact very few times when the car actually travels at 50 km/h. The driver can describe the trip to work as having an average speed of 50 km/h, or she can look at the speedometer at any time to see the current (instantaneous) speed. Over a specific time interval, the average and instantaneous speeds can vary significantly depending on the conditions. In this lesson, you will look at the relationship between average and instantaneous speeds. Estimated Hours for Completing This Lesson Part A: Instantaneous Speed 1 Part B: Speed as the Tangent to a Curve 1 Part C: Properties of the Tangent to a Curve 1.5 Key Questions 1
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Lesson 17, page 2 Advanced Functions MHF4U-B Copyright © 2012 The Ontario Educational Communications Authority. All rights reserved. ilc.org What You Will Learn After completing this lesson, you will be able to calculate average and instantaneous rates of change calculate average and instantaneous speed from a graph explain the relationships between secants and tangents use a tangent to a curve to determine if it is increasing or decreasing test the tangency of a line and curve using the quadratic formula For this lesson, there is a supplementary interactive tutorial called Slope of a Tangent that you may find useful for review. (You can also view it from its link on the Course Materials page.) The Introduction, pages 1 and 2 of Part 1, and page 1 of Part 2 are relevant to this course.
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Lesson 17, page 3 Advanced Functions MHF4U-B Copyright © 2012 The Ontario Educational Communications Authority. All rights reserved. ilc.org Part A: Instantaneous Speed The average speed is the distance travelled divided by the elapsed time. Average speed is defined as v ave = Δ s Δ t . s represents the change in position. t represents the change in time. From here on in the course, speed will mean instantaneous speed rather than average speed. Instantaneous speed is what the speedometer of your car says at any instant as you speed up, slow down, or travel at a constant rate. If a person drives the 50 km to work in 1 hour and then turns around to go back home in 90 minutes: average speed on the way to work is 50 km/h 50 km 1 hour average speed on the way back is 33.3 km/h 50 km 1.5 hours average speed for the entire trip is 40 km/h 100 km 2.5 hours ( Note : Average speed is not the average of the speeds.) The instantaneous speeds at any time could vary between 0 and
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