Calculus

Chapter 2 Limits and Continuity

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We live in a world of constant change. When one quantity changes with respect to another we call that a rate of

change. When you drive in your car the distance you travel changes as the time you travel changes. This rate of

change is the speed of your car. Rate of change is a central topic in calculus and you will begin your journey

into calculus investigating this idea.

2.1 Average and Instantaneous Velocity

1. Suppose you drive in your car from Hinsdale to Milwaukee. The cities

are about 90 miles apart and it takes you 2 hours. Recall that velocity is the

change of position with respect to the change in time. In this activity we

will be looking at various velocities during the trip.

a. What was your average velocity for the trip?

b. Does the average velocity indicate that this was the velocity traveled at each moment during the trip?

Explain.

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The graph below is a plot of distance vs. time for your trip.

Let’s call the function that pairs the distance traveled d in time t function f. So in function notation we have

)(tfd .

c. Looking at the graph, describe this trip in words. That is, when were you speeding up, slowing down,

stopped, and traveling at a constant velocity?

d. From the graph find the values for )0(f and )2(f . Describe what these values mean in terms of your trip.

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e. On the graph, connect the points )0(,0 f and )2(,2 f . Find the slope of this line. What does the slope of

this line tell you about the trip?

f. What was the average velocity during the second half of the trip where 21 t ? Describe this average

velocity in terms of slope.

In general, the average rate of change for any function over any interval is the slope of the line joining the

endpoints of that interval. This line connecting the endpoints is called the secant line. So we see that the slope

of the secant line represents the average rate of change.

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2. Clearly you were not traveling at 45 mph throughout the entire trip. At some point you might be traveling 30

mph or 75 mph or 0 mph. Your car’s speedometer indicates the instantaneous velocity at each moment in time.

a. Suppose you want to know how fast you were traveling at exactly 1 hour 13 minutes and 28 seconds into

your trip. How does calculating this instantaneous velocity pose a problem for us?

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b. As you have discussed in the part a calculating the instantaneous velocity is not as simple as calculating the

average velocity. So let’s think of some strategies for estimating the instantaneous velocity of our car at some

moment in time say t = 1 hour into the trip. We will start with a graphical approach to this problem by zooming

in on the graph around the 1 hour mark as shown below.

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Zooming in we get the following portion of the curve.

The table below gives distance values for our trip at time values 0.25 hours before and after 1 hour.

Point B A C

time (hr) 0.75 1.00 1.25

distance (miles) 25 40.00 46

Remember that we want to determine our instantaneous velocity at the target point A. Since slopes of secant

lines are easy to find and represent average rates of changes perhaps then we can use the slopes of secant lines

to estimate the instantaneous rate of change. Draw a secant line connecting points A and B and another secant

line connecting points A and C. Extend both of these secant lines through the graph. Calculate the slopes of

these two secant lines.

What do these slope values represent?

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c. Do you think that either slope value is an accurate estimate for the instantaneous velocity at A? Explain.

d. Sketch a line in the graph through the target point A whose slope you believe represents the instantaneous

velocity at A. What type of line does this line appear to be?

e. Estimate the coordinates for a second point on the tangent line that you drew in part d and then using the

target point at A calculate the slope of this line. Interpret this value in the context of this problem.

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3. From the previous question we now know that the instantaneous velocity is graphically the slope of the

tangent line at the target point 1,40 . Let’s practice this strategy on our original graph for our trip to

Milwaukee.

a. On the graph below, use a ruler to draw a tangent line to the curve at 1 hours. Look along this tangent line

for another point on the line that is close to or right on the intersection of a pair of grid lines on the graph. Write

the coordinates of this point.

b. Using the coordinates of the target point and the other point on your tangent line, calculate the slope of your

tangent line. Compare this value with others in your group. Interpret this value in the context of this problem.

c. Find an equation for your tangent line (use point-slope form).

d. Using a similar method, estimate the instantaneous velocity at 1.75 hours into the trip

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e. Estimate when the instantaneous velocity of our car was the same as the average velocity for the entire trip.

Discuss how you can find these values graphically.

4. Suppose a tennis ball is dropped from a tall building and d the number of feet it falls in t seconds. The

following table shows some of the values for distance and time.

t (sec) 0 1 2 3 4 5

d (meters) 0 15 58 127 202 333

a. Plot these six data points and connect them with a smooth curve. Appropriate label the axis and indicate the

units used.

b. Calculate the slope of the secant line from 1,15 4,202to and interpret its meaning.

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c. Suppose we want to calculate the instantaneous velocity of the ball at 2t seconds. Graphically what are

we interested in finding?

d. You should realize from part c that we need to estimate the slope of the

tangent line at 2 seconds. We could carefully draw the tangent line at 2 seconds

and then try to estimate a second point on that tangent line to calculate the slope.

We did this in the previous problem to estimate our instantaneous velocity at 1

hour and 1.75 hours. However, since we have a table of values for the position

of the ball at various times we will make use of these table values and use a

different strategy for estimating the slope of the tangent line. Let’s make three

estimates for the instantaneous velocity of the ball at 2t seconds. by finding

the slope of the secant lines for the following (in each case carefully sketch the

corresponding secant line using a straight edge):

i. from 1t to 2t

ii. from 2t to 3t

iii. from 1t to 3t

e. Of the three estimates from part d which do you believe is the best estimate for the velocity of the ball at

2t seconds? Explain your reasoning graphically.

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So we have seen that one possible strategy for finding the slope of the tangent line is to simply draw the tangent

line to the curve at the target point and estimate a second point on this line. Then use this point together with the

target point to calculate the slope. This is a reasonable graphical approach however, as you have probably

noticed, we find that it’s not very accurate since it depends on how each of us draws the tangent line and on our

estimates for points on this line. We have also seen that another possible strategy is to pick two known points

on the curve that when connected produce a secant line that “looks” parallel to the tangent line our target point.

We then use the slope of this secant to estimate the slope of the tangent line since lines that are parallel have the

same slope. You might have also averaged the slopes of secant lines connecting points on either side of the

target point with the target point as an estimate for the slope of the tangent line. These are all valid graphical

strategies but they don’t seem to be very mathematical and depend too much on estimates. Clearly we need a

more mathematical strategy.

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5. Let’s return to the graph below for our trip to Milwaukee and lets investigate a more mathematical strategy

for finding the instantaneous rate of change (slope of the tangent line) at 1 hour into our trip.

a. We have already seen that the slopes of the secant lines AB and AC are not good estimates for the slope of

the tangent line at point A. But maybe we can improve these estimates.

Let’s consider the secant line AB . Draw that secant line above and discuss how you might move point B so

that the slope of this secant line is a better estimate for the slope of the tangent line at A.

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b. Open the Sketchpad file HCtoMilwaukee. Display the secant line AB by clicking the button

. Now grab point B and move it closer to point A. Describe what appears to be

happening to this secant line.

c. . What is a good estimate for the instantaneous velocity at 1 hour into our trip?

d. Click the button . What is the slope of the tangent line? How good was your

estimate from part c? How close is the secant line to this tangent line?

e. As you have seen in this activity, if we move point B “really close” to point A we see that the secant line

becomes nearly coincident to the tangent line and that the slope of the secant line is nearly equal to the slope of

the tangent line. So the big question now is how close is “really close” and when is it close enough? Discuss

this idea.

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Chapter 2 Limits and Continuity

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Notes