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Cell[CellGroupData[{
Cell["Calculus: The Language of Change", "Title"],
Cell[TextData[{
StyleBox["Exercise Set 10.0\n",
FontVariations->{"Underline"->True}],
"Velocity"
}], "Subtitle"],
Cell["by John Robeson", "Subsubtitle"],
Cell[CellGroupData[{
Cell[TextData[StyleBox["Exercise 1) 9.2.1 Again", "Subtitle"]], "Subtitle"],
Cell["\<\
Look up your solution to Exercise 9.2.1 or resolve it. Be sure to include \
the features of stopping at stop signs and at Grandmother's house in your \
graph. How do the speeds of 65 mph and 25 mph appear on your solution? Be \
especially careful with the slope and shape of your graph. We want to \
connect slope and speed and bend and acceleration later in the chapter and \
will ask you to refer to your solution.\
\>", "Text"],
Cell[CellGroupData[{
Cell["Solution", "Section"],
Cell["Exercise 9.2.1 is repeated below.", "Text"],
Cell[CellGroupData[{
Cell[TextData[StyleBox["Exercise 9.2.1 Again", "Text"]], "Subsection"],
Cell["\<\
Make a qualitative rough sketch of a graph of the distance traveled as a \
function of time on the following hypothetical trip:\
\>", "Text"],
Cell["(a) You travel a total of 100 miles in 2 hours.", "Text"],
Cell["\<\
(b) Most of the trip is on rural interstate highway at the 65 mph speed \
limit. What qualitative feature or shape does the graph of distance vs. time \
have when speed is 65 mph?\
\>", "Text"],
Cell["\<\
(c) You start from your house at rest, gradually increase your speed to 25 \
mph, slow down, and stop at a stop sign.\
\>", "Text"],
Cell["\<\
(d) What shape is the graph of distance vs. time while you are stopped?\
\>", "Text"],
Cell["\<\
(e) You speed up again to 25 mph, travel a while, and enter the interstate. \
At the end of the trip, you exit, slow to 25 mph, stop at a stop sign, and \
proceed to your final destination.\
\>", "Text"]
}, Closed]],
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Cell["(a)", "Subsection"],
Cell["(a) You travel a total of 100 miles in 2 hours.", "Text",
Cell[CellGroupData[{
Cell["Solution to (a)", "Subsubsection"],
Cell["\<\
Our graph of distance vs. time should start at (0, 0) and end at (2, 100), \
where (0, 0) is the start of the trip at your house and (2, 100) is our \
destination 100 miles away after 2 hours.\
\>", "Text"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell["(b)", "Subsection"],
Cell["\<\
(b) Most of the trip is on rural interstate highway at the 65 mph speed \
limit. What qualitative feature or shape does the graph of distance vs. time \
have when speed is 65 mph?\
\>", "Text"],
Cell[CellGroupData[{
Cell["Solution to (b)", "Subsubsection"],
Cell[TextData[{
"Recall that speed is ",
Cell[BoxData[
\(distance\/time\)]],
". So our speed at any instant is \nreflected in the slope of our distance \
vs. time graph. Therefore, for this portion of the trip our graph should be \
a straight line with a positive slope of 65 ",
Cell[BoxData[
\(miles\/hour\)]],
"."
}], "Text"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell["(c)", "Subsection"],
Cell["\<\
(c) You start from your house at rest, gradually increase your speed to 25 \
mph, slow down, and stop at a stop sign.\
\>", "Text"],
Cell[CellGroupData[{
Cell["Solution to (c)", "Subsubsection"],
Cell["\<\
As stated in part (a) our graph starts at (0, 0). The slope at this point is \
0 since initially we are stopped and our speed is zero. The graph then \
curves upward until our slope becomes 25 mph. The graph continues upward \
linearly at this rate until we begin to slow down, at which time the graph \
curves to a flat section for the time we are stopped.\
\>", "Text"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell["(d)", "Subsection"],
Cell["\<\
(d) What shape is the graph of distance vs. time while you are stopped?\
\>", "Text"],
Cell[CellGroupData[{
Cell["Solution to (d)", "Subsubsection"],
Cell["\<\
As indicated in part (c), the graph is flat when we are stopped, since our \
speed, and hence the slope of our graph, is zero.\
\>", "Text"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell["(e)", "Subsection"],
Cell["\<\
(e) You speed up again to 25 mph, travel a while and enter the interstate. \
At the end of the trip, you exit, slow to 25 mph, stop at a stop sign, and \
proceed to your final destination.\
\>", "Text"],
Cell[CellGroupData[{
Cell["Solution to (e)", "Subsubsection"],
Cell["\<\
Again, our graph curves upward from a slope of zero to a slope of 25 mph and \
continues at this rate until we enter the interstate at which time the graph \
curves upward more until our slope becomes 65 mph. It is linear at this \
slope of 65 mph until we exit and the graph curves again to a slope of 25 \
mph, then curves even more to a flat section when we are again stopped. The \
final section of our graph ends at the coordinate (2, 100) where our slope is \
again zero (stopped).\
\>", "Text"],
Cell["\<\
Note that our graph is always going up or is flat since we never go \
backwards, but of course, the slope changes many times and is much steeper at \
65 mph than at 25 mph.\
\>", "Text"]
}, Closed]]
}, Closed]]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[TextData[StyleBox[
"Exercise 2) Small-Scale Plot", "Subtitle"]], "Subtitle"],
Cell["\<\
A very small-scale plot of distance traveled vs. time will appear straight \
because this is a magnified graph of a smooth function. What feature of this \
straight line represents the speed? In particular, how fast is the person \
going at t = 0.5 for the graph in Figure 10.3? What feature of the \
large-scale graph does this represent?\
\>", "Text"],
Cell[CellGroupData[{
Cell["Solution", "Section"],
Cell["\<\
The slope of our magnified graph represents the instantaneous speed. In \
particular, the instantaneous speed in the graph of Figure 10.3 at t = 0.5 is\
\
\>", "Text"],
Cell[CellGroupData[{
Cell[BoxData[
\(\((18.5281 - 12.9094)\)/\((0.55 - 0.45)\)\)], "Input"],
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Cell["\<\
or 56.187 mph. This represents the slope of the tangent line of the graph at \
t = 0.5.
\
\>", "Text"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[TextData[StyleBox[
"Exercise 3) Straight Line Movement", "Subtitle"]], "Subtitle"],
Cell[TextData[{
"An object moves along a straight line such as a straight level railroad \
track. Suppose the time is denoted t, with t = 0 when the train leaves the \
station. Let s represent the distance the train has traveled. The variable \
s is a function of t, s = s[t]. We need to set units and a direction. Why? \
Explain in your own words why the derivative ",
Cell[BoxData[
\(ds\/dt\)]],
" represents the instantaneous velocity of the object. What does a \
negative value of ",
Cell[BoxData[
\(ds\/dt\)]],
" mean? Could this happen? How does the train get back?"
}], "Text"],
Cell[CellGroupData[{
Cell["Solution", "Section"],
Cell["\<\
We need to set units for distance because 50 miles is certainly different \
that 50 feet. And movement away from the station is different that movement \
toward the station. We will consider movement away as positive distance and \
movement toward the station as negative distance.\
\>", "Text",
Evaluatable->False],
Cell[TextData[{
"Instantaneous velocity is determined by instantaneous speed and \
instantaneous direction. We know from Exercise 2 above that instantaneous \
speed is the slope of the tangent line, ",
Cell[BoxData[
\(ds\/dt\)]],
", at any \ninstant of time. We have defined direction in terms of \
movement away or toward the station (positive and negative respectively.) An \
example of instantaneous velocity is 54 mph which means the train is \
traveling at a speed of 54 mph away from the station."
}], "Text"],
Cell[TextData[{
"A negative value of ",
Cell[BoxData[
\(ds\/dt\)]],
" means that the train is traveling toward \nthe staion. This could happen \
if the train is moving backward on the tracks. Another possibiltiy is that \
the train goes to a destination, turns around, and travels back."
}], "Text"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[TextData[StyleBox[
"Exercise 4) Krazy Kousin Keith", "Subtitle"]], "Subtitle"],
Cell["\<\
Krazy Kousin Keith drove to Grandmother's, and the reading on his odometer is \
graphed in Figure 10.4. What was he doing at time t = 0.7?\
\>", "Text"],
Cell[CellGroupData[{
Cell["Solution", "Section"],
Cell["\<\
At t = 0.7 the slope of the graph has just turned negative, which means that \
for this time period he is moving back toward home. Apparently he has \
forgotten something, is lost, or is taking a detour.\
\>", "Text",
Evaluatable->False],
Cell["", "Text"]
}, Closed]]
}, Closed]],
Cell[CellGroupData[{
Cell[TextData[StyleBox[
"Exercise 5) Trip to Grandmother's", "Subtitle"]], "Subtitle"],
Cell["\<\
Portions of a trip to Grandmother's look like the two graphs in Figure 10.5. \
Which one is \"gas,\" and which one is \"brakes\"? Sketch two tangent lines \
on each of these graphs and estimate the speeds at these points of tangency. \
That is, which one shows slowing down and which speeding up?\
\>", "Text"],
Cell[CellGroupData[{
Cell["Solution", "Section"],
Cell["\<\
You should find that the first graph is \"gas\" because the speed (slope) \
increases with time and the second graph is \"brakes\" because its speed \
(slope) gets smaller as time increases.\
\>", "Text",
Evaluatable->False]
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