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You're about to climb the ladder when \ your calculus instructor, who is also at the pool for some reason, stops you \ and says, \"That diving board is 32 feet above the surface of the water. Are \ you sure you want to do a belly flop? Do you know how fast you'll be going \ when you hit the water?\" (Your calculus instructor cares about your \ learning ", StyleBox["and", FontSlant->"Italic"], " your water safety.)" }], "Text"], Cell[TextData[{ "Let's try to find the answer to that question. How fast will you be going \ when you hit the water? To get us started, let's say that after ", StyleBox["t ", FontSlant->"Italic"], "seconds, your height above the water in feet is given by the function ", Cell[BoxData[ \(TraditionalForm\`h \((t)\) = \(-16\) t\^2 + 32\)]], "." }], "Text"], Cell[TextData[{ StyleBox["Question 1:", FontWeight->"Bold"], " At what height will you be when you hit the water?" }], "Text"], Cell[TextData[{ StyleBox["Question 2:", FontWeight->"Bold"], " After how many seconds will you hit the water?" }], "Text"], Cell[TextData[{ "To find your ", StyleBox["instantaneous velocity ", FontSlant->"Italic"], "when you hit the water, we'll start by looking at your ", StyleBox["average velocity", FontSlant->"Italic"], " over a few time intervals. Your average velocity over a given time \ interval is your change in position divided by the change in time." }], "Text"], Cell[TextData[{ StyleBox["Question 3:", FontWeight->"Bold"], " What is your average velocity over the entire dive? (That is, what is \ your average velocity from time ", Cell[BoxData[ \(TraditionalForm\`t = 0\)]], " to the time value you found in Question 2.)" }], "Text"], Cell["\<\ The velocity you found in Question 3 gives you a rough estimate of your \ instantaneous velocity at the end of your dive. To get a better estimate of \ your final velocity, let's look at your average velocity over increasingly \ smaller time intervals.\ \>", "Text"], Cell[TextData[{ StyleBox["Question 4:", FontWeight->"Bold"], " What is your average velocity from time ", Cell[BoxData[ \(TraditionalForm\`t = a\)]], " to the time value you found in Question 2? (Express your answer in terms \ of ", StyleBox["t", FontSlant->"Italic"], ". Do not simplify.)" }], "Text"], Cell["\<\ Enter the formula you found in Question 4 after the equals sign in the next \ cell. Then select the next cell and hit Shift-Return.\ \>", "Text"], Cell[BoxData[ \(\(\(AV[t_]\)\(:=\)\)\)], "Input"], Cell["\<\ To check your formula, select the next cell and hit Shift-Return. You should \ get the same number you found in Question 3.\ \>", "Text"], Cell[BoxData[ \(AV[0]\)], "Input"], Cell[TextData[{ "To find better estimates of your final velocity, find your average \ velocity over increasingly smaller time intervals. Pick four ", StyleBox["t", FontSlant->"Italic"], "-values between 0 and the impact time you found in Question 2. Make sure \ that the ", StyleBox["t", FontSlant->"Italic"], "-values you pick get closer and closer to the impact time. Replace the ", StyleBox["t'", FontSlant->"Italic"], "s in the next four cells with the ", StyleBox["t-", FontSlant->"Italic"], "values you picked and hit Shift-Return in each cell to find your average \ velocity over each of these smaller time intervals." }], "Text"], Cell[BoxData[ \(AV[t]\)], "Input"], Cell[BoxData[ \(AV[t]\)], "Input"], Cell[BoxData[ \(AV[t]\)], "Input"], Cell[BoxData[ \(AV[t]\)], "Input"], Cell[TextData[{ StyleBox["Question 5:", FontWeight->"Bold"], " Write down a table with the ", StyleBox["t", FontSlant->"Italic"], "-values you chose in one column and the corresponding average velocities \ in the other. Based on your data, what do you think your impact velocity \ is?" }], "Text"], Cell[TextData[{ "Your instantaneous velocity at impact time is the limit of your average \ velocities between time ", StyleBox["t", FontSlant->"Italic"], " and your impact time as ", StyleBox["t", FontSlant->"Italic"], " approaches your impact time.", " " }], "Text"], Cell[TextData[{ StyleBox["Question 6:", FontWeight->"Bold"], " Using the formula you found in Question 4, express your impact velocity \ as a limit. Then use the techniques we learned in Section 2.3 to evaluate \ that limit. Was your guess in Question 5 accurate?" }], "Text"], Cell[TextData[{ StyleBox["Question 7: ", FontWeight->"Bold"], " The limit you evaluated in Question 6 gave you a velocity in feet per \ second. What is this velocity in miles per hour? Would you want to do a \ belly flop knowing what you know now?" }], "Text"], Cell[TextData[{ StyleBox["Homework:", FontWeight->"Bold"], " Section 2.6 #7-19 odd" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["The Derivative", "Section"], Cell[TextData[{ "In part one of this lab, you were given a position function (", Cell[BoxData[ \(TraditionalForm\`h \((t)\) = \(-16\) t\^2 + 32\)]], ") and you found the instantaneous velocity at time ", Cell[BoxData[ \(TraditionalForm\`t = \@2\)]], ". Let's generalize. Suppose an object's position at any time ", StyleBox["t", FontSlant->"Italic"], " is given by a function ", Cell[BoxData[ \(TraditionalForm\`s(t)\)]], "." }], "Text"], Cell[TextData[{ StyleBox["Question 8:", FontWeight->"Bold"], " In terms of ", StyleBox["s, t, ", FontSlant->"Italic"], "and ", StyleBox["a", FontSlant->"Italic"], ", find the average velocity of the object between time ", StyleBox["t ", FontSlant->"Italic"], "and time ", StyleBox["a.", FontSlant->"Italic"] }], "Text"], Cell[TextData[{ StyleBox["Question 9:", FontWeight->"Bold"], " The instantaneous velocity of the object at time ", StyleBox["a", FontSlant->"Italic"], " is equal to the limit of the average velocities between time ", StyleBox["t ", FontSlant->"Italic"], "and time ", StyleBox["a", FontSlant->"Italic"], " as ", StyleBox["t ", FontSlant->"Italic"], "approaches ", StyleBox["a. ", FontSlant->"Italic"], "Express the instantaneous velocity of the object at time ", StyleBox["a", FontSlant->"Italic"], " as a limit using your answer to Question 8." }], "Text"], Cell[TextData[{ "In last week's lab, you looked at the function ", Cell[BoxData[ \(TraditionalForm\`f(x) = x\^2\)]], ", and you saw that the slope of the tangent line to the curve ", Cell[BoxData[ \(TraditionalForm\`y = f(x)\)]], " is the limit of the slopes of the secant lines through the points ", Cell[BoxData[ \(TraditionalForm\`\((x, x\^2)\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\((1, 1)\)\)]], " as ", StyleBox["x", FontSlant->"Italic"], " approaches 1. Let's generalize. Suppose you have a function ", Cell[BoxData[ \(TraditionalForm\`f(x)\)]], "." }], "Text"], Cell[TextData[{ StyleBox["Question 10:", FontWeight->"Bold"], " In terms of ", StyleBox["f, x, ", FontSlant->"Italic"], "and ", StyleBox["a", FontSlant->"Italic"], ", find the slope of the secant line through the points ", Cell[BoxData[ \(TraditionalForm\`\((a, f(a))\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\((x, f(x))\)\)]], "." }], "Text"], Cell[TextData[{ StyleBox["Question 11:", FontWeight->"Bold"], " The slope of the tangent line to the curve ", Cell[BoxData[ \(TraditionalForm\`y = f(x)\)]], " at the point ", Cell[BoxData[ \(TraditionalForm\`\((a, f(a))\)\)]], " is equal to the limit of the slopes of the secant lines through the \ points ", Cell[BoxData[ \(TraditionalForm\`\((a, f(a))\)\)]], " and ", Cell[BoxData[ \(TraditionalForm\`\((x, f(x))\)\)]], " as ", StyleBox["x", FontSlant->"Italic"], " approaches ", StyleBox["a", FontSlant->"Italic"], ". Express this tangent line slope as a limit using your answer to \ Question 10." }], "Text"], Cell["\<\ Your answers to Questions 9 and 13 should look very similar. Limits of this \ form are so useful that mathematicians give them a special name.\ \>", "Text"], Cell[TextData[{ StyleBox["Definition: ", FontWeight->"Bold"], "The ", StyleBox["derivative", FontSlant->"Italic"], " of a function ", StyleBox["f", FontSlant->"Italic"], " at a number ", StyleBox["a", FontSlant->"Italic"], " is denoted by ", Cell[BoxData[ \(TraditionalForm\`f' \((a)\)\)]], " and is defined to be ", Cell[BoxData[ \(TraditionalForm\`f' \((a)\) = lim\+\(x \[Rule] a\)\(f(x) - f(a)\)\/\(x - a\)\)]], ", if this limit exists. " }], "Text"], Cell[TextData[{ "Another, equivalent, definition of ", Cell[BoxData[ \(TraditionalForm\`f' \((a)\)\)]], " is ", Cell[BoxData[ \(TraditionalForm\`f' \((a)\) = lim\+\(h \[Rule] 0\)\(f(a + h) - f(a)\)\/h\)]], "." }], "Text"], Cell[TextData[{ StyleBox["Question 12:", FontWeight->"Bold"], " Show that these two limits are the same. (Hint: Let ", Cell[BoxData[ \(TraditionalForm\`h = x - a\)]], " and use this in the first limit.)" }], "Text"], Cell[TextData[{ StyleBox["Fact: ", FontWeight->"Bold"], "If an object has position function ", Cell[BoxData[ \(TraditionalForm\`s(t)\)]], ", then its (instantaneous) velocity at time ", StyleBox["a", FontSlant->"Italic"], " is ", Cell[BoxData[ \(TraditionalForm\`s' \((a)\)\)]], ", the derivative of ", StyleBox["s", FontSlant->"Italic"], " at ", StyleBox["a.", FontSlant->"Italic"] }], "Text"], Cell[TextData[{ StyleBox["Fact:", FontWeight->"Bold"], " The tangent line to the curve ", Cell[BoxData[ \(TraditionalForm\`y = f(x)\)]], " at the point ", Cell[BoxData[ \(TraditionalForm\`\((a, f(a))\)\)]], " is the line through ", Cell[BoxData[ \(TraditionalForm\`\((a, f(a))\)\)]], " with slope ", Cell[BoxData[ \(TraditionalForm\`f' \((a)\)\)]], ", the derivative of ", StyleBox["f", FontSlant->"Italic"], " at ", StyleBox["a", FontSlant->"Italic"], "." }], "Text"], Cell[TextData[{ StyleBox["Homework:", FontWeight->"Bold"], " Section 3.1 #3, 5, 7, 13-25 odd" }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["The Derivative as a Function", "Section"], Cell[TextData[{ "Let ", Cell[BoxData[ \(TraditionalForm\`f(x) = x\^2 - 2 x\)]], "." }], "Text"], Cell[TextData[{ StyleBox["Question 13:", FontWeight->"Bold"], " Using the definition of a derivative, find ", Cell[BoxData[ \(TraditionalForm\`f' \((0)\)\)]], "." }], "Text"], Cell[TextData[{ StyleBox["Question 14:", FontWeight->"Bold"], " Using the definition of a derivative, find ", Cell[BoxData[ \(TraditionalForm\`f' \((1)\)\)]], "." }], "Text"], Cell[TextData[{ "Suppose we also wanted to find ", Cell[BoxData[ \(TraditionalForm\`f' \((2)\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`f' \((3)\)\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`f' \((4)\)\)]], ". We could compute three more limits, but it would be simpler to find a \ formula for ", Cell[BoxData[ \(TraditionalForm\`f' \((a)\)\)]], " and then substitute 2, 3, and 4 in for ", StyleBox["a", FontSlant->"Italic"], "." }], "Text"], Cell[TextData[{ StyleBox["Question 15:", FontWeight->"Bold"], " Using the definition of a derivative, find ", Cell[BoxData[ \(TraditionalForm\`f' \((a)\)\)]], " in terms of ", StyleBox["a", FontSlant->"Italic"], "." }], "Text"], Cell[TextData[{ "We have just defined a new function -- one that take ", StyleBox["x", FontSlant->"Italic"], " to ", Cell[BoxData[ \(TraditionalForm\`f' \((x)\)\)]], ". Since we ", StyleBox["derived", FontSlant->"Italic"], " this function from ", StyleBox["f", FontSlant->"Italic"], ", we call it the derivative of ", StyleBox["f", FontSlant->"Italic"], "." }], "Text"], Cell[TextData[{ StyleBox["Definition:", FontWeight->"Bold"], " Given a function ", StyleBox["f", FontSlant->"Italic"], ", the ", StyleBox["derivative ", FontSlant->"Italic"], "of ", StyleBox["f", FontSlant->"Italic"], " is the function ", Cell[BoxData[ \(TraditionalForm\`f'\)]], " defined by ", Cell[BoxData[ \(TraditionalForm\`f' \((x)\) = lim\+\(h \[Rule] 0\)\(f(x + h) - f(x)\)\/h\)]], "." }], "Text"], Cell[TextData[{ StyleBox["Question 16:", FontWeight->"Bold"], " Using your answer to Question 15, find ", Cell[BoxData[ \(TraditionalForm\`f' \((x)\)\)]], "." }], "Text"], Cell["\<\ Let's look at a few graphs. Type the formula you found in Question 16 in the \ next cell after the equals sign. 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Sketch the tangent line to ", StyleBox["f ", FontSlant->"Italic"], "at the point ", Cell[BoxData[ \(TraditionalForm\`\((2, 0)\)\)]], ". Find the slope of this tangent line using the grid lines on the graph. \ How does this slope compare with the value of ", Cell[BoxData[ \(TraditionalForm\`f' \((2)\)\)]], "?" }], "Text"], Cell[TextData[{ StyleBox["Question 18:", FontWeight->"Bold", FontSlant->"Plain"], " ", StyleBox["Draw a few tangent lines to ", FontSlant->"Plain"], "f", StyleBox[" to the right of ", FontSlant->"Plain"], Cell[BoxData[ \(TraditionalForm\`x = 1\)]], ". ", StyleBox["What can you say about the slopes of these tangent lines? 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