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Recall that to find an equation of a line, we need a ", StyleBox["point", FontSlant->"Italic"], " on the line and the ", StyleBox["slope", FontSlant->"Italic"], " of the line. We have a point on the tangent line -- the point (1,1). \ However, we don't have the slope and we can't find it directly. To find the \ slope of the line, we need ", StyleBox["two", FontSlant->"Italic"], " points on the line. We only have ", StyleBox["one", FontSlant->"Italic"], "." }], "Text"], Cell["\<\ Instead of finding the slope of the tangent line, we'll find the slope of a \ line that's pretty close to the tangent line. Note that the points (1,1) and \ (2,4) are on the curve. 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o`04o`L0oooooooo00L07?ooo`0300Oooooooooo03Coool000goool00`070?ooooooo`27oooo0P07 08Goool00ol70?ooo`07000Loooo00<01oooooooool0=?ooo`003Oooo`0300L0oooooooo08Ooool0 0`070?ooooooo`24oooo00Co1`3oooooool01`0Loooo00<01oooooooool0 oooo00<01`3oool01`00U?ooo`Go1`1:oooo002?oooo0P0709Coool5o`L0B_ooo`00ooooogGoool0 0?oooomeoooo003oooooMOooo`00\ \>"], ImageRangeCache->{{{73.4375, 370.5}, {424.438, 157.188}} -> {-3.3496, \ 2.1875, 0.0137998, 0.0137998}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[TextData[{ "We call this line a ", StyleBox["secant line ", FontSlant->"Italic"], "because it intersects the curve in two points. Note that the slope of the \ secant line (in red) is pretty close to the slope of the tangent line (in \ blue). Since we know two points on the secant line -- (1,1) and (2,4) -- we \ can find its slope." }], "Text"], Cell[TextData[{ StyleBox["Question 1:", FontWeight->"Bold"], " What is the slope of the secant line to the curve ", Cell[BoxData[ \(TraditionalForm\`y = x\^2\)]], " through the points (1,1) and (2,4)?" }], "Text"], Cell[TextData[{ "This gives us an ", StyleBox["approximation ", FontSlant->"Italic"], "of the slope of the tangent line. However, it is a pretty rough \ approximation, since the red secant line and the blue tangent line are close, \ but not that close. ", "Instead of looking at a secant line through (1,1) and (2,4), let's ", "find a secant line that is even closer to the tangent line. Our new \ secant line should still go through (1,1) because that's the only point we \ know on the tangent line." }], "Text"], Cell[TextData[{ StyleBox["Question 2:", FontWeight->"Bold"], " What is another point on the curve ", Cell[BoxData[ \(TraditionalForm\`y = x\^2\)]], " our secant line could go through that would give us a better \ approximation to the tangent line?" }], "Text"], Cell[TextData[{ "Let's call the point you chose ", Cell[BoxData[ \(TraditionalForm\`\((x, x\^2)\)\)]], "." }], "Text"], Cell[TextData[{ StyleBox["Question 3:", FontWeight->"Bold"], " In terms of ", StyleBox["x", FontSlant->"Italic"], ", what is the slope of the secant line through (1,1) and ", Cell[BoxData[ \(TraditionalForm\`\((x, x\^2)\)\)]], "? (Don't simplify your answer.)" }], "Text"], Cell["\<\ Type the formula you just found in the next cell after the equals sign. Then \ select the cell and hit Shift-Enter.\ \>", "Text"], Cell[BoxData[ \(m[x_] := \((x^2 - 1)\)/\((x - 1)\)\)], "Input"], Cell[TextData[{ "To find the slope of the secant line through (1,1) and ", Cell[BoxData[ \(TraditionalForm\`\((x, x\^2)\)\)]], ", replace the ", StyleBox["x", FontSlant->"Italic"], " in the next cell with the ", StyleBox["x", FontSlant->"Italic"], "-coordinate of the point you chose in Question 2. Then select the cell \ and hit Shift-Enter." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(m[1.5]\)], "Input"], Cell[BoxData[ \(2.5`\)], "Output"] }, Open ]], Cell[TextData[{ "If you've entered everything correctly, that's the slope of the secant \ line through (1,1) and ", Cell[BoxData[ \(TraditionalForm\`\((x, x\^2)\)\)]], ". Let's look at the graph now. Don't worry about what the next cell \ says, just select it and hit Shift-Enter." }], "Text"], Cell[BoxData[ \(showGraph[x_] := Show[Graphics[{RGBColor[0, 0, 1], Disk[{1, 1}, .05], RGBColor[1, 0, 0], Disk[{x, x^2}, .05]}], Plot[t^2, {t, \(-2\), 2}, DisplayFunction \[Rule] Identity], Plot[2 t - 1, {t, \(-2\), 3}, DisplayFunction \[Rule] Identity, PlotStyle \[Rule] RGBColor[0, 0, 1]], Plot[m[x]\ \((t - 1)\) + 1, {t, \(-2\), 3}, DisplayFunction \[Rule] Identity, PlotStyle \[Rule] RGBColor[1, 0, 0]], PlotRange \[Rule] {{\(-2\), 3}, {\(- .5\), 4}}, Axes \[Rule] True, AspectRatio \[Rule] Automatic]\)], "Input"], Cell[TextData[{ "Replace the ", StyleBox["x ", FontSlant->"Italic"], "in the next cell with the ", StyleBox["x", FontSlant->"Italic"], "-coordinate of the point you chose in Question 2. Then select the cell \ and hit Shift-Enter. 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0;3oool006koool200L0/?ooo`00ooooob7oool00?oooolQoooo003ooooo8Oooo`00\ \>"], ImageRangeCache->{{{73.4375, 302.813}, {615.5, 409.188}} -> {-3.76076, \ 8.67774, 0.018004, 0.018004}}], Cell[BoxData[ TagBox[\(\[SkeletonIndicator] Graphics \[SkeletonIndicator]\), False, Editable->False]], "Output"] }, Open ]], Cell[TextData[{ StyleBox["Question 4:", FontWeight->"Bold"], " How can we get an even better approximation of the slope of the tangent \ line?" }], "Text"], Cell[TextData[{ "Let's find five successively better approximations of the slope of the \ tangent line. Pick five ", StyleBox["x", FontSlant->"Italic"], " values with the property that each one should determine a secant line \ whose slope is closer to the slope of the tangent line than the previous. \ Replace the ", StyleBox["x", FontSlant->"Italic"], "'s in the next five cells with your picks and hit Shift-Enter in each \ cell." }], "Text"], Cell[BoxData[{ \(m[x]\), "\[IndentingNewLine]", \(showGraph[x]\)}], "Input"], Cell[BoxData[{ \(m[x]\), "\[IndentingNewLine]", \(showGraph[x]\)}], "Input"], Cell[BoxData[{ \(m[x]\), "\[IndentingNewLine]", \(showGraph[x]\)}], "Input"], Cell[BoxData[{ \(m[x]\), "\[IndentingNewLine]", \(showGraph[x]\)}], "Input"], Cell[BoxData[{ \(m[x]\), "\[IndentingNewLine]", \(showGraph[x]\)}], "Input"], Cell[TextData[{ StyleBox["Question 5:", FontWeight->"Bold"], " Based on your calculations above, what do you think the slope of the \ tangent line actually is?" }], "Text"], Cell[TextData[{ StyleBox["Question 6:", FontWeight->"Bold"], " Now that we have a point on the tangent line and think we know the slope \ of the tangent line, what is an equation of the tangent line?" }], "Text"], Cell[TextData[{ "Before proceeding to part two of today's lab, have Mr. Bruff look over \ your ", StyleBox["Mathematica", FontSlant->"Italic"], " notebook (that's this file) to make sure you're on the right track." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Cue the Limit", "Section"], Cell[TextData[{ "We found in the first part of today's lab that the slope of the secant \ line to ", Cell[BoxData[ \(TraditionalForm\`y = x\^2\)]], " through the points (1,1) and ", Cell[BoxData[ \(TraditionalForm\`\((x, x\^2)\)\)]], " is given by ", Cell[BoxData[ \(TraditionalForm\`m(x) = \(x\^2 - 1\)\/\(x - 1\)\)]], ". We also saw that as ", StyleBox["x", FontSlant->"Italic"], " got closer and closer to 1, the slope of the corresponding secant line \ got closer and closer to the slope of the tangent line to ", Cell[BoxData[ \(TraditionalForm\`y = x\^2\)]], " at the point (1,1)." }], "Text"], Cell["\<\ For fun and to make sure you've got the concept, hit Shift-Enter on the next \ cell. \ \>", "Text"], Cell[BoxData[{ \(\(m[x_] := \((x^2 - 1)\)/\((x - 1)\);\)\), "\[IndentingNewLine]", \(\(showGraph[x_] := Show[Graphics[{RGBColor[0, 0, 1], Disk[{1, 1}, .05], RGBColor[1, 0, 0], Disk[{x, x^2}, .05]}], Plot[t^2, {t, \(-2\), 2}, DisplayFunction \[Rule] Identity], Plot[2 t - 1, {t, \(-2\), 3}, DisplayFunction \[Rule] Identity, PlotStyle \[Rule] RGBColor[0, 0, 1]], Plot[m[x]\ \((t - 1)\) + 1, {t, \(-2\), 3}, DisplayFunction \[Rule] Identity, PlotStyle \[Rule] RGBColor[1, 0, 0]], PlotRange \[Rule] {{\(-2\), 3}, {\(- .5\), 4}}, Axes \[Rule] True, AspectRatio \[Rule] Automatic];\)\), "\[IndentingNewLine]", \(\(Table[showGraph[x], {x, 2, 1.01, \(- .01\)}];\)\)}], "Input"], Cell[TextData[{ "Now double-click on the bracket second from the left. Then under the Cell \ menu above, select Animate Selected Graphics -- or just hit Ctrl-Y. As ", StyleBox["x", FontSlant->"Italic"], " gets closer and closer to 1, t", "he red dot, located at ", Cell[BoxData[ \(TraditionalForm\`\((x, x\^2)\)\)]], ", will get closer and closer to the point (1,1) and the secant line \ determined by ", StyleBox["x ", FontSlant->"Italic"], "(also in red) will get closer and closer to the tangent line." }], "Text"], Cell[TextData[{ "We say that the slope of the tangent line is the ", StyleBox["limit", FontSlant->"Italic"], " of the slopes of the secant lines as ", StyleBox["x", FontSlant->"Italic"], " approaches 1", ". Since the slope of the secant line through (1,1) and ", Cell[BoxData[ \(TraditionalForm\`\((x, x\^2)\)\)]], " is given by ", Cell[BoxData[ \(TraditionalForm\`\(x\^2 - 1\)\/\(x - 1\)\)]], ", we write this limit symbolically as \n\t\n\t\t", StyleBox["m", FontSlant->"Italic"], " = ", Cell[BoxData[ \(TraditionalForm\`\(lim\+\ \)\+\(x \[Rule] 1\)\ \(x\^2 - 1\)\/\(x - \ 1\)\)]], "." }], "Text"], Cell[TextData[{ "As ", StyleBox["x", FontSlant->"Italic"], " gets closer and closer to 1, ", Cell[BoxData[ \(TraditionalForm\`\(x\^2 - 1\)\/\(x - 1\)\)]], " gets closer and closer to the number we want -- the slope of the tangent \ line. We'll talk more about limits tomorrow when we look at Section 2.2 in \ your calculus book. For now, try to answer the following questions." }], "Text"], Cell[TextData[{ StyleBox["Question 7:", FontWeight->"Bold"], " In today's lab, we tried to find the slope of the tangent line to ", Cell[BoxData[ \(TraditionalForm\`y = x\^2\)]], " at the point (1,1) and we arrived at the limit you see above. Now \ suppose we were looking for the slope of the tangent line to ", Cell[BoxData[ \(TraditionalForm\`y = x\^2\)]], " at the point ", Cell[BoxData[ \(TraditionalForm\`\((a, a\^2)\)\)]], " instead. At what limit would we arrive in that case?" }], "Text"], Cell[TextData[{ StyleBox["Question 8:", FontWeight->"Bold"], " Let's generalize this one step further. Instead of trying to find the \ slope of the tangent line to ", Cell[BoxData[ \(TraditionalForm\`y = x\^2\)]], " at the point ", Cell[BoxData[ \(TraditionalForm\`\((a, a\^2)\)\)]], ", suppose we were looking for the slope of the tangent line to the curve \ ", Cell[BoxData[ \(TraditionalForm\`y = f(x)\)]], " at the point ", Cell[BoxData[ \(TraditionalForm\`\((a, f(a))\)\)]], ". At what limit would we arrive in that case?" }], "Text"], Cell[TextData[{ StyleBox["Homework:", FontWeight->"Bold"], " Section 2.1 #3." }], "Text"] }, Open ]] }, Open ]] }, FrontEndVersion->"4.2 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 941}}, WindowToolbars->{}, WindowSize->{882, 807}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic} ] (******************************************************************* Cached data follows. 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