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week1.html
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<body><div class="sidenav normalwidth"><button style="border:none; background-color: Transparent;" onclick="showhidemenu()" title="Toggle toc"><span style="font-size: 30px;">☰</span></button><button class="openbs" title="Open buttons">+</button><button class="closebs" title="Close buttons">-</button><ul class="leftmenu" style="display: none;"><li><a class="kap" href="introduction.html#159ca076-dc4e-4bb9-83f5-a5ddb8ff2299"><b>1</b> Introduction</a></li><a href="#kap1" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap1" class="collapse"><li><a href="introduction.html#ae495219-240d-4cf1-9a9b-29918804ef56">1.1 Readings</a></li><li><a href="introduction.html#26671802-fb3b-4273-8e82-12d04b3b35a4">1.2 Notes</a></li><li><a href="introduction.html#3fc447d4-9ac7-439e-afc9-6c5ad35b2e24">1.3 Exercises</a></li></ul><li><a class="kap" href="week1.html#3a6133ac-4a7b-47d1-9ea5-47ca81f168f2"><b>2</b> Course week 1</a></li><a href="#kap2" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap2" class="collapse"><li><a href="week1.html#cbb4e3bf-894c-49ad-90c5-58d0ed1e7cc3">2.1 Readings</a></li><li><a href="week1.html#d018418e-a17c-4b70-b799-f34e0298333b">2.2 Notes</a></li><li><a href="week1.html#e010e5dd-5121-4b83-8e89-d32234dee55e">2.3 Exercises</a></li></ul><li><a class="kap" href="week2.html#5c85fc37-321e-4707-9e48-bdb3f9e4a5bd"><b>3</b> Course week 2</a></li><a href="#kap3" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap3" class="collapse"><li><a href="week2.html#8ec6af3d-7c28-4353-89f8-1992db17e7e8">3.1 Readings</a></li><li><a href="week2.html#b18e048c-25be-4300-9ed0-4407e46d540c">3.2 Notes</a></li><li><a href="week2.html#cca75863-ec45-459f-a0f4-3a965a60422e">3.3 Exercises</a></li><li><a href="week2.html#ee2537cc-717b-4463-8904-87085c60123a">3.4 Feedback</a></li></ul><li><a class="kap" href="week3.html#9beeae3e-d65c-4020-b00a-c526ceadf899"><b>4</b> Course week 3</a></li><a href="#kap4" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap4" class="collapse"><li><a href="week3.html#6bef6f20-56d9-4f57-803f-0ba61b2540ac">4.1 Readings</a></li><li><a href="week3.html#7a92877c-98c7-4bdd-9e2f-762cb37563e8">4.2 Notes</a></li><li><a href="week3.html#b4fcf09c-686b-41fc-8a20-854b9c21bf31">4.3 Exercises</a></li><li><a href="week3.html#fe6dc55b-aaac-49aa-9a44-55212b41273d">4.4 Feedback</a></li></ul><li><a class="kap" href="week4.html#2402c1cb-8568-4fb6-bdbe-b6485e20d1ca"><b>5</b> Course week 4</a></li><a href="#kap5" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap5" class="collapse"><li><a href="week4.html#1e2ed3d8-2100-4005-899d-ebde86500961">5.1 Readings</a></li><li><a href="week4.html#34298628-e47f-4a13-a2e8-ee6d069fc8b5">5.2 Notes</a></li><li><a href="week4.html#23a8a6e0-68cc-464c-a101-71cdfbbe4353">5.3 Problems</a></li><li><a href="week4.html#497d6801-a7db-445d-b745-3c9c166483b5">5.4 Feedback</a></li></ul><li><a class="kap" href="week5.html#881f0ad5-1f5b-406f-b1f3-cf97397cdfe3"><b>6</b> Course week 5</a></li><a href="#kap6" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap6" class="collapse"><li><a href="week5.html#f420a408-715e-462b-b2c1-59a979e8b14e">6.1 Readings</a></li><li><a href="week5.html#718a53d7-d8cf-4f5e-8ecf-6561f6f394b8">6.2 Notes</a></li><li><a href="week5.html#08659956-822b-4a08-9e5f-f727bd863256">6.3 Problems</a></li><li><a href="week5.html#74ef3f61-c1c2-42f7-b3c5-dbf69578da08">6.4 Feedback</a></li></ul><li><a class="kap" href="week6.html#b028fe91-09aa-484e-af70-e887bdccbf90"><b>7</b> Course week 6</a></li><a href="#kap7" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap7" class="collapse"><li><a href="week6.html#5f5a6489-a309-4ba4-9cc2-e37611e8e350">7.1 Readings</a></li><li><a href="week6.html#bebc3711-369c-44c2-a9a1-23c8bc9900b0">7.2 Notes</a></li><li><a href="week6.html#2619640f-84f5-4826-8212-f383e4451cb3">7.3 Problems</a></li><li><a href="week6.html#8626687f-527d-42cc-94eb-5f19d20ba029">7.4 Feedback</a></li></ul><li><a class="kap" href="week7.html#c4dc25ad-24a4-45e7-b4c5-e59016fd3dcc"><b>8</b> Course week 7</a></li><a href="#kap8" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap8" class="collapse"><li><a href="week7.html#ee3b5046-f23c-41e8-beb7-08e80a78ebca">8.1 Readings</a></li><li><a href="week7.html#98cafeeb-e279-4a89-a571-d5fc17e6824b">8.2 Notes</a></li><li><a href="week7.html#fdd931f4-395e-40ca-81b8-c3186c52d21d">8.3 Problems</a></li><li><a href="week7.html#4de6c567-85d9-4936-92e4-4f323d10a16b">8.4 Feedback</a></li></ul><li><a class="kap" href="week8.html#a6129057-0dca-47e2-a908-acd3fc725bca"><b>9</b> Course week 8</a></li><a href="#kap9" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap9" class="collapse"><li><a href="week8.html#91a7f924-2cc3-46e0-99d5-84b72cd25982">9.1 Readings</a></li><li><a href="week8.html#2227f711-3610-4e0e-b835-539a1c67559b">9.2 Notes</a></li><li><a href="week8.html#a580698f-3a0d-489f-a786-09490c52da6a">9.3 Problems</a></li><li><a href="week8.html#f636cddc-14ea-46dc-921a-fc8de482f125">9.4 Feedback</a></li></ul><li><a class="kap" href="week9.html#1673477a-155b-4ee5-8c3c-524c4da8f0da"><b>10</b> Course week 9</a></li><a href="#kap10" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap10" class="collapse"><li><a href="week9.html#8d4c9c5d-c542-4cd9-8401-e1a52df4286f">10.1 Readings</a></li><li><a href="week9.html#4dec455a-332a-4ae1-99a8-9c55b704483d">10.2 Notes</a></li><li><a href="week9.html#af8ac1d5-5cc3-4c7d-8eeb-bd6086a2beb9">10.3 Problems</a></li><li><a href="week9.html#9fb03b93-76bb-4df9-ba3b-3076a33a3c84">10.4 Feedback</a></li></ul><li><a class="kap" href="week10.html#9a44ce9a-039b-43ea-bab2-a8ca6d6ad271"><b>11</b> Course week 10</a></li><a href="#kap11" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap11" class="collapse"><li><a href="week10.html#0a130e84-47b6-42a1-b4a0-c045425900f4">11.1 Readings</a></li><li><a href="week10.html#e0c9a4d0-1c3b-4411-9a42-5eac771e0265">11.2 Notes</a></li><li><a href="week10.html#13ab74d2-e187-4b74-a77c-c9599ae7c0ed">11.3 Problems</a></li><li><a href="week10.html#7b803349-1c7b-40d6-9855-33a3e7b15c17">11.4 Feedback</a></li></ul><li><a class="kap" href="week11.html#bad3f0c8-0554-443d-bd04-1392582ea095"><b>12</b> Course week 11</a></li><a href="#kap12" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap12" class="collapse"><li><a href="week11.html#8a687e45-dd04-4a19-a682-b8573340d40e">12.1 Readings</a></li><li><a href="week11.html#03a71c8c-2af2-4746-bbb6-8efe3d3e5e6b">12.2 Notes</a></li><li><a href="week11.html#da7ad12d-8b44-4cfe-a32f-9f8954d6f1bb">12.3 Problems</a></li><li><a href="week11.html#f729eb4e-feb9-4840-a28e-4524cefeda12">12.4 Feedback</a></li></ul><li><a class="kap" href="week12.html#4edeeb01-8694-4c0d-bab3-cea029123630"><b>13</b> Course week 12</a></li><a href="#kap13" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap13" class="collapse"><li><a href="week12.html#3a9c26cd-489b-4a83-84e0-b7ce17925a99">13.1 Readings</a></li><li><a href="week12.html#ae4d4c1f-1399-464e-af22-d908c76bb143">13.2 Notes</a></li><li><a href="week12.html#caf55b0e-7ea3-4ccd-84fb-0c2dfaa25472">13.3 Problems</a></li><li><a href="week12.html#ac0eb5fa-eb33-436e-95b9-d3bf33460ad2">13.4 Feedback</a></li></ul><li><a class="kap" href="week13.html#f0e1b4c2-d02a-4740-bc27-8b653fedf832"><b>14</b> Course week 13</a></li><a href="#kap14" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap14" class="collapse"><li><a href="week13.html#36ebe4f7-f0c7-46e4-8404-b0bb54f694c9">14.1 Readings</a></li><li><a href="week13.html#b6ef6597-50c4-47d2-a2d0-15334207b7d0">14.2 Notes</a></li><li><a href="week13.html#7b891666-2d5c-4f9a-80f9-adfff76ce1f4">14.3 Problems</a></li><li><a href="week13.html#988cdc9c-6611-42fc-b4eb-370b70fb44c1">14.4 Feedback</a></li></ul><li><a class="kap" href="week14.html#96e609e7-8f1b-4f46-8653-af953c4e9823"><b>15</b> Course week 14</a></li><a href="#kap15" data-toggle="collapse"><span class="downtick">❯</span></a><ul id="kap15" class="collapse"><li><a href="week14.html#90ef85b9-1287-4e2c-9a13-5b8f4bca1584">15.1 Readings</a></li><li><a href="week14.html#3350aecb-60a7-4229-97a1-c063c9336eeb">15.2 Notes</a></li><li><a href="week14.html#aeadc6cf-2833-4923-b0f5-14a9ab76ce1e">15.3 Problems</a></li></ul></ul></div><div class="main normalmargin"><div style="margin-top:20px"></div><div style="margin-top:20px"></div><h1 id="3a6133ac-4a7b-47d1-9ea5-47ca81f168f2">2<span style="float:right;">Course week 1</span></h1>
<span id="sec2.1"></span><h2 id="cbb4e3bf-894c-49ad-90c5-58d0ed1e7cc3">2.1 Readings</h2>
Read sections 6.1 – 6.7 in the textbook.<div style="margin-top:20px"></div><span id="sec2.2"></span><h2 id="d018418e-a17c-4b70-b799-f34e0298333b">2.2 Notes</h2>
Below, you will find some important results from last week's curriculum which
might prove useful when you work on this week's problems (further below).<div style="margin-top:20px"></div><b>Rules for limits</b>
<div class="frameit">
If <span class="math"></span><script type="math/tex">\displaystyle A = \lim_{x \to a} f(x)</script> and <span class="math"></span><script type="math/tex">\displaystyle B = \lim_{x \to a} g(x)</script>, then the following rules apply:
<div class="math"></div><script type="math/tex; mode=display">\begin{aligned}
\lim_{x \to a} \left( f(x) \pm g(x) \right) &= A \pm B \\
\lim_{x \to a} \left( f(x) g(x) \right) &= AB \\
\lim_{x \to a} \frac{f(x)}{g(x)} &= \frac{A}{B}, \quad \text{if~} B \neq 0 \\
\lim_{x \to a} \left( f(x) \right)^r &= A^r
\end{aligned}</script>
</div><div style="margin-top:20px"></div><b>Equality of limits</b>
<div class="frameit">
If two functions <span class="math"></span><script type="math/tex">f(x)</script> and <span class="math"></span><script type="math/tex">g(x)</script> satisfy <span class="math"></span><script type="math/tex">f(x) = g(x)</script> for <span class="math"></span><script type="math/tex">x</script> close to <span class="math"></span><script type="math/tex">a</script>, but not necessarily at <span class="math"></span><script type="math/tex">x=a</script>,
and if one of the limits exist, then:
<div class="math"></div><script type="math/tex; mode=display">
\lim_{x \to a} f(x) = \lim_{x \to a} g(x)
</script>
</div><div style="margin-top:20px"></div><b>Derivatives of selected functions</b>
<div class="frameit">
<div class="math"></div><script type="math/tex; mode=display">\begin{aligned}
\underline{f(x)} \qquad {}& \underline{f'(x)} \qquad {}& \underline{\text{Note}}\\
x^a \qquad {}& a x^{a-1} \qquad {}& \\
e^x \qquad {}& e^x \quad {}& \\
\ln(x) \qquad {}& 1/x \quad {}& x > 0 \\
e^{ax} \qquad {}& ae^{ax} \quad {}& \\
a^{x} \qquad {}& \ln(a) a^{x} \quad {}& a > 0
\end{aligned}</script>
</div><div style="margin-top:20px"></div><b>Rules for derivatives</b>
<div class="frameit">
<div class="math"></div><script type="math/tex; mode=display">\begin{aligned}
{}&\frac{\mathrm{d}\phantom{x}}{\mathrm{d}x} \left( A + f(x) \right) = \frac{\mathrm{d}f}{\mathrm{d}x} = f'(x)\\[3ex]
{}&\frac{\mathrm{d}\phantom{x}}{\mathrm{d}x} \left( A \cdot f(x) \right) = A \frac{\mathrm{d}f}{\mathrm{d}x} = A f'(x)\\[3ex]
{}&\frac{\mathrm{d}\phantom{x}}{\mathrm{d}x} \left( f(x) \pm g(x) \right) = \frac{\mathrm{d}f}{\mathrm{d}x} \pm \frac{\mathrm{d}g}{\mathrm{d}x} = f'(x) \pm g'(x)\\[3ex]
{}&\frac{\mathrm{d}\phantom{x}}{\mathrm{d}x} \left( f(x) g(x) \right) = \frac{\mathrm{d}f}{\mathrm{d}x} g(x) + f(x) \frac{\mathrm{d}g}{\mathrm{d}x} = f'(x)g(x) + f(x)g'(x)\\[3ex]
{}&\frac{\mathrm{d}\phantom{x}}{\mathrm{d}x} \left( \frac{f(x)}{g(x)}\right) = \frac{\frac{\mathrm{d}f}{\mathrm{d}x} g(x) - f(x) \frac{\mathrm{d}g}{\mathrm{d}x}}{(g(x))^2} = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}
\end{aligned}</script>
</div><div style="margin-top:20px"></div><b>Equation for the tangent to a curve in a point</b>
<div class="frameit">
The equation for the tangent to the graph of <span class="math"></span><script type="math/tex">f(x)</script> in the point <span class="math"></span><script type="math/tex">(x_0, f(x_0))</script> is given
by the equation
<span id="equ2.1"></span><div class="math"></div><script type="math/tex; mode=display">
y = f(x_0) + f'(x_0) (x - x_0)
\tag{2.1}</script>
</div><div style="margin-top:20px"></div><span id="sec2.3"></span><h2 id="e010e5dd-5121-4b83-8e89-d32234dee55e">2.3 Exercises</h2><div style="margin-top:20px"></div><span id="env2.1"></span><a class="Exerciseno" data-count="2.1"></a><a href="#6e831dad-553c-4cf8-8cba-78bba4286ccd" class ="btn btn-default Exercisebutton" data-toggle="collapse"></a><div id=6e831dad-553c-4cf8-8cba-78bba4286ccd class = "collapse Exercise envbuttons">
The pupose of this exercise is to illustrate that the tangent is the limit of a secant when <span class="math"></span><script type="math/tex">h \to 0</script>.<div style="margin-top:20px"></div>The definition of the derivative <span class="math"></span><script type="math/tex">f'(x_0)</script> is the limit of the Newton quotient
<span id="equ2.2"></span><div class="math"></div><script type="math/tex; mode=display">
f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h}
\tag{2.2}</script>
But what does that mean graphically? We can interpret the Newton quotient as the slope of the straight line
through the points
<div class="math"></div><script type="math/tex; mode=display">
(x_0, f(x_0)) \quad \text{and}\quad (x_0 + h, f(x_0 + h)).
</script>
This line is called a secant. We obtain the derivative in <span class="math"></span><script type="math/tex">x_0</script> when the point <span class="math"></span><script type="math/tex">x_0 + h</script> comes infinitesimally close to <span class="math"></span><script type="math/tex">x_0</script>, i.e., the limit <span class="math"></span><script type="math/tex">h \to 0</script>.<div style="margin-top:20px"></div>Consider a particular function:
<div class="math"></div><script type="math/tex; mode=display">
f(x) = -x^3 + 5x^2 + 2.
</script>
In order to calculate <span class="math"></span><script type="math/tex">f'(2)</script>, we can use equation <a href=#equ2.2 class="labelref">2.2</a> with <span class="math"></span><script type="math/tex">x_0 = 2</script>.
Instead of calculating the limit, we can explore the problem graphically for different values of <span class="math"></span><script type="math/tex">h</script>, e.g., by using a calculator or a spreadsheet to calculate the slope of the secant as <span class="math"></span><script type="math/tex">h</script> becomes smaller and smaller.
In the interactive plot below, you can see how the secant changes, as you change <span class="math"></span><script type="math/tex">h</script> by pulling the point <span class="math"></span><script type="math/tex">x_0 + h</script> along the <span class="math"></span><script type="math/tex">x</script>-axis. At the same time you can see how the value of the slope of the secant (the Newton quotient) also changes.<div style="margin-top:20px"></div><ol class="number"><li id="ite2.1"> Describe what happens as <span class="math"></span><script type="math/tex">h</script> becomes smaller and smaller
</li><li id="ite2.2"> Calculate <span class="math"></span><script type="math/tex">f'(x)</script> by using rules for derivatives
</li><li id="ite2.3"> Calculate the value <span class="math"></span><script type="math/tex">f'(2)</script>
</li><li id="ite2.4"> Compare the exact value of <span class="math"></span><script type="math/tex">f'(2)</script> to the value in the plot for the smalles value of <span class="math"></span><script type="math/tex">h</script>
</li><li id="ite2.5"> The plot has deliberately been made so that you can not set <span class="math"></span><script type="math/tex">h = 0</script>. What would happen if <span class="math"></span><script type="math/tex">h = 0</script>?
</li></ol>
It is exactly because we cannot set <span class="math"></span><script type="math/tex">h = 0</script> in the Newton quotient, that we need to take the limit. Calculus is a smart and systematic way of taking these limits without having to go back to the definition in equation <a href=#equ2.2 class="labelref">2.2</a> every time.<div style="margin-top:20px"></div><center>
<link rel="stylesheet" type="text/css" href="https://jsxgraph.org/distrib/jsxgraph.css" />
<script type="text/javascript" src="https://jsxgraph.org/distrib/jsxgraphcore.js"></script>
<div id="plotbox" class="jxgbox" style="width:500px; height:500px;"></div>
<script type="text/javascript">
board = JXG.JSXGraph.initBoard('plotbox', {
boundingbox: [-1, 24, 6, -4],
axis: true,
showClearTrace: true,
showFullscreen: false, /* does not work in qadil */
showCopyright: false});
var pol = function(x){
return -x*x*x + 5*x*x + 2;
}
var graph = board.create('functiongraph', [pol], {strokeWidth: 2, name:"f", withLabel: true});
var x0 = board.create('glider', [2, 0, board.defaultAxes.x], {name: 'x_0', size:4});
var fx0 = board.create('point', [
function() { return x0.X(); },
function() { return pol(x0.X()); }
], {name: '', color: 'grey', fixed: true, size:3});
var x = board.create('glider', [5, 0, board.defaultAxes.x], {name: 'x', size:4});
var fx = board.create('point', [
function() { return x.X(); },
function() { return pol(x.X()); }
], {name: '', color: 'grey', fixed: true, size:3});
var line = board.create('line',[fx0, fx],{strokeColor:'blue',dash:2});
/*
// Move this to another linked board and plot secant slope as function of h
var f1 = board.create('point', [
function() { return x.X(); },
function() { return (fx.Y()-fx0.Y())/(fx.X()-fx0.X() + 0.0000001); }],
{ size: 1, name: 'f_1', color: 'black', fixed: true, trace: true});
*/
var hTxt = board.create('text', [0.5, -2, function() {
return 'h = ' + (fx.X() - fx0.X()).toFixed(5);
}]);
var secTxt = board.create('text', [0.5, -3, function() {
return 'Slope of secant = ' + ((fx.Y() - fx0.Y()) / (fx.X() - fx0.X())).toFixed(3);
}]);
</script>
<div style="margin-top:20px"></div></center>
</div><div style="margin-top:20px"></div>
<span id="env2.2"></span><a class="Exerciseno" data-count="2.2"></a><a href="#7cf2b11e-d3cb-4eb5-9a9a-f32d03e67232" class ="btn btn-default Exercisebutton" data-toggle="collapse"></a><div id=7cf2b11e-d3cb-4eb5-9a9a-f32d03e67232 class = "collapse Exercise envbuttons">
In this exercise we consider the same function as in exercise <a href=#env2.1 class="labelref">2.1</a>.<div style="margin-top:20px"></div>In the graph below you can zoom in on the curve by marking a rectangular area. Try to zoom in on the point <span class="math"></span><script type="math/tex">(2, f(2))</script>.
If you zoom in enough, it will become impossible to distinguish the curve from the tangent (dashed line). This is true
for all differentiable functions, so in other words: locally any differentiable function is indistinguishable from a
straight line (which is identican to the tangent when you have zoomed and infinite amount).<div style="margin-top:20px"></div>To go back and start over, simply double click anywhere in the graph.
<center>
<iframe src="./img/week1/diff_zoom.html" title="plot" width=900 height=600 style="border:none;" ></iframe><div style="margin-top:20px"></div></center>
</div><div style="margin-top:20px"></div><span id="env2.3"></span><div class="exercise" data-count="2.3">
a) Find the slope of the curve at the point <span class="math"></span><script type="math/tex">(3, f(3))</script>.
<div class="centerimg"><img src="img/week1/exercise_1.png" width="50%"></div>
<span id="{labelname}"></span><div id="quiz1" style="width:100%;"><div class="row quizquestion">
b) Where does the curve have a slope of zero?
</div><div class="row quizboxes"><div class="col-xs-5 quizanswer" datav="F">
<span class="math"></span><script type="math/tex">x \approx -1.72</script>
</div><div class="col-xs-5 quizanswer" datav="F">
<span class="math"></span><script type="math/tex">x = 1.666\ldots</script>
</div><div class="col-xs-5 quizanswer" datav="T">
<span class="math"></span><script type="math/tex">x = 0</script> and <span class="math"></span><script type="math/tex">x = 1.333\ldots</script>
</div><div class="col-xs-5 quizanswer" datav="F">
<span class="math"></span><script type="math/tex">x \approx -1.72</script> and <span class="math"></span><script type="math/tex">x = 1.666\ldots</script>
</div></div></div><button id="quizbutton1" class = "btn btn-default">Check</button><script>$("#quizbutton1").click(function(){quizshowanswers(quiz1, quizbutton1);});</script>
</div><div style="margin-top:20px"></div><span id="env2.4"></span><div class="exercise" data-count="2.4">
A function <span class="math"></span><script type="math/tex">f(x)</script> and its derivative <span class="math"></span><script type="math/tex">f'(x)</script> have the particular values
<div class="math"></div><script type="math/tex; mode=display">
\begin{array}{rl}
f(-1) &=& 0.65 \\
f'(-1) &=& 0.45
\end{array}
</script>
What is the equation for the tangent to the graph at the point <span class="math"></span><script type="math/tex">(-1, f(-1))</script> shown in the figure below?
<div class="centerimg"><img src="img/week1/exercise_2.png" width="50%"></div>
</div><div style="margin-top:20px"></div><span id="env2.5"></span><div class="exercise" data-count="2.5">
The tangent to the graph for the function <span class="math"></span><script type="math/tex">f(x)</script> at the point <span class="math"></span><script type="math/tex">(2, f(2))</script> is given by the equation
<div class="math"></div><script type="math/tex; mode=display">
y = 2.1 x - 3.1
</script>
What are the values of <span class="math"></span><script type="math/tex">f(2)</script> and <span class="math"></span><script type="math/tex">f'(2)</script>?
<div class="centerimg"><img src="img/week1/exercise_3.png" width="50%"></div>
</div><div style="margin-top:20px"></div><span id="env2.6"></span><div class="exercise" data-count="2.6">
The plot below shows the graph for a function <span class="math"></span><script type="math/tex">f(x)</script>
<div class="centerimg"><img src="img/week1/exercise_4a.png" width="50%"></div>
<span id="{labelname}"></span><div id="quiz2" style="width:100%;"><div class="row quizquestion">
Which of the following graphs shows a plot of the derivative of <span class="math"></span><script type="math/tex">f(x)</script>?
</div><div class="row quizboxes"><div class="col-xs-5 quizanswer" datav="F">
<div class="centerimg"><img src="img/week1/exercise_4c.png" width="75%"></div>
</div><div class="col-xs-5 quizanswer" datav="T">
<div class="centerimg"><img src="img/week1/exercise_4b.png" width="75%"></div>
</div><div class="col-xs-5 quizanswer" datav="F">
<div class="centerimg"><img src="img/week1/exercise_4e.png" width="75%"></div>
</div><div class="col-xs-5 quizanswer" datav="F">
<div class="centerimg"><img src="img/week1/exercise_4d.png" width="75%"></div>
</div></div></div><button id="quizbutton2" class = "btn btn-default">Check</button><script>$("#quizbutton2").click(function(){quizshowanswers(quiz2, quizbutton2);});</script>
</div><div style="margin-top:20px"></div><span id="env2.7"></span><div class="exercise" data-count="2.7">
The figure below shows the graphs for a function <span class="math"></span><script type="math/tex">f(x)</script> (solid blue) and its derivative <span class="math"></span><script type="math/tex">f'(x)</script> (dashed red).
<div class="centerimg"><img src="img/week1/exercise_5.png" width="50%"></div>
Find the equation for the tangent to <span class="math"></span><script type="math/tex">f(x)</script> for <span class="math"></span><script type="math/tex">x = 4</script>.<div style="margin-top:20px"></div><a href="#722fe3e2-513b-419e-b9e2-60e0c841c7ab" class ="btn btn-default Hintbutton"data-toggle="collapse"></a> <div id=722fe3e2-513b-419e-b9e2-60e0c841c7ab class = "collapse Hint envbuttons">
What are the values of <span class="math"></span><script type="math/tex">f(4)</script> and <span class="math"></span><script type="math/tex">f'(4)</script>?<div style="margin-top:20px"></div><a href="#62baa586-8806-4128-b398-8ddfd0edd0ce" class ="btn btn-default Hintbutton"data-toggle="collapse"></a> <div id=62baa586-8806-4128-b398-8ddfd0edd0ce class = "collapse Hint envbuttons">
Use equation <a href=#equ2.1 class="labelref">2.1</a>.
</div>
</div>
</div><div style="margin-top:20px"></div><span id="env2.8"></span><div class="exercise" data-count="2.8">
Find the limits below.
<ol class="number"><li id="ite2.6"> <span class="math"></span><script type="math/tex">\displaystyle \lim_{x \to -3} \left( x^2 + 4x + 2 \right)</script>
</li><li id="ite2.7"> <span class="math"></span><script type="math/tex">\displaystyle \lim_{x \to 3/2} \left( (x - \frac{2}{3}) (x + \frac{1}{2}) \right)</script>
</li><li id="ite2.8"> <span class="math"></span><script type="math/tex">\displaystyle \lim_{x \to 0} \left( \sqrt{(2-x)(8+x)} + \frac{1}{x-2} \right)</script>
</li><li id="ite2.9"> <span class="math"></span><script type="math/tex">\displaystyle \lim_{x \to 3} \frac{2x^2-2x-2}{x+2} </script>
</li></ol>
</div><div style="margin-top:20px"></div>
<span id="env2.9"></span><div class="exercise" data-count="2.9">
Calculate the limit
<div class="math"></div><script type="math/tex; mode=display">
\lim_{x \to 2} \frac{3x^2 - 3x - 6}{2x^2 + 2x - 12}
</script>
<a href="#97a8dd19-cc71-4991-a78e-3ac082af4746" class ="btn btn-default Hintbutton"data-toggle="collapse"></a> <div id=97a8dd19-cc71-4991-a78e-3ac082af4746 class = "collapse Hint envbuttons">
Are you sure you don't want to think about it first? Use hints only when you need them!<div style="margin-top:20px"></div><a href="#2aef69a4-6150-4f48-96c6-6f061c092020" class ="btn btn-default" data-toggle="collapse">Yes, I'm stuck. Show me the hint!</a><div id=2aef69a4-6150-4f48-96c6-6f061c092020 class="collapse">
Use the result <em>equality of limits</em> above.<div style="margin-top:20px"></div><a href="#2920235e-f5ac-4e2b-903d-9a6f80c36908" class ="btn btn-default Hintbutton"data-toggle="collapse"></a> <div id=2920235e-f5ac-4e2b-903d-9a6f80c36908 class = "collapse Hint envbuttons">
Rewrite both the numerator and the denominator by factorising (see the box <em>Quadratic Equations</em> in section <a href="introduction.html#sec1.2.1">1.2.1</a>).
</div>
</div>
</div>
</div><div style="margin-top:20px"></div>
<span id="env2.10"></span><div class="exercise" data-count="2.10">
Calculate the derivatives of the following functions:
<ol class="number"><li id="ite2.10"> <span class="math"></span><script type="math/tex">f(x) = 3x^6</script>
</li><li id="ite2.11"> <span class="math"></span><script type="math/tex">g(x) = x^{3/2}</script>
</li><li id="ite2.12"> <span class="math"></span><script type="math/tex">h(y) = \frac{2}{3} y^{-3}</script>
</li><li id="ite2.13"> <span class="math"></span><script type="math/tex">q(x) = 1 / (x^{-7/3})</script>
</li></ol>
</div><div style="margin-top:20px"></div><span id="env2.11"></span><div class="exercise" data-count="2.11">
Calculate the derivatives of the following functions:
<ol class="number"><li id="ite2.14"> <span class="math"></span><script type="math/tex">f(x) = 2ax^3</script>, where <span class="math"></span><script type="math/tex">a</script> is a constant.
</li><li id="ite2.15"> <span class="math"></span><script type="math/tex">f(z) = 2\sqrt{z} - \frac{1}{7}z^5</script>.
</li><li id="ite2.16"> <span class="math"></span><script type="math/tex">f(x) = ax^2 + bx + c</script>, where <span class="math"></span><script type="math/tex">a</script>, <span class="math"></span><script type="math/tex">b</script> and <span class="math"></span><script type="math/tex">c</script> are constants.
</li><li id="ite2.17"> <span class="math"></span><script type="math/tex">g'(y)</script>, where <span class="math"></span><script type="math/tex">g(y) = 5y^3 - 3y^2</script>.
</li><li id="ite2.18"> <span class="math"></span><script type="math/tex">h(y) = \exp (-10 y)</script>.
</li></ol>
</div><div style="margin-top:20px"></div>
<span id="env2.12"></span><div class="exercise" data-count="2.12">
Calculate <span class="math"></span><script type="math/tex">y'(x)</script> when <span class="math"></span><script type="math/tex">y</script> is given by:
<ol class="number"><li id="ite2.19"> <span class="math"></span><script type="math/tex">y = 5x^2 - 3x + 7</script>
</li><li id="ite2.20"> <span class="math"></span><script type="math/tex">y = x^7 + 3 \sqrt{x} + \frac{1}{2 \sqrt{x}}</script>
</li><li id="ite2.21"> <span class="math"></span><script type="math/tex">y = x \cdot (x -2x^{-1})</script>
</li></ol>
</div><div style="margin-top:20px"></div><span id="env2.13"></span><div class="exercise" data-count="2.13">
Calculate the derivative of the following functions:
<ol class="number"><li id="ite2.22"> <span class="math"></span><script type="math/tex">f(x) = 10x^3 (1 - \sqrt{x})</script>
</li><li id="ite2.23"> <span class="math"></span><script type="math/tex">f(x) = (2x - x^2)(x^3 + x^2) + 1000</script>
</li><li id="ite2.24"> <span class="math"></span><script type="math/tex">f(x) = (x + 2)\cdot (\sqrt{x} -3x)</script>
</li></ol>
</div><div style="margin-top:20px"></div><span id="env2.14"></span><div class="exercise" data-count="2.14">
Calculate the derivatives of the following expressions:
<ol class="number"><li id="ite2.25"> <span class="math"></span><script type="math/tex"> \frac{\sqrt{x} - 1}{x^2} </script>
</li><li id="ite2.26"> <span class="math"></span><script type="math/tex"> \frac{x-2}{x+2} </script>
</li><li id="ite2.27"> <span class="math"></span><script type="math/tex"> \frac{2x-3}{5x+4} </script>
</li></ol>
</div><div style="margin-top:20px"></div></div></body>