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Chapter 2: Differentiation
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Section 2.1: What Is a Derivative?
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Introduction


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Straight lines have "slope." In fact, the slope of a straight line is constant. The "straightness" of a straight line is characterized by the constancy of its slope.
Do curves have "slope?" Curves are not straight like lines, so how can the property of "slope" be attached to a curve? The answer to the question is that "slope" can be defined for a curve, but it will not be constant along the curve, it will change from point to point. This notion of a positionvarying slope along a curve is essentially that of the derivative. In this first section of Chapter 2, the idea or concept of the derivative as giving the slope at points along a curve will be explored.
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Essentials



Local Linearity


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On the graph of $f\left(x\right)\=3{x}^{3}\+9{x}^{2}7$ in Figure 2.1.1, the green dot is at the point $\left(1\,f\left(1\right)\right)\=\left(1\,5\right)$. To see the effect of "zooming in" on this point, move the slider:

${}$$k\=$
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The graphing interval $1{2}^{3k}\le x\le 1\+{2}^{3k}\/4$ shrinks about $x\=1$ as $k$ increases.

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As $k$ increases, the graph near $\left(1\,5\right)$ increasingly tends to resemble that of a straight line. This phenomenon is known as "local linearity." In other words, over a small enough interval, any smooth curve is well approximated by a straight line.


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Figure 2.1.1 Zooming in upon the point $\left(1\,5\right)$






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The slope of this straightline approximation obtained by zooming in at a point, is taken as the slope of the curve at that point. The process by which this "slope at a point" is determined is called differentiation, and the expression for the slope along the curve is called the "derivative."

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The Tangent Line


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As result of the elementary geometry course, a student may well consider that a line tangent to a curve is one that touches the curve in a single point. This notion of a tangent is probably sufficient for the calculus student, provided that it does not prevent the student from later accepting that the tangent line is the limit of secant lines.

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The slope of a line tangent to a curve is the same as the slope obtained by zooming in at the point of contact.

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Figure 2.1.2 shows, at $x\=1$, a line tangent to the curve graphed in Figure 2.1.1. Near $x\=1$, the "locally linear" portion of the curve has nearly the same slope as the tangent line, and exactly at $x\=1$, the slope of the curve is that of the tangent line.


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Student:Calculus1:Tangent(3*x^3+9*x^27,1,3..2,output=plot, caption="",view=[3..2,20..30],pointoptions=[symbol=solidcircle,symbolsize=15,color=green]);


Figure 2.1.2 A curve, and its tangent at $x\=1$






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Tangents as Limits of Secant Lines


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Figure 2.1.3 contains an animation in which blue secant lines are drawn between $\left(1\,5\right)$ and another point on the curve, a point that moves closer and closer to $\left(1\,5\right)$.

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As the moving point gets closer and closer to the fixed point $\left(1\,5\right)$, the secant lines approach their limiting position, that of the tangent line.

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The slopes of the secant lines approach the slope of the tangent line.

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Figure 2.1.4 contains a screenshot of the
applied to the function graphed in Figure 2.1.3.


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Student:Calculus1:NewtonQuotient(3*x^3+9*x^27,1,.5..1.5,h=1/2,output=animation,caption="",iterations=12,title="");


Figure 2.1.3 Animation: Secants become a tangent






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Figure 2.1.4 The Tangent tutor configured for the animation in Figure 2.1.3; note the Plot Options panel to the right



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In Figure 2.1.4 the Tangent tutor is configured for the animation seen in Figure 2.1.3. The animation in the tutor is launched by pressing the "Animate" button in the lower portion of the tutor. Alternatively, pressing the "Display" button would display a collection of secant lines whose endpoints and slopes are listed in the table in the tutor. Note that the slope and equation of the tangent line are also given in the tutor. The NewtonQuotient command at the bottom of the tutor is the command used to generate Figure 2.1.3.
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The Explore command also provides an animation of the secant line approaching a tangent line.
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$\mathrm{Explore}\left(\mathrm{plot}\left(\left[3{x}^{3}\+9{x}^{2}7\,\left(3{h}^{2}\+18h\+27\right)\cdot \left(x1\right)\+5\right]\,x\=.5..1.5\,y\=5..20\right)\,\mathrm{parameters}\=\left[h\=0..0.4\right]\,\mathrm{initialvalues}\=\left[h\=0.4\right]\,\mathrm{placement}\=\mathrm{right}\right)$
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It takes a bit more syntax to include "dots" for the fixed and moving points, and for the tangent line itself. The tutor provides those amenities at "no cost" but the Explore command provides a slider that allows the secant to approach the tangent line smoothly.


Slope of Tangent Line as Limit of Slopes of Secant Lines


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Figures 2.1.3 and 2.1.4 provide a graphical insight into the process whereby a secant line approaches a tangent line. The algebraic equivalent, illustrated below, results in the slope of the tangent line by finding the limit of the slopes of the approaching secant lines.${}$
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Define $f\left(x\right)$

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Controldrag $f\left(x\right)\=\dots$

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Context Panel: Assign Function


$f\left(x\right)\=3{x}^{3}\+9{x}^{2}7$$\stackrel{\text{assign as function}}{\to}$${f}$

Slopes of secant lines through $\left(1\,5\right)$ and $\left(1\+h\,f\left(1\+h\right)\right)$

$\frac{f\left(1\+h\right)f\left(1\right)}{\left(1\+h\right)1}$

$\=\frac{f\left(1\+h\right)5}{h}$

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$\=\frac{3{\left(1\+h\right)}^{3}\+9{\left(1\+h\right)}^{2}75}{h}$

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$\=\frac{3{h}^{3}plus;18{h}^{2}plus;27h}{h}$

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$\=\frac{h\left({h}^{2}\+18hplus;27\right)}{h}$

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$\={h}^{2}\+18hplus;27$



The limiting value of these slopes of secant lines is 27, attained as $h\to 0$.



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Average Velocity as the Slope of a Secant Line


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Let $s\=s\left(t\right)$ represent, at time $t$, the displacement from a fixed origin O, as an object moves along a straight line. For example, the value of $s$ could be the reading on the odometer in an automobile as it is driven along some road. It represents how far the car has moved from the start time up until time $t$.
The average velocity would then be the distance driven divided by the time it took to traverse that distance. In other words, the formula "distance equals rate times time" is solved for "rate" so that the rate (or average velocity) is distance divided by time.
Let $\mathrm{\Δ}tequals;{t}_{2}{t}_{1}$ represent a time interval between two times ${t}_{1}<{t}_{2}$. Correspondingly, let $\mathrm{\Δ}sequals;s\left({t}_{2}\right)s\left({t}_{1}\right)$ be the distance traversed during the time interval $\left[{t}_{1}\,{t}_{2}\right]$. Then the average velocity over this time interval is
${v}_{\mathrm{avg}}$ = $\frac{\mathrm{\Δ}s}{\mathrm{\Δ}t}$ = $\frac{s\({t}_{2}\)s\({t}_{1}\)}{{t}_{2}{t}_{1}}$
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Thus, the average velocity is simply the slope of the secant line on the graph of $y\=s\(t\)$, that is, the line through the points $\({t}_{1}\,s\({t}_{1}\)\)$ and$\({t}_{2}\,s\({t}_{2}\)\)$.
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Instantaneous Velocity as the Slope of a Tangent Line


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If the average velocity has a limit as ${t}_{2}$ approaches ${t}_{1}$, then this limit is the slope of the line tangent to $y\=s\(t\)$ at $t\={t}_{1}$. This limit is also known as the instantaneous velocity at time $t\={t}_{1}$. Letting $v\(t\)$ represent the instantaneous velocity at time $t$, these results are summarized via the equalities
$v\({t}_{1}\)$ = $\underset{{t}_{2}\to {t}_{1}}{lim}\phantom{\rule[0.0ex]{0.4em}{0.0ex}}{v}_{\mathrm{avg}}$ = $\underset{h\→0}{lim}\frac{\mathrm{\Δ}s}{\mathrm{\Δ}t}$ = $\underset{{t}_{2}\to {t}_{1}}{lim}\phantom{\rule[0.0ex]{0.4em}{0.0ex}}\frac{s\({t}_{2}\)s\({t}_{1}\)}{{t}_{2}{t}_{1}}$
The instantaneous velocity is the slope of the tangent line that the secant lines, whose slopes give the average velocities, approach.
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The instantaneous velocity measures the rate at which displacement is changing. Hence, the phrase "rate of change" is applied to the slope of the tangent line, even when the independent variable is not the time $t$.
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The rate of change of the velocity is the acceleration.


Précis


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Table 2.1.1 summarizes the main points of Section 2.1.
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At a point on a curve, its slope is that of the line tangent to the curve at that point.

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Zooming in on the graph of a curve shows that locally, in the small, a curve is nearly a straight line.

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Secant lines through two points that get closer and closer tend to a tangent line.

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The slopes of these secant lines tend, in the limit, to the slope of the tangent line.

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The average velocity is the slope of a secant line along a graph of displacement vs. time.

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The instantaneous velocity is the slope of a tangent line on a graph of displacement vs. time.

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At any point on a curve, the slope of its tangent line is the rate of change of the dependent variable, even if the independent variable is not time.


Table 2.1.1 Main points of Section 2.1



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Examples


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Example 2.1.1

The displacement of an object from a fixed origin O is described by the function $s\left(t\right)\=1820tplus;9{t}^{2}{t}^{3}comma;t\ge 0$, where the object is moving along a straight line. Determine the position and instantaneous velocity at time $t\=3$, and the average velocity for the timeinterval $2\le t\le 4$.

Example 2.1.2

Show that at $x\=2$ a unique slope cannot be assigned to the graph of $f\left(x\right)\=\sqrt{\left{x}^{2}4\right\+9}$. Consequently, a tangent line does not exist at $\left(2\,3\right)$ on this curve.

Example 2.1.3

Show that at $x\=0$ a unique slope cannot be assigned to the graph of $f\left(x\right)\={\leftx\right}^{1\/3}$.
Does this curve have a tangent line at $\left(0\,0\right)$?

Example 2.1.4

At $x\=1$, what can be said about the slope of the graph of $f\left(x\right)\=\{\begin{array}{cc}{x}^{2}& x\le 1\\ x\left(3x\right)& xgt;1\end{array}$ ?



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