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<h2 class="hd hd-2 unit-title">1. Geometric interpretation of the derivative</h2>
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<h2 class="hd hd-2 unit-title">2. Objectives</h2>
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<p><b class="bfseries">Geometric Interpretation of the Derivative: Tangent and Secant Lines</b></p><ul class="itemize"><li><p>
Instantaneous rate of change </p></li><li><p><b class="bf"><span style="color:#27408C">Slope of the tangent line at that point</span></b></p></li></ul><p><b class="bfseries">Objectives</b></p><p>
At the end of this lecture, and after some practice, you should be able to: </p><ul class="itemize"><li><p>
Understand the correspondence between rates of change (both average and instantaneous) and <span style="color:#99182C"><b class="bf">slopes of secant or tangent lines</b></span>. </p></li><li><p>
Be able to <span style="color:#99182C"><b class="bf">estimate a derivative</b></span> of a function at a point, given its graph. </p></li><li><p>
Determine when a derivative does not exist. </p></li><li><p>
Use [mathjaxinline]\Delta x[/mathjaxinline] and [mathjaxinline]\Delta y[/mathjaxinline] to denote small changes. </p></li></ul><p><b class="bfseries">Contents: 18 pages</b></p><p>
8 videos (21 minutes 1x speed) 25 questions </p>
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<h2 class="hd hd-2 unit-title">3. Tangent lines</h2>
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<h3 class="hd hd-2">Introduction to tangent lines</h3>
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<p>
We've seen that we can draw a tangent line. There is some sense that the tangent line goes through a point of a function in a way that it doesn't cut across the function at that point. But we have to be careful. </p><p>
Let's take a moment to think about what tangent lines are not. First, though, let's start with a <b class="bfseries">bad</b> way to think about tangent lines. A lot of students learn in high school that a tangent line is “a line that touches the curve in only one point." This is true if your curve is a circle, but for many other curves and functions, this is a <b class="bfseries">terrible</b> definition. Let's see why. </p>
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Intersecting once
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<h2 class="hd hd-2 unit-title">4. Intuition for tangent lines</h2>
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<h2 class="hd hd-2 unit-title">6. Average rate of change and secant slopes</h2>
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Average rate of change
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We want to bring in the graph of the function [mathjaxinline]\ f[/mathjaxinline], and connect it to derivatives and rates of change. Let's recall where we were: </p>
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The average rate of change of a function [mathjaxinline]\ f[/mathjaxinline] between [mathjaxinline]x = a[/mathjaxinline] and [mathjaxinline]x = b[/mathjaxinline] is given by what formula? </p>
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(Enter your answer in terms of the function [mathjaxinline]\ f[/mathjaxinline]. Type + and - for addition and subtraction. Type [mathjaxinline]*[/mathjaxinline] for multiplication; e.g. 2[mathjaxinline]*[/mathjaxinline]x for [mathjaxinline]2x[/mathjaxinline]. Type / for division; e.g. 2/x for [mathjaxinline]\displaystyle \frac{2}{x}[/mathjaxinline]. Use parentheses to designate order of operations.) </p>
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Average rate of change, geometrically
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<img alt="Increasing, concave up function with two labeled points at x equals a and x equals b." src="/assets/courseware/v1/57c4d0c81f1c532d7bc3e98e5d86ede6/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der2_twopoints.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
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On the graph of [mathjaxinline]\ f[/mathjaxinline], [mathjaxinline]x = a[/mathjaxinline] and [mathjaxinline]x = b[/mathjaxinline] correspond to the points [mathjaxinline](a, f(a))[/mathjaxinline] and [mathjaxinline](b,f(b))[/mathjaxinline]. The average rate of change [mathjaxinline]\frac{f(b)-f(a)}{b-a}[/mathjaxinline] then represents which of these quantities? </p>
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<text> The overall distance [mathjaxinline]\sqrt {\Delta x^2 + \Delta y^2}[/mathjaxinline] between the two points</text>
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<text> The slope of the secant line connecting the two points.</text>
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<text> The angle between the line connecting the two points and the [mathjaxinline]x[/mathjaxinline] axis.</text>
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<h2 class="hd hd-2 unit-title">7. Slopes of secant lines</h2>
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<h2 class="hd hd-2 unit-title">8. Secant and tangent lines</h2>
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<p>
Now, when we defined the derivative of [mathjaxinline]\ f[/mathjaxinline] at [mathjaxinline]a[/mathjaxinline], [mathjaxinline]\ f'(a)[/mathjaxinline], we took the limit of [mathjaxinline]\displaystyle \frac{f(b)-f(a)}{b-a}[/mathjaxinline] as [mathjaxinline]b[/mathjaxinline] approached [mathjaxinline]a[/mathjaxinline]. We've just seen that this quantity is the slope of the secant line that connects the points [mathjaxinline](a,f(a))[/mathjaxinline] and [mathjaxinline](b,f(b))[/mathjaxinline]. </p><p>
So what is the limit of the secant line as [mathjaxinline]b[/mathjaxinline] approaches [mathjaxinline]a[/mathjaxinline]? Note that looking at [mathjaxinline]b[/mathjaxinline] approaching [mathjaxinline]a[/mathjaxinline] is the same as looking at what happens as [mathjaxinline]\Delta x[/mathjaxinline] approaches 0. </p><iframe src="https://mathlets1801.surge.sh/secantApproximation.html" width="1100 px" height="630 px" style="border:0px; display: block;"/><p>
Directions: Pick a point [mathjaxinline]a[/mathjaxinline] using the horizontal slider below the graph. Start with [mathjaxinline]\Delta x = 1[/mathjaxinline]. Play with moving the point [mathjaxinline]\Delta x[/mathjaxinline] closer to [mathjaxinline]0[/mathjaxinline] and observe how the line changes. You can try changing the function with the dropdown function menu, and changing the fixed point [mathjaxinline]a[/mathjaxinline]. </p><p><span style="color:#99182C"><b class="bf">Food for thought:</b></span> What happens to the secant line as [mathjaxinline]\Delta x[/mathjaxinline] approaches [mathjaxinline]0[/mathjaxinline]? How is the secant line related to the tangent line? </p>
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<h2 class="hd hd-2 unit-title">9. Limits of secant lines</h2>
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The derivative [mathjaxinline]\ f'(a)[/mathjaxinline] is which of the following? (Pick all options that work.) </p>
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<text>An average rate of change</text>
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<text>A secant line</text>
</label>
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<text>A tangent line</text>
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<text>The slope of a secant line</text>
</label>
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<h2 class="hd hd-2 unit-title">12. Estimating derivatives</h2>
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Estimating derivatives
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Here is the graph of a function [mathjaxinline]g[/mathjaxinline]. </p>
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Is the slope of the tangent line at [mathjaxinline]x=1[/mathjaxinline] positive, negative, or zero? </p>
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Estimate derivatives
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Find good estimates for the following derivatives at the prescribed points. </p>
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<p style="display:inline">[mathjaxinline]g'(1)\approx[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">13. Linear function</h2>
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Secant line for a line
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<p>
What if the graph of [mathjaxinline]\ f[/mathjaxinline] is a line? For instance, here is a line with slope 1/2. </p>
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<img alt="Line with slope 0.5 intersecting the origin." src="/assets/courseware/v1/82f36f6b466ba9f15837b65f99e1bc6b/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der2_line.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
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<p>
If we take two points on this line, what will be the slope of the secant line joining them? </p>
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What is [mathjaxinline]\ f'(3)[/mathjaxinline]? </p>
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<h2 class="hd hd-2 unit-title">14. When the tangent line doesn't exist</h2>
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<p>
The slope of the tangent line, which is known as the derivative, ONLY exists if the tangent line exists! Let's explore some cases when the tangent line (and hence the derivative) does not exist. </p>
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Absolute value
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<img alt="For x less than 0, the function has slope negative 1. For x greater than 0, the function has slope 1." src="/assets/courseware/v1/059ae5a0212798a651870b8cd192673c/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der2_absolute.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
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Here's the graph of the absolute value function [mathjaxinline]\ f(x)=|x|[/mathjaxinline]. Will the graph look more and more like a line as you zoom in towards the origin? </p>
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Right and left limits
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Determine the right and left hand limits of the slopes of secant lines at the origin. </p>
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<img alt="For x less than 0, the function has slope negative 1. For x greater than 0, the function has slope 1." src="/assets/courseware/v1/059ae5a0212798a651870b8cd192673c/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der2_absolute.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
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<p style="display:inline">[mathjaxinline]\lim _{x \to 0^{+}} \frac{|x|-0}{x-0}=[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\lim _{x \to 0^{-}} \frac{|x|-0}{x-0}=[/mathjaxinline]</p>
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Jump discontinuity
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Corners aren't the only points where a function might not be differentiable. Here's a graph of a function called the step function. </p>
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<img alt="Function equals 0 for x less than 0. Function equals 1 for x greater than or equal to 1." src="/assets/courseware/v1/1e5f34dbb6a6b196e0dc3b8d29273671/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der2_heavyside.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
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What do you think the graph will look like if we zoom in at the point (0,1)? </p>
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<text> If [mathjaxinline]\ f[/mathjaxinline] is not continuous at [mathjaxinline]x = a[/mathjaxinline], then [mathjaxinline]\ f[/mathjaxinline] is not differentiable at [mathjaxinline]a[/mathjaxinline].</text>
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<text> If [mathjaxinline]\ f[/mathjaxinline] is continuous at [mathjaxinline]x = a[/mathjaxinline], then [mathjaxinline]\ f[/mathjaxinline] is differentiable at [mathjaxinline]a[/mathjaxinline].</text>
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<h2 class="hd hd-2 unit-title">15. Wrap up</h2>
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<h2 class="hd hd-2 unit-title">16. More estimation of derivatives</h2>
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More estimation of derivatives
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Each question refers to the graph of a function below. At each specified point [mathjaxinline]x_0[/mathjaxinline], give an estimate of [mathjaxinline]\ f'(x_0)[/mathjaxinline] or say that the derivative <em>does not exist</em> by typing DNE. </p>
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<img alt="When x equals negative 1.2, the function is approximately negative 0.7. When x equals negative 1.1, the function is approximately negative 0.9. When x equals negative 1, the function equals 0. When x equals negative 0.9, the function is approximately 0.9. When x equals negative 0.8, the function is approximately 0.7. When x equals negative 0.2, the function is approximately 0.01. When x equals negative 0.1, the function is approximately 0.005. When x equals 0, the function is 0 and approximately horizontal. When x equals 0.1, the function is approximately 0.005. When x equals 0.2, the function is approximately 0.01. When x equals 0.8, the function is approximately negative 0.7. When x equals 0.9, the function is approximately negative 0.9. When x equals 1, the function is negative 1 and has a sharp corner. When x equals 1.1, the function is approximately 0.9. When x equals 1.2, the function is approxiamtely 0.95. When x equals 1.8, the function is approximately negative 0.1. When x equals 1.9, the function is approximately negative 0.001. When x equals 2, the function is 0 and resembles a vertical line. When x equals 2.1, the function is approximately 0.001. When x equals 2.2, the function is approxiamtely 0.1. When x equals 2.8, the function is approximately 0.94. When x equals 2.9, the function is approxiamtely 0.97. When x equals 3, the function is 1. When x equals 3.1, the function is approximately 1.03. When x equals 3.2, the function is approxiamtely 1.06." src="/assets/courseware/v1/7893839684749044660384ec6c73f32d/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der2_tangentlines.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
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<p>
<p style="display:inline">[mathjaxinline]\ f'(-1) \approx[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\ f'(0) \approx[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\ f'(1) \approx[/mathjaxinline]</p>
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<p style="display:inline">[mathjaxinline]\ f'(2) \approx[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">17. Equation of a tangent line</h2>
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We can now answer the question that we posed at the beginning of the learning sequence. By using what we know about finding the derivative, we can determine the equation of a tangent line. </p>
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Determine point on tangent line
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Let's find the tangent line to the function [mathjaxinline]j(x) = 2x^2 + 3x[/mathjaxinline] through the point [mathjaxinline]x=1[/mathjaxinline]. </p>
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What point do we automatically know that this tangent line goes through? </p>
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(Enter the point as an ordered pair: (a, b). ) </p>
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Find tangent line slope
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Use the definition of the derivative to calculate the slope of the tangent line to the graph of [mathjaxinline]j(x) = 2x^2 + 3x[/mathjaxinline] at the point [mathjaxinline]x = 1[/mathjaxinline]. </p>
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Find the equation of the tangent line
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Now that we have the slope of the tangent line, and a point that it goes through, find the equation of the line. </p>
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(Enter your equation in the form [mathjaxinline]y=m*x+b[/mathjaxinline].) </p>
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<p style="display:inline">[mathjaxinline]y=[/mathjaxinline]</p>
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<tr class="fiptitle">
<th class="formulainput" scope="col">Allowable Entries</th>
<th class="formulainput" scope="col">Descriptions</th>
<th class="formulainput" scope="col">Example Entries</th>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Numbers</th>
<td class="formulainput">Integers</td>
<td class="formulainput">
<font color="#0078b0">2520</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Fractions</td>
<td class="formulainput">
<font color="#0078b0">2/3</font>
</td>
</tr>
<tr class="formulainput">
<td class="formulainput">Decimals </td>
<td class="formulainput"><font color="#0078b0">3.14</font>, <font color="#0078b0">.98</font></td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="4" scope="row">Operators</th>
<td class="formulainput">+ - * / (add, subtract, multiply, divide)</td>
<td class="formulainput">Enter <font color="#0078b0"> (x+2*y)/(x-1)</font> for \( \displaystyle \frac{x+2y}{x-1} \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">^ (raise to a power)</td>
<td class="formulainput">Enter <font color="#0078b0"> x^(n+1) </font> for \( x^{n+1} \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">_ (add a subscript)</td>
<td class="formulainput">Enter <font color="#0078b0"> v_0 </font> for \( v_0 \) </td>
</tr>
<tr class="formulainput">
<td class="formulainput">Use ( ) to clarify order of operations</td>
<td class="formulainput"> Enter <font color="#0078b0">(2+3)*2 </font> for 10 <br/>
Enter <font color="#0078b0"> 2+3*2 </font> for 8 </td>
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<th class="formulainput" scope="row">Greek letters</th>
<td class="formulainput">Enter (english) name of letter</td>
<td class="formulainput">Enter <font color="#0078b0">alpha </font> for \( \alpha \)<br/>
Enter <font color="#0078b0">lambda </font> for \(\lambda \)
</td>
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<tr class="formulainput">
<th class="formulainput" scope="row">Mathematical <br/> constants</th>
<td class="formulainput">e, pi</td>
<td class="formulainput">Enter <font color="#0078b0">e^x </font> for \( e^x \)<br/>
Enter <font color="#0078b0">2*pi </font> for \( 2\pi \)
</td>
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<tr class="formulainput">
<th class="formulainput" scope="row">Basic functions</th>
<td class="formulainput">abs, ln, log, log_2, sqrt</td>
<td class="formulainput">Enter <font color="#0078b0">abs(x+y) </font> for \( \left|x+y \right| \)<br/>
Enter <font color="#0078b0">sqrt(x^2-y) </font> for \( \sqrt{x^2-y} \)
</td>
</tr>
<tr class="formulainput">
<th class="formulainput" rowspan="3" scope="row">Trigonometric <br/> functions</th>
<td class="formulainput">sin, cos, tan, sec, csc, cot</td>
<td class="formulainput">Enter <font color="#0078b0">sin(4*x+y)^2 </font> for \(\sin^2(4x+y) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput">arcsin, arccos, arctan, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">arctan(x^2/3) </font> for \(\tan^{-1}\left(\frac{x^2}{3}\right) \)</td>
</tr>
<tr class="formulainput">
<td class="formulainput"> sinh, cosh, arcsinh, etc.</td>
<td class="formulainput">Enter <font color="#0078b0">cosh(4*x+y) </font> for \(\cosh(4x+y) \)</td>
</tr>
<tr class="formulainput">
<th class="formulainput" scope="row">Differentials</th>
<td class="formulainput">dx, dy</td>
<td class="formulainput">Enter a function followed by differential. You must multiply by the differential. <br/> Enter <font color="#0078b0">e^x*dx </font> for \( e^xdx \)<br/>
Enter <font color="#0078b0">(2*pi+y)*dy </font> for \( (2\pi+y)dy \)
</td>
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<h2 class="hd hd-2 unit-title">18. Review questions</h2>
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Review question 1
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If it exists, the derivative of a function at a point is a &#8195;&#8195;&#8195;&#8195; . </p>
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True or false: If [mathjaxinline]\ f[/mathjaxinline] is continuous at [mathjaxinline]x = a[/mathjaxinline], then it must be differentiable there. </p>
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True or false: A tangent line can only intersect the graph of [mathjaxinline]\ f[/mathjaxinline] in one point. </p>
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True or false: A tangent line to the graph of [mathjaxinline]\ f[/mathjaxinline] at [mathjaxinline](a,f(a))[/mathjaxinline] can cross the graph of [mathjaxinline]\ f[/mathjaxinline] at that point. </p>
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<img alt="On day 8, the volume is approximately 526800 gallons. On day 10, the volume is approximately 550000 gallons. On day 12, the volume is approxiamtely 782000 gallons. On day 38, the volume is approximately 975000 gallons. On day 40, the volume is approximately 960000 gallons. On day 42, the volume is approxiamtely 944900 gallons." src="/assets/courseware/v1/95383b2c5adfa58dab98ea9769a2dd51/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der2_watertower.svg" style="margin: 10px 25px 25px 25px" width="400px"/>
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Here's a graph of the volume of water in a water tower over the course of a summer. Estimate the instantaneous rate of change of the volume with respect to time on day 10, and on day 40. Don't forget your units! </p>
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<p style="display:inline">Day 40: Instantaneous rate of change is </p>
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Suppose that the graph of [mathjaxinline]\ f[/mathjaxinline] has a tangent line at [mathjaxinline]x = 3[/mathjaxinline]. That tangent line has equation [mathjaxinline]y = -2x + 1[/mathjaxinline]. What is [mathjaxinline]\ f(3)[/mathjaxinline]? What is [mathjaxinline]\ f'(3)[/mathjaxinline]? </p>
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<p style="display:inline">[mathjaxinline]\ f(3)=[/mathjaxinline]</p>
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<h2 class="hd hd-2 unit-title">19. Summary</h2>
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<p><b class="bfseries">Secant lines</b></p><p>
The <span style="color:#27408C"><b class="bf">secant line</b></span> of a function [mathjaxinline]\ f(x)[/mathjaxinline] over the interval [mathjaxinline]a \leq x \leq b[/mathjaxinline], is the line that passes through the points [mathjaxinline](a, f(a))[/mathjaxinline] and [mathjaxinline](b, f(b)).[/mathjaxinline] </p><ul class="itemize"><li><p>
The slope of the secant line is [mathjaxinline]\displaystyle \frac{f(b)-f(a)}{b-a}[/mathjaxinline], which is the average rate of change of the function [mathjaxinline]f(x)[/mathjaxinline] over the interval [mathjaxinline]a\leq x \leq b[/mathjaxinline]. </p></li><li><p>
The equation for the secant line is [mathjaxinline]\displaystyle y = \frac{f(b)-f(a)}{b-a}(x-a) + f(a).[/mathjaxinline] </p></li></ul><p><b class="bfseries">Tangent lines</b></p><p>
The <span style="color:#27408C"><b class="bf">tangent line</b></span> to a function [mathjaxinline]\ f(x)[/mathjaxinline] at the point [mathjaxinline]x=a[/mathjaxinline] is the line that passes through the point [mathjaxinline](a, f(a))[/mathjaxinline], and whose slope is the instantaneous rate of change of [mathjaxinline]\ f(x)[/mathjaxinline] at the point [mathjaxinline]x=a[/mathjaxinline]. This slope is the slope of the line you get if you imagine zooming in on the function until it looks like a line. </p><ul class="itemize"><li><p>
The slope of the tangent line is [mathjaxinline]\displaystyle f'(a)[/mathjaxinline]. </p></li><li><p>
The equation for the tangent line is [mathjaxinline]\displaystyle y = f'(a)(x-a) + f(a).[/mathjaxinline] </p></li></ul><p><b class="bfseries">Properties of tangent lines</b></p><p>
If the derivative of [mathjaxinline]\ f(x)[/mathjaxinline] exists at [mathjaxinline]x=a[/mathjaxinline], then the tangent line exists. The tangent line may exist if the derivative is undefined at [mathjaxinline]x=a[/mathjaxinline] though. (Example [mathjaxinline]\ f(x) = \sqrt [3]{x}[/mathjaxinline] has a vertical tangent line at [mathjaxinline]x=0[/mathjaxinline].) </p><p><b class="bfseries">What a tangent line is not</b></p><p>
A lot of students learn in high school that a tangent line is “a line that touches the curve in only one point." This is true if your curve is a circle, but for many other curves and functions, this is a <b class="bfseries">terrible</b> definition. See the examples below. </p><center><img src="/assets/courseware/v1/6f3463326c04acf6e8cc50a2a0ec7a9e/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der2_sincrossline.svg" width="250px" alt="Sine curve and a line with slope negative 1 passing through the origin." style="margin: 10px 25px 25px 25px"/> <img src="/assets/courseware/v1/47424bac8b8de7121697de0570ee864e/asset-v1:MITx+18.01.1x+2T2019+type@asset+block/images_u1der2_infiniteintersectiontanline.svg" width="250px" alt="Sine curve whose amplitude increases at either side of a point at which a horizontal line just touches the local maximum" style="margin: 10px 25px 25px 25px"/> </center>
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