Tangent LinesIn this project, we will consider finding equations of lines tangent to the graph of a function. Finding the slope of such a line will require us to use an approximation procedure.Move the cursor to the "restart" command below and depress the "enter" key.restart:
with(plots):<Text-field layout="Heading 1" style="Heading 1"><Font family="Times New Roman">Tangent Lines</Font></Text-field>We should have some notion of what a point of tangency is from geometry. Figure #1 below shows a line that is tangent to a circle at point P. That is, this line "touches" the circle at one point. We now wish to generalize the idea of tangency to other graphs; in particular, we wish to consider finding the equation of lines tangent to the graph of a function.<Text-field layout="Heading 2" style="Heading 2"><Font family="Times New Roman">Example 1</Font></Text-field>Find the equation of the line tangent to the graph of f(x) = 1/x when x = 2. Figure #2 below shows the graph of f together with the tangent line whose equation we seek.<Text-field layout="Heading 3" style="Heading 3"><Font family="Times New Roman">Description of the Method</Font></Text-field>To find the equation of any line, we need a point that the line passes through and the slope of that line. Since the point of tangency lies on the graph of f and has x-coordinate NiMvJSJ4RyIiIw==, we find the y-coordinate of the point of tangency by evaluating f at NiMvJSJ4RyIiIw== to obtain f(2) = 1/2. Thus the point of tangency has coordinates (2, f(2)) = (2,1/2).Finding the slope of the tangent line is a much more complicated problem. We recall that the definition of the slope of a line requires that we have two points that lie on the line. That is, if points (x1,y1) and (x2,y2) lie on a line, then the slope of the line is defined by
slope = NiMqJiwmSSN5Mkc2IiIiIkkjeTFHRiYhIiJGJywmSSN4MkdGJkYnSSN4MUdGJkYpRik=.However, for the tangent line problem depicted in Figure 2, we only have one point that we can be certain lies on this tangent line. So how can we find or at least closely approximate the slope of the tangent line in this case? The following sections develop an approach for closely approximating the slope of the tangent line.<Text-field layout="Heading 4" style="Heading 4"><Font family="Times New Roman">First Approximation</Font></Text-field>Our approach will be to find a second point (labeled A below) on the graph of f and consider the slope of the line passing through this point and the point of tangency. This line will not have the same slope as the tangent line, but we will take this slope as our first approximation to the slope of the tangent line. If point A has coordinates (1,1), what is the slope of the blue line (answer = -0.5)?<Text-field layout="Heading 4" style="Heading 4"><Font family="Times New Roman">Second Approximation</Font></Text-field>To obtain a better approximation to the slope of the tangent line, we will choose another point (labeled B below) on the graph of f closer to the point of tangency. The line passing through this point and the point of tangency will have a slope that is closer to the actual slope of the tangent line. We will take this slope as our second approximation to the slope of the tangent line. If point B has coordinates (5/4 , 4/5), what is the slope of the blue line (answer = -2/5)?
<Text-field layout="Heading 4" style="Heading 4"><Font family="Times New Roman">Successive Approximations</Font></Text-field>We will then continue choosing points on the graph of f progressively closer to the point of tangency. The slopes of the lines through each of these points and the point of tangency will become progressively better approximations to the slope of the tangent line. We hope to recognize a pattern or trend in these approximate slope values, so that we may infer the value of the actual slope. Click on the graph below and use the play controls that will appear near the top of the screen to view a demonstration of this process of approximating the slope of the tangent 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layout="Heading 3" style="Heading 3"><Font family="Times New Roman">Implementing the Method</Font></Text-field>We now use Maple to implement the method just described for estimating the slope of a tangent line. We will restate the problem: Find the equation of the line tangent to the graph of when x = 2.We know the line passes through the point (2, f(2)) = (2,1/2), so we simply need to determine the slope to complete this problem. <Text-field layout="Heading 4" style="Heading 4"><Font family="Times New Roman">Finding the Slope</Font></Text-field>We will approximate the slope by choosing any other point that lies on the graph of f and then determining the slope of the line that passes through that point and the point of tangency. <Text-field layout="Heading 4" style="Heading 4"><Font family="Times New Roman">First Approximation</Font></Text-field>Take as the second point the point that corresponds to x = 1/2. The coordinates of this point are (1/2, f(1/2)) = (1/2,2). The slope of the line that passes through (2,1/2) (the point of tangency) and (1/2,2) is given by the formula
m = NiMqJiwmIiIjIiIiKiZGJkYmRiUhIiJGKEYmLCZGJ0YmRiVGKEYo = -1
The following Maple commands perform the above calculations. Move the cursor to the "m : =" command line and depress enter key. Also, notice the two places where the value x = 1/2 is inserted in the formula:f:=1/x:
m:=(subs(x=1/2,f)-subs(x=2,f))/(1/2-2);We therefore have our first approximation to the slope to be -1.<Text-field layout="Heading 4" style="Heading 4"><Font family="Times New Roman">Second Approximation</Font></Text-field>We now choose a point on the graph closer to the point of tangency and again approximate the slope. The easiest way to ensure that we are choosing a point closer to the point of tangency is to choose an x-value between x = 1/2 and x = 2. That is, if we choose our x-value closer to NiMvJSJ4RyIiIw==, then the corresponding point on the graph of this function will be closer to the point (2,1/2). Take the next point to be the point that corresponds to x = 1. The coordinates of this point are (1, f(1)) = (1,1). The slope of the line that passes through (2,1/2) (the point of tangency) and (1,1) is given by the formula m = NiMqJiwmIiIiRiUqJkYlRiUiIiMhIiJGKEYlLCZGJUYlRidGKEYo = NiMsJComIiIiRiUiIiMhIiJGJw==.The appropriate Maple command to perform this calculation is as follows:
m:=(subs(x=1,f)-subs(x=2,f))/(1-2);We therefore have our second approximation to the slope to be -1/2 or -0.5.<Text-field layout="Heading 4" style="Heading 4"><Font family="Times New Roman">Third Approximation</Font></Text-field>Choosing an x-value still closer to NiMvJSJ4RyIiIw==, say NiMvJSJ4RyQiIzohIiI=, we wish to find the slope of the line that passes through (2,1/2) (the point of tangency) and (1.5, f(1.5)) = (1.5,2/3). To calculate this slope we execute the Maple command that follows:m:=(subs(x=1.5,f)-subs(x=2,f))/(1.5-2);We therefore have our third approximation to the slope to be -1/3 or approximately -0.3333.<Text-field layout="Heading 4" style="Heading 4"><Font family="Times New Roman">Successive Approximations</Font></Text-field>We continue choosing x-values closer to NiMvJSJ4RyIiIw== by choosing the values that follow: NiMvJSJ4RyQiJHYiISIj, NiMvJSJ4RyQiIz4hIiI=, NiMvJSJ4RyQiJCY+ISIj, and NiMvJSJ4RyQiJCo+ISIj. Fill in the x-values in the command that follows to find the corresponding approximations to the slope of the tangent line. There is nothing special about the x values that we picked other than that they get progressively closer to the number 2. You may perform other substitutions if you wish in order to obtain even better approximations for the slope of the tangent line.m:=(subs(x=( ),f)-subs(x=2,f))/( ( )-2);You should obtain (approximately) -0.2857, -0.2632, -0.2564, and -0.2513, respectively.<Text-field layout="Heading 4" style="Heading 4"><Font family="Times New Roman">Results</Font></Text-field>Below is a summary of the results we have obtained.x = 1/2 m = -1x = 1 m = -0.5x = 1.5 m = -0.333x = 1.75 m = -0.2857x = 1.9 m = -0.2632
x = 1.95 m = -0.2564x = 1.99 m = -0.2513From these results we conclude that as x approaches 2, the slope appears to approach -0.25. Therefore, we will estimate that the line tangent to the graph of at the point (2,1/2) has slope m = -0.25. The equation of this line is given by
NiMvSSJ5RzYiLCYqJi1JJkZsb2F0R0kqcHJvdGVjdGVkR0YqNiQiI0QhIiMiIiJJInhHRiVGLiEiIkYuRi4=We have utilized values of x that are less than 2, but we could have utilized values of x that are greater than 2 and get successively closer to 2.<Text-field layout="Heading 2" style="Heading 2"><Font family="Times New Roman">Example 2</Font></Text-field>We consider a second example of finding the equation of a tangent line. As we shall see, if we take away the majority of the discussion for this problem, the solution process is relatively short.Find the equation of the line tangent to the graph of NiMvLSUiZkc2IyUieEctJSRzaW5HRiY= at x = NiMlI1BpRw==/4.We first note that the point through which the tangent line passes is (NiMlI1BpRw==/4, f(NiMlI1BpRw==/4)) = (NiMlI1BpRw==/4 , 1/NiMqJCIiIyMiIiJGJA==).Below is a demonstration of the approximation procedure for estimating the slope of this tangent line. Click on the graph that follows and use the play controls that will appear near the top of the screen to view the 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seen in the demonstration, we will approximate the slope of the tangent line by choosing points on the graph of f successively closer to x = NiMlI1BpRw==/4. Noting that NiMlI1BpRw==/4 ~ 0.7854 (i.e. is approximately equal to), the values we will choose for x are as follows: x = 3NiMlI1BpRw==/4 ~ 2.3562, x = NiMlI1BpRw==/2 ~ 1.5708, x = 1, x = 0.9, x = 0.8, and x = 0.79. Fill in the x-values in the Maple command below in order to find the corresponding approximations to the slope of the tangent line.f:=sin(x);
m:=(subs(x=( ),f)-subs(x=Pi/4,f))/(( )-Pi/4);
evalf(%);You should obtain (approximately) 0, 0.3729, 0.6261, 0.6651, 0.7019, and 0.7055, respectively. Below is a summary of the results we have obtained.x = 2.3562 m = 0x = 1.5708 m = 0.3729x = 1 m = 0.6261x = 0.9 m = 0.6651x = 0.8 m = 0.7019
x = 0.79 m = 0.7055It is not clear exactly what the slope values are approaching as the x-values approach Pi/4, but we should feel pretty confident that the last approximation to the slope of the tangent line is the best estimate. Thus, we will say that the line tangent to f(x) = sin(x) at (NiMlI1BpRw==/4 , 1/NiMqJCIiIyMiIiJGJA==) has an approximate slope of m = 0.7055. Using this slope we obtain the following equation for the tangent line:
y = 0.7055x + 0.153008<Text-field layout="Heading 1" style="Heading 1"><Font family="Times New Roman">Assignment</Font></Text-field>Open and complete the Tangent Lines worksheet (Microsoft Word document) found on the departmental webpage. You may use Maple to help you complete the assignment.<Text-field layout="Heading 1" style="Heading 1"><Font family="Times New Roman">Alternative Viewpoint About Slope of Tangent Line</Font></Text-field>The above sections demonstrate a procedure for finding the slope of the tangent line at a particular point on the graph of the function. Much of this semester will be spent refining this procedure. However in this section, we wish to present another way of thinking about the slope of the tangent line. The viewpoint described in this section is less practical to implement but has tremendous value in helping us develop our intuition about the geometrical meaning of the slope to the tangent line. An example should help clarify.<Text-field layout="Heading 2" style="Heading 2"><Font family="Times New Roman">Example 3</Font></Text-field>The Figure 2 is reproduced below and the point of tangency "T" is shown. If we now zoom in on the graph around point T we get the following:As we zoom in even more, what do you notice is happening?Notice in the last graph that the graph of the function and the tangent line are indistinguishable and the graph of the function appears to be linear. This is the same phenomenon that you experience as you walk in the desert, that is, we know that we live on a sphere called Earth but it sure looks flat like a Euclidean plane. This illusion of the Earth looking flat is a result of our scaling. The larger our scale, our being in outer space for instance, then the easier it is to discern curvature; however, the smaller our scale, our walk in the desert for example, then the harder it is to see the curvature of a function. The consequence of all this is that we can think of the tangent line as that line whose graph becomes more indistinguishable from the graph of the function the more you zoom in. In other words, the more you zoom in the more accurately the tangent line approximates the function.Copyright 2001 Karl HavlakRevised 2006 Andrew Siefker