1 Tangents and Secants

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1 MTH 11 Web Based Material Essex County College Division of Mathematics and Physics Worksheet #, Last Update July 15, Tangents and Secants The idea of a it is central to calculus and an intuitive grasp of this concept is essential to further study. Initially we will examine two special types of its: tangents and velocities. These two its gives rise to the central idea in differential calculus the derivative. The word tangent is derived from the Latin word tangens, which means touching. Thus, a tangent to a curve is a line that touches the curve and has the same direction as the curve at the point of contact. Let s take a look at the curve y = x3 3 x to this curve at the point (0, 5). x + 5 and a line y = x + 5, which is tangent Figure 1: y = x3 3 x x + 5 and y = x + 5 This particular tangent line also happens to be a secant line, because it touches this particular curve twice. 3 Again, take a look! Let s take a look at a very simple example. Find an equation of the tangent line to the parabola y = x at the point P (1, 1). First we will make a rough sketch and then try to make a guess at what this tangent line is. Next we will take a sequence of secant lines that approach the tangent line, compute their slopes and make a prediction where these slopes are going. Since we want 1 This document was prepared by Ron Bannon (ron.bannon@mathography.org) using L A TEX ε. A secant is a straight line that cuts a curve in two or more parts. A tangent is a straight line that touches a curve at a point, but if extended does not cross it at that point. 3 You should be able to visually note that there are at least two points of intersection, but can you show that there s only two? What are the two points? 1

2 to move towards the point P (1, 1), along the curve y = x, we can select points from both the right and left of P and compute the slope of the secant line using the slope formula. Let s not be timid (stay close) when choosing values to the right and left of x = 1. Here s a nice pattern for the right of x = 1: 1.1, 1.01, 1.001; now lets compute the slope of the secant using P (1, 1) for these three values for x. Likewise, let s start from the left of P and compute slopes. This time you should choose three vales for x, and we ll discuss possible choices in class. Okay, let s make a prediction using the sequence of the secant slopes from both left and right about the possible value of the slope of the tangent line at the point P (1, 1). Finally, does your initial guess agree with our prediction?

3 Now let s take another example, but this time we will look at a function that has units. Suppose that an object is dropped from a platform that is 400 meters above the ground, and its position (s in meters) above the ground is a function of time (t is seconds), where s = s (t) = 4.9t Clearly this is a parabola, and if we draw a tangent line at the point where t = 5, we will be able to approximate the slope. 1. what is the unit of this slope?. Using t = 5, find values to the left and right of t = 5, compute the corresponding slope of the secant lines, and make a prediction about what the slope of the tangent line at this point. 3. What is the equation of this tangent line? 3

4 Introductory Limits Definition: We write f (x) = L x a and say, the it of f (x), as x approaches a, equals L, if we can make the values of f (x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of a) but not equal to a. Example: Find the it, if it exists. x 1 x + 1 x 1 Use values close to x = 1, from both left and right, and use a graph. Example: Find the it, if it exists. x 0 x + 4 Use values close to x = 0, from both left and right, and use a graph. 4 x 4 Use: ±1, ±0.5, ±0.1, ±0.05, ±

5 Example: Find the it, if it exists. x 0 cos π x Use values close to x = 0, from both left and right, and use a graph. 5 Example: Find the it, if it exists. tan x x x 0 x 3 Use values close to x = 0, from both left and right, and use a graph. 6 5 Use: ±1, ±1/, ±1/3, ±1/4, ±1/6. 6 Use: ±1, ±0.5, ±0.1, ±0.05, ±0.01, ±

6 Assignment: 1. Get the book! We re using the sixth edition of James Stewart s Single Variable Calculus: Early Transcendetals. You should buy the version that gives you access to WebAssign because you will need to complete online homework our Class Key is essex Here are your options: Option 1: Paperback version available only in ECC s bookstore includes textbook, Enhanced WebAssign access code with ebook and access code for the online student solutions manual. ISBN: ; Option : Buy a WebAssign/eBook access code directly from WebAssign, the URL is: < Option 1: Buy directly from Cengage < Just search for ISBN: ,. You should read.1 and., and then be able to do the following problems..1 The Tangent and Velocity Problems: 1,, 4, 5, 7, 9.. The Limit of a Function: 3, 4, 7, 8, 10, 15, 17,, 7, 34, 36, 37, 40. WebAssign problems, similar to the ones above, will be posted and you need to get started right away! Again the URL is < and our Class Key is essex Get started right away! 3. Don t fall behind! If you desire an education you re just going to have to put a considerable amount of time in. However, if you re looking to be called educated, without considerable effort, get a t-shirt that says, Harvard. 6