Tangent Lines

8 Key excerpts on "Tangent Lines"

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  eBook - PDF

  Calculus

  Single Variable

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  58 CHAPTER 2 Henglein and Steets/Getty Images One of the crowning achievements of calculus is its ability to capture continuous motion mathematically, allowing that motion to be analyzed instant by instant. The Derivative Many real-world phenomena involve changing quantities—the speed of a rocket, the inflation of currency, the number of bacteria in a culture, the shock intensity of an earthquake, the voltage of an electrical signal, and so forth. In this chapter we will develop the concept of a “derivative,” which is the mathematical tool for studying the rate at which one quantity changes relative to another. The study of rates of change is closely related to the geometric concept of a tangent line to a curve, so we will also be discussing the general definition of a tangent line and methods for finding its slope and equation. 2.1 Tangent Lines and Rates of Change In this section we will discuss three ideas: Tangent Lines to curves, the velocity of an object moving along a line, and the rate at which one variable changes relative to another. Our goal is to show how these seemingly unrelated ideas are, in actuality, closely linked. Tangent Lines In Example 1 of Section 1.1 we used an informal argument to find the equation of a tangent line to a curve. However, at that stage in the text we did not have a precise definition of a tangent line. Now that limits have been defined precisely we can give a mathematical definition of the tangent line to a curve y = f (x) at a point P(x 0 , f (x 0 )) on the curve. As illustrated in Figure 2.1.1, the slope m PQ of the secant line through P and a second point Q(x, f (x)) on the graph of f is m PQ = f (x) − f (x 0 ) x − x 0 If we let x approach x 0 , then the point Q will move along the curve and approach the point P. Suppose the slope m PQ of the secant line through P and Q approaches a limit as x →x 0 . In that case we can take the value of the limit to be the slope m tan of the tangent line at P.

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  Calculus

  Late Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  59 2 One of the crowning achievements of calculus is its ability to capture continuous motion mathematically, allowing that motion to be analyzed instant by instant. THE DERIVATIVE Many real-world phenomena involve changing quantities—the speed of a rocket, the inflation of currency, the number of bacteria in a culture, the shock intensity of an earthquake, the voltage of an electrical signal, and so forth. In this chapter we will develop the concept of a “derivative,” which is the mathematical tool for studying the rate at which one quantity changes relative to another. The study of rates of change is closely related to the geometric concept of a tangent line to a curve, so we will also be discussing the general definition of a tangent line and methods for finding its slope and equation. 2.1 Tangent Lines AND RATES OF CHANGE In this section we will discuss three ideas: Tangent Lines to curves, the velocity of an object moving along a line, and the rate at which one variable changes relative to another. Our goal is to show how these seemingly unrelated ideas are, in actuality, closely linked. Tangent Lines In Example 1 of Section 1.1 we used an informal argument to find the equation of a tangent line to a curve. However, at that stage in the text we did not have a precise definition of a tangent line. Now that limits have been defined precisely we can give a mathematical definition of the tangent line to a curve y = f (x) at a point P(x 0 , f (x 0 )) on the curve. As illustrated in Figure 2.1.1, the slope m PQ of the secant line through P and a second point Q(x, f (x)) on the graph of f is m PQ = f (x) − f (x 0 ) x − x 0 If we let x approach x 0 , then the point Q will move along the curve and approach the point P. Suppose the slope m PQ of the secant line through P and Q approaches a limit as x → x 0 . In that case we can take the value of the limit to be the slope m tan of the tangent line at P. Thus, we make the following definition. Figure 2.1.1

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  eBook - PDF

  Calculus

  Early Transcendentals

  • Howard Anton, Irl C. Bivens, Stephen Davis(Authors)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  79 2 One of the crowning achievements of calculus is its ability to capture continuous motion mathematically, allowing that motion to be analyzed instant by instant. THE DERIVATIVE Many real-world phenomena involve changing quantities—the speed of a rocket, the inflation of currency, the number of bacteria in a culture, the shock intensity of an earthquake, the voltage of an electrical signal, and so forth. In this chapter we will develop the concept of a “derivative,” which is the mathematical tool for studying the rate at which one quantity changes relative to another. The study of rates of change is closely related to the geometric concept of a tangent line to a curve, so we will also be discussing the general definition of a tangent line and methods for finding its slope and equation. 2.1 Tangent Lines AND RATES OF CHANGE In this section we will discuss three ideas: Tangent Lines to curves, the velocity of an object moving along a line, and the rate at which one variable changes relative to another. Our goal is to show how these seemingly unrelated ideas are, in actuality, closely linked. Tangent Lines In Example 1 of Section 1.1 we used an informal argument to find the equation of a tangent line to a curve. However, at that stage in the text we did not have a precise definition of a tangent line. Now that limits have been defined precisely we can give a mathematical definition of the tangent line to a curve y = f (x) at a point P(x 0 , f (x 0 )) on the curve. As illustrated in Figure 2.1.1, the slope m PQ of the secant line through P and a second point Q(x, f (x)) on the graph of f is m PQ = f (x) − f (x 0 ) x − x 0 If we let x approach x 0 , then the point Q will move along the curve and approach the point P. Suppose the slope m PQ of the secant line through P and Q approaches a limit as x → x 0 . In that case we can take the value of the limit to be the slope m tan of the tangent line at P. Thus, we make the following definition. Figure 2.1.1

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  eBook - ePub

  What is Calculus?

  From Simple Algebra to Deep Analysis

  • R Michael Range(Author)
  • 2015(Publication Date)
  • WSPC

    (Publisher)

  analytic version of calculus based on limits, and it will allow us to focus from the very beginning on the principal new ideas in their natural context, where they truly are indispensable.

  We shall freely use standard concepts and formulas familiar from typical high school geometry and algebra courses, including, for example, the slope of a line. The most important background material will be thoroughly reviewed in Chapter I , in a form that will include and highlight the critical concepts that are necessary to understand the new central ideas related to limits. The reader is encouraged to refer to appropriate sections in Chapter I as needed in order to follow the discussion in the Prelude.

  2      Tangents to Circles

  The construction of Tangent Lines to circles, parabolas, and similar classical curves has a long history, going back to Euclid (4th century B.C.), Apollonius (3rd century B.C.), and other Greek geometers over 2300 years ago. In antiquity a tangent was defined as “a line which touches a curve but does not cut it” [Victor J. Katz, A History of Mathematics , 3rd. ed., Addison-Wesley, New York 2009, p. 120]. The tangent appears to fit the curve near the point of contact in an optimal way. The situation is particularly simple for a circle C , where the tangent at a point P on C is that unique line that is perpendicular to the radial line connecting P to the center of the circle. In general, the line that is perpendicular to a tangent at a point P on a curve is called the normal to the curve at P . Circles are special, since all normals go through the center and consequently are easy to draw for any point P on the circle. However, when one considers more general curves, there is no obvious way to construct normals and/or tangents. Finding either one immediately determines the other.1 The main problem then is to turn the intuitive but vague ancient idea of “tangent” recalled above into a precise definition that can be used to identify tangents and determine their slopes for arbitrary curves. Intuitively, we recognize that (in a small neighborhood) a tangent intersects the curve under consideration only at one point P —the point of tangency —while most small perturbations (i.e., changes) of the tangent will intersect the curve at two distinct points close to P . (See Figure 2

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  Math Dictionary With Solutions

  A Math Review

  • Chris Kornegay(Author)
  • 1999(Publication Date)
  • SAGE Publications, Inc

    (Publisher)

  472 I Tt >· Table, in trigonometry See Trigonometry Table, how to use. >· Tangent, in trigonometry The tangent is one of the six functions of trigonometry. De-fined for a right triangle, it is a fraction of the opposite side to the adjacent side. From angle A, tan A = | . From angle B, tanS = ^. For more, see Trigonometry, right triangle definition of. >• Tangent Line In relation to a curve, a tangent line is any line that intersects the curve in one point without passing through the curve. It touches the curve at one point. In geometry, if we consider a circle, a tangent line is any line that contains a point of the circle but no interior points of circle. It touches the circle at one point. In the figure, we say that (read the line PQ) is tangent to the circle Ο at the point P. ^ Term A term is a number, a variable, or the product of a number and a variable(s). The concept of a term is usually used to delineate one term from another. In the expression 6x^ — — Axy + 5, there are four terms: dx^, —x^, -4xy, and 5. Each term is separated by, yet includes, a positive (-I-) and a negative (—) sign. The term 6x^ is composed of the number -|-6, called the numerical coefficient or coefficient, and a variable with an exponent of 3. The term —x^ is composed of the coefficient — 1 and a variable with an exponent of 2. The — 1 in this term is usually not written since (—l)^^ is the same as —x^. The term —Axy is composed of the number —4 and the variables χ and y. Finally, the last term 5, which has no variable, is referred to as a constant term. For more, see Coefficient or Combining Like Terms. ^ Terminal Side In trigonometry, the terminal side of an angle is a ray that is determined by rotating the initial side of the angle clockwise or counterclockwise. See Angle, in trigonometry. >• Terminating Decimals See Decimal Numbers, terminating. THIRTY-SIXTY-NINETY DEGREE TRIANGLE • 473 I > Terms See Term.

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  Single Variable Calculus

  Early Transcendentals, Metric Edition

  • James Stewart, Daniel K. Clegg, Saleem Watson, , James Stewart, James Stewart, Daniel K. Clegg, Saleem Watson(Authors)
  • 2020(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  How can this idea be made precise? For a circle we could simply follow Euclid and say that a tangent is a line  that inter- sects the circle once and only once, as in Figure 1(a). For more complicated curves this definition is inadequate. Figure l(b) shows a line  that appears to be a tangent to the curve C at point P, but it intersects C twice. To be specific, let’s look at the problem of trying to find a tangent line  to the parab- ola y - x 2 in the following example. EXAMPLE 1 Find an equation of the tangent line to the parabola y - x 2 at the point Ps1, 1d. SOLUTION We will be able to find an equation of the tangent line  as soon as we know its slope m. The difficulty is that we know only one point, P, on , whereas we need two points to compute the slope. But observe that we can compute an approximation to m by choosing a nearby point Qs x, x 2 d on the parabola (as in Figure 2) and computing the slope m PQ of the secant line PQ. (A secant line, from the Latin word secans, meaning cutting, is a line that cuts [intersects] a curve more than once.) We choose x ± 1 so that Q ± P. Then m PQ - x 2 2 1 x 2 1 For instance, for the point Qs1.5, 2.25d we have m PQ - 2.25 2 1 1.5 2 1 - 1.25 0.5 - 2.5 The tables in the margin show the values of m PQ for several values of x close to 1. The closer Q is to P, the closer x is to 1 and, it appears from the tables, the closer m PQ is to 2. This suggests that the slope of the tangent line  should be m - 2.

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  eBook - ePub

  Painless Calculus

  • Barron's Educational Series, Christina Pawlowski-Polanish(Authors)
  • 2021(Publication Date)
  • Barrons Educational Services

    (Publisher)

  Chapter 4

  Applications of Derivatives

  Knowing how to calculate the derivative of a function and what it represents leads to some very useful applications of derivatives. Equation of a Tangent Line

  Discovering that the derivative of a function at a value represents the slope of the tangent line can be extended further to writing the equation of a tangent line to a curve at a point. There are a few ways to write the equation of a line, including the slope-intercept form, y = mx + b, and the point-slope form, y – y1 = m(x – x1 ). Typically, you will be given the equation of the function and the x-value at which the tangent line and the curve intersect. With this given information, using the point-slope form to write the equation of a tangent line will be more convenient.

  Writing the equation of a tangent line is painless. There are three steps to follow.

  Step 1: Find the coordinates of the point of tangency; these are the x- and y-coordinates of the point of intersection of the tangent line and the curve.

  Step 2: Find the slope of the tangent line by calculating the derivative at the point of tangency.

  Step 3: Substitute the point of tangency and the slope of the tangent into the point-slope form of a line.

  Example 1:

  Given the function y = x3 , write the equation of the tangent line to the curve at x = 2.

  Solution:

  Step 1: Find the coordinates of the point of tangency.

  Given y = x3 and x = 2:

  y = (2)3 = 8

  The point of tangency is (2, 8).

  Step 2: Find the slope of the tangent line.

  Calculate the derivative:

  Evaluate the derivative at x = 2:

  This represents the slope of the tangent line at x = 2.

  Step 3: Use point-slope form, y – y1 = m(x – x1 ), to find the equation:

  Point: (x1 , y1 ) = (2, 8)

  Slope: m = 12

  y – 8 = 12(x – 2) or y = 12x – 16

  PAINLESS TIP

  Unless otherwise stated, the equation for a tangent line can be left in point-slope form. Since the point of tangency will very rarely be a y

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  eBook - ePub

  Advanced Calculus

  Differential Calculus and Stokes' Theorem

  • Pietro-Luciano Buono(Author)
  • 2016(Publication Date)
  • De Gruyter

    (Publisher)

  3Tangent Spaces and 1-forms

  We define the concept of tangent space using specific examples: curves, the spaces ℝn and finally two-dimensional surfaces given by z = f (x, y ). The tangent space to a geometric object is an important topic in advanced calculus and differential geometry. The second part introduces a formal definition of differential and the construction of the so-called 1-forms.

  3.1Tangent spaces

  We begin by discussing how Tangent Lines lead to tangent space and build more general tangent spaces for ℝn and surfaces.

  3.1.1From Tangent Lines to Tangent Spaces

  The previous section shows how to determine the tangent line to a curve C at some point p ∈ C using the derivative of a vector function r (t ) defining C . The goal of this section is to extract the vectors which lie on the Tangent Lines so that we can add them together without leaving the tangent line.

  Example 3.1.1. Consider the plane curve C given by x (t ) = t 2 , y (t ) = 1 − t for t ∈ [0, 1]. We obtain the tangent line equation at t = 1/ 2 by the formula

  where s ∈ ℝ is a parameter. The tangent line is not a subspace, if one adds two elements of 𝓁 (s ), then the result is not in 𝓁 (s ) anymore:

  However, if we decide to fix the base point (x (1/ 2), y (1/ 2)) and only add the parametrized part and add the base point after, this leaves us on the tangent line:

  and so (x (1/ 2), y (1/ 2))+(s 1 +s 2 )(1, −1)

  is a point of the tangent line. See Figure 3.1 for an illustration.

  With this in mind, we can now introduce the concept of tangent space, which is a crucial aspect of advanced calculus and of differential geometry in general.

  Definition 3.1.2. Let C be a smooth space curve (in ℝ

  n ). If p

  ∈ C then the tangent space of C at

  p, denoted by Tp C, is the set of all vectors tangent to C at the point p.

  Fig. 3.1. Curve C given by r (t ) = (t 2 , 1 − t ) and the tangent line at p = (1/ 4, 1/ 2). The orientation is from (0, 1) to (1,

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