What is Secant: Definition and 61 Discussions

In numerical analysis, the secant method is a root-finding algorithm that uses a succession of roots of secant lines to better approximate a root of a function f. The secant method can be thought of as a finite-difference approximation of Newton's method. However, the secant method predates Newton's method by over 3000 years.

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  1. N

    How can I overcome the lack of support for my passion for mathematics?

    Slope of a Tangent Line For f (x) = x^2 − 1: Find the slope of the secant line containing the points P = (−1, f (−1)) and Q = (−1 + h, f (−1 + h)). Solution: I think I need to find f(-1). f(-1) = (-1)^2 - 1 f(-1) = 1 - 1 f(-1) = 0 Point P becomes (-1, 0). I now must find f(-1 + h)...
  2. N

    Can you clarify your understanding for parts (a) and (b)?

    Slope of a Tangent Line For f (x) = x^3. (a) Find the slope of the secant line containing the points (2, 8) and (3, 27). (b) Find the slope of the secant line containing the points (2, 8) and (x, f (x)), where x does not equal 2. For (a), I just have to find m, the slope using m = delta...
  3. brotherbobby

    To prove a trigonometric identity with tan() and cot()

    Attempt : I could not progress far, but the following is what I could do. $$\begin{align*} \mathbf{\text{LHS}} & = (\tan A+\tan B+\tan C)(\cot A+\cot B+\cot C) \\ & = 3+\tan A \cot B+\tan B \cot A+\tan A \cot C+\tan C \cot A+\tan B \cot C+\tan C \cot B\\ & = 3+\frac{\tan^2A+\tan^2B}{\tan A \tan...
  4. P

    MHB Hayldiburasomas' question via email about Secant Method

    The Secant Method is a numerical scheme to solve equations of the form $\displaystyle f\left( x \right) = 0 $, so we must rewrite the equation as $\displaystyle 0 = \frac{1}{2}\,x^2 - 10 - \sin{ \left( 1.8\,x \right) } $. Thus $\displaystyle f\left( x \right) = \frac{1}{2}\,x^2 - 10 - \sin{...
  5. karush

    MHB 3.2.15 mvt - Mean value theorem: graphing the secant and tangent lines

    $\tiny{3.2.15}$ Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function the secant line through the endpoints, and the tangent line at $(c,f(c))$. $f(x)=\sqrt{x} \quad [0,4]$ Are the secant line and the tangent line parallel...
  6. L

    MHB Solving Equations Using Bisection, Newton, and Secant Methods

    Task Use the Bisection, Newton, and Secant methods to solve (to at least 8 signicant figures) the equation sin(x) = 0.98 cos(2x2) over the interval [0, 2.5], in radiant units. For the Newton method, try with several different initial guesses including x0 = 1. With the Secant method, use the same...
  7. C

    MHB Secant and Tangent Angles in Circles, finding an arc length.

    I got 138.17, but that isn't correct. I don't know how to do it, since the only way I thought, gave me the wrong answer. Can anyone help?
  8. K

    Derivative of Secant: Find dy/dx

    Homework Statement Find dy/dx for ##~y=\sec^2(5x)## Homework Equations Secant and it's derivative: $$\sec\,x=\frac{1}{\cos\,x}$$ $$\sec'\,x=\tan\,x\cdot\sec\,x$$The Attempt at a Solution $$y=\sec^2(5x)~\rightarrow~y'=2\cdot 5 \cdot \sec(5x)\tan(5x)\sec(5x)=10\tan(5x)\sec^2(5x)$$ The answer...
  9. S

    Integration with hyperbolic secant

    Homework Statement Solve ##\displaystyle{d\sigma = \frac{d\rho}{\cosh\rho}.}## Homework Equations The Attempt at a Solution The answer is ##\displaystyle{\sigma = 2 \tan^{-1}\text{sinh}(\rho/2)}##. See equation (10.2) in page 102 of the lecture notes in...
  10. M

    Geometry - Arcs created by Secant Lines

    Homework Statement This picture: http://i.imgur.com/n015WjU.png It's drawn with exactly the amount of information from the worksheet. Specifically, the two secants meet at a point, with an angle of 28 degrees between them. Both secants partition off an arc of 120 degrees. The goal is to...
  11. stevendaryl

    Insights Solve Integrals Involving Tangent and Secant with This One Trick - Comments

    stevendaryl submitted a new PF Insights post Solve Integrals Involving Tangent and Secant with This One Weird Trick Continue reading the Original PF Insights Post.
  12. K

    What is the Derivative of Inverse Secant and its Graph Representation?

    Homework Statement The two graphs are possible legitimate representations of ##y=\sec^{-1}(x)##. The derivative is positive on all the domain and so is graph A, but graph B has negative tangent when x<-1 Homework Equations Derivative of inverse secant...
  13. A

    I What is the derivative of the inverse secant function?

    Please refer to the below image (Example 5). Do anyone know how 5x^4 > 1 > 0?
  14. R

    Secant Method Minimum Estimation

    Homework Statement I am having some trouble understanding the following solved problem: Use two iterations of the Secant method to estimate where ##x^2 -x## has a local minimum. Start with ##x^{(0)} =1##, ##x^{(1)} =1/2.## Answer: Homework Equations Secant iteration: $$x_k = x_{k-1} -...
  15. C

    Finding the Slope of Secant and Tangent Lines

    Homework Statement The points P (2,-1) and Q (3,-4) lie on the parabola y = -x2+2x-1 a) Find the slope of the secant line PQ. b) Find the slope of the tangent line to the parabola to the parabola at P. c) Find the equation of the tangent line at P. Homework EquationsThe Attempt at a Solution I...
  16. J

    Finding error of secant method empirically

    Homework Statement I need to find the roots of a given equation using the secant method and matlab. I have already found all the roots, but I am also asked to find the rate of convergence for the method empirically, meaning that they want me to find the order of convergence through the set...
  17. W

    Complexifying the integral of the secant function

    Since learning about being able to complexify differential equations (I am doing the MIT OCW course by Arthur Mattuck), I have tried to apply to this to particular problems in integration as well. Specifically, I have tried to integrate the secant of some function to see if it would lead to the...
  18. A

    Quick question on power series of secant

    Hey, everyone. I am trying to find the power series of secant from the known power series of cosin, but my answer does not match up with Wolfram and Wikipedia. I know: cos(\theta) = 1 - \frac{1}{2}x^2 + \frac{1}{4!}x^4 + ... So, using the first two terms (assuming a small angle)...
  19. V

    MHB Find the slope of the secant to the curve f(x)=-3logx+2 between these points:

    The question was too long to post in the title so I just wrote down the first part. I hope this is alright. Here is the question that I am doing right now: This is the graphical representation (thanks to Desmos Graphing Calculator): So I have substituted the points in the equation to get...
  20. N

    Solve a Trig Equation Containing secant

    Homework Statement sec(5x)-5=0 Homework Equations The Attempt at a SolutionI turned it into 1/cos(5x)=5 x=θ/5 then i switched it to cos(5x)=/1/5 Then cos-1(1/5)=θ which = 1.36 (one solution) then divided by 5 which is .2738 which works as one solution since the positive cos is in the 1st...
  21. P

    Slope of Secant Homework: y=x^2+x, P(1,2) & Q(2,6)

    Homework Statement 1. For the curve y = x^2+ x, and the point P (1,2) a) determine the slope of the secant mPQ if Q(2, 6) b) what is the slope of the secant if Q is at the following values of x 1.5 1.1 1.01 1.001 c) Estimate the value of the tangent to the curve at P. Reason...
  22. M

    How do I integrate sec(x) sin(nx) over a specific interval?

    So I know that sec(x) has period 2Pi, and it's even so I don't need to figure out coefficients for bn. Let's take the limits of the integral to go from -3/2 Pi to 1/2 Pi. How do I integrate sec(x) sin(nx) dx?! Am I on the right path? PS: I know that this doesn't satisfy the Dirichlet...
  23. M

    Fourier transform of the hyperbolic secant function

    Homework Statement Hi there! I'm just trying to figure out the Fourier transform of the hyperbolic secant function... I already know the outcome: 4\sum\ ((-1)^n*(1+2n))/(ω^2*(2n+1)^2) But sadly, I cannot figure out how to work round to it! :( maybe one of you could help me... Homework...
  24. S

    Discovering the Slope of a Secant Line: A Step-by-Step Guide with Examples

    Homework Statement Find the slope of the line that joins P (0,2) and Q (x, 2^x+1) for x ≠0. type the expression using x as a variable. Homework Equations Rise over Run? The Attempt at a Solution I have a picture of the graph but i don't really know what to do, Please help. I graphed it at x=...
  25. N

    Integrals involving Secant & Tangent Derivation

    Homework Statement If the power of the secand is even and positive.. \int sec^{2k} x tan^{n} x dx = \int (sec^2 x)^{k-1} tan ^n x sec^2 x dx The Attempt at a Solution The way I see it, sec^{2k} x = sec^2 x dx * sec^k x dx the next step seems to be to break down sec^k, but on closer...
  26. B

    What is the Derivative of Inverse Secant?

    Homework Statement Find the derivative of sec^{-1}(\frac{\sqrt{1+x^{2}}}{x}) Homework Equations sec^{-1}=\frac{U'}{U\sqrt{U^{2}-1}} The Attempt at a Solution U'=-\frac{1}{x^{2}\sqrt{1+x^{2}}} U\sqrt{U^{2}-1}= \frac{\sqrt{1+x^{2}}}{x^{2}} Therefore the derivative is...
  27. B

    Secant line in Fermat's Theorem

    I'm trying to understand something in Fermat's Theorem. I can't really phrase it in words, but I will write what my textbook says. Apparently if \lim_{x→c}\frac{f(x)-f(c)}{x-c} > 0 then there exists an open interval (a,b) containing c such that \frac{f(x)-f(c)}{x-c} > 0 for all...
  28. B

    Finding secant with calculator

    Homework Statement what is sec(pi/4) what is sec(0) The Attempt at a Solution First let me make sure that cos^-1, acos and sec, they are all the same right? I put in cos^-1(pi/4) in my calculator and the answer I get is .667, which I think is the same as sec (pi/4). The book...
  29. B

    Why don't the secants cancel in this equation?

    Homework Statement Homework Equations The Attempt at a Solution I don't see how they get from step one to two. I would think both secants would cancel since one is positive and the other is negative but that doesn't happen. i think i understand the manipulations of the tangents but not...
  30. C

    Partial Sums resembling sums of secant hyperbolic

    Homework Statement Compute the following partial sum \sum_{k=0}^n\frac{1}{2^{2^k}+2^{-2^k}} Homework Equations The Attempt at a Solution So far, I've tried transforming the terms into secant hyperbolic functions...
  31. D

    Trig: Writing the equation for vertical asymptote of a secant function?

    Homework Statement Homework Equations How did they come up with \frac{1}{2}+k for the equation of the vertical asymptote? I understand everything else except this.The Attempt at a Solution On this particular exercise, I graphed it and saw that each of my vertical dashed lines were all one whole...
  32. M

    Finding Inverse Hyperbolic secant in terms of logarithms ?

    The Problem is when I Compute the Inverse I have to solutions sech^{-1}(x) = ln(\frac{1\pm \sqrt{1-x^{2}}}{x}) : 0<x\leq 1 And this not function which of them I will choose Another Question is how can I prove without the graph that csch (x) is one - to -one thanks
  33. A

    Secant method, bisection or NR

    Hi so I have attached a graph, and I am trying to find the first root of the function. I began using Newton Raphson however due to the oscillatory nature of the curve, sometimes NR would not find the first route but an nth route. I need the first. Therefore what I did was compute the first time...
  34. C

    Why are Cosine and Secant Even Functions?

    Homework Statement Why is Cosin and Secant even? Cos (-t) = cos t, Sec (-t) = sec t Why don't they equal - sec t instead like the rest of the functions? Thanks Homework Equations The Attempt at a Solution
  35. S

    Is Secant Inversely Proportional to Cosine?

    cos = 5/5.38 so secant is = 5.38/5? thats my solution anyway, just want to check if i got it right.
  36. J

    Solving Inverse Secant Problem: Finding y = sec^-1(9s^4 + 7)

    Homework Statement Find the value of y = sec^-1(9s^4 + 7) with respect to s Homework Equations The Attempt at a Solution secy = 9s^4 + 7 but where do I go from here?
  37. J

    Finding Slopes and Equations of Secant and Tangent Lines for a Given Curve

    Homework Statement Given a point P (3, 10) and the equation of a curve as x^2 -5x-4, find the slope of the secant and the equation of the tangent line to the curve Homework Equations The Attempt at a Solution I tried using y = f(x + h) -f(x) all divided by h and got (x + h)^2 -...
  38. C

    MATLAB Absolute and Relative Approximate Errors for Secant Method in MATLAB

    I try to solve this equation with secant method in MATLAB. fn=40*n^1.5-875*n+35000 my initial guess is n1=60; n2=68; I want to find root and absolute relative approximate error at the end of each iteration. I have an infinite loop. Can you help me repair my file? This is my m-file...
  39. R

    Understanding Secant Modulus: Definition, Importance, and Applications

    what is secant modulus?
  40. Z

    Finding the Gradient of the Secant on a Curve: P and Q Coordinates Given

    Homework Statement There are two parts to this problem. On the curve 2x^2-5 lie two points P and Q. Let the abscissa of P be "x" and the abscissa of Q be "x+h". No numerical coordinates are given. a) State the coordinates of P and Q. b) Using these points find the gradient of...
  41. A

    Solving Equation 1 with Secant Method

    So I have studied some of the Numberical methods such as Newton Raphson etc, but I want to try and solve this particular problem using the secant method. The problem is something I am reading in an article, and I am trying to follow the maths through. I am slightly wondering though how it is...
  42. D

    Real world apps for the secant function

    On a project i have i need to find a real world application for a function. I chose secant. It's too late to turn back and i need help.
  43. C

    Real World Example for Cosecant or Secant

    I am math teacher and was asked the classic question, "Where am I ever going to use this in life" I usually have a real word example, but for the cosecant function I am stuck (Besides higher level math classes.) Anyone know a specific example?
  44. S

    Finding the Zeroes of a Secant Graph

    This is a continuation of my previous post, so please bear with me Homework Statement Find the zeroes of the following function: y= 2 sec (-2x+180deg) + 3Homework Equations Break down the equation into: y= 2 sec -2(x-90deg) + 3; Finding zeroes, means finding the x values when y=0...
  45. S

    TI-83 Problem: Finding the Domain of a Secant Graph

    TI-83 problem...Secant Graphs Homework Statement What are the properties of: y= 2 sec(-2x + 90deg) + 3 Homework Equations sec (x) = (1/cos (x)) The Attempt at a Solution I have a problem with finding the Domain of y= 2 sec(-2x + 180deg) + 3. First of all, I have to put the...
  46. V

    Fourier transform (FT) of a Hyperbolic secant

    Hi, Did anyone know how to do the Fourier transform of the hyperbolic secant? I know the answer; it's given in the text (I'm reading Ablowitz, Fokas, Complex Variables), it's another hyperbolic secant, but I want to know how to do it. My dilemma is: a) what contour to use? I'm having...
  47. R

    Efficiently Solve a Complex Secant Problem | Homework Help

    Homework Statement Reduce to a single term: I really have been spending a wretched time to solve that tan (x/2)*(1+sec x)(1+sec 2x)(1+sec 4x).......(1+sec (2^n)x) Homework Equations tan (x/2)=2 tan (x/4) / 1- tan^2 (x/4) can this ever help 1+tan^2=sec^2 The Attempt at a...
  48. E

    MATLAB MATLAB: Understanding the Secant Rule

    I am working on a MATLAB program to find the optimum heating of a heater. I am having a hard time applying the secant rule and really have no idea how to apply this rule in general. If someone could shine some light on the subject that would be great.
  49. 2

    Comp Sci C++ program for secant method

    so i worte this and it works somtimes. the function I am trying to find the roots of is 2x^3 - 6x^2 + 3x + 1 the program i wrote will find two of the roots but i can't get the third. I know the secant method it sensitive to the interval that you pick so i was thinking that was the...
  50. J

    Slope of Secant Line at PQ

    I have not seen math in years and just started claculus last week. Homework Statement The point p(25,7) lies on the curve y=\sqrt{}x+2. Let Q be the point (x, \sqrt{}x+2). Find the slope of the secant line PQ for the following values of x: If x=25.1, the slope of PQ is: Homework...
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