Tractrix A person moves from the origin along the positive y-axis pulling a weight at the end of a 12-meter rope (see figure). Initially, the weight is located at the point (12, 0).
(a) Show that the slope of the tangent line of the path of the weight is
(b) Use the result of part (a) to find the equation of the path of the weight. Use a graphing utility to graph the path and compare it with the figure.
(c) Find any vertical asymptotes of the graph in part (b).
(d) When the person has reached the point (0, 12), how far has the weight moved?
Want to see the full answer?
Check out a sample textbook solutionChapter 8 Solutions
Calculus
- Van der Waals Equation In Exercise 18 at the end of Section 2.3, we discussed the ideal gas law, which shows the relationship among volume V, pressure p, and temperature T for a fixed amount 1 mole of a gas. But chemists believe that in many situations, the van der Waals equation gives more accurate results. If we measure temperature T in kelvins, volume V in liters, and pressure p in atmosphere 1 atm is the pressure exerted by the atmosphere at sea level, then the relationship for carbon dioxide is given by p=0.082TV0.0433.592V2atm What volume does this equation predict for 1 mole of carbon dioxide at 500 kelvins and 100 atm?Suggestion: Consider volumes ranging from 0.1 to 1 liter.arrow_forwardVelocity What is the rate of change in directed distance?arrow_forwardTangent line given Determine the constants b and c such that the line tangent to f(x) = x 2 + bx + c at x = 1 is y = 4x + 2.arrow_forward
- Finding tangent lines with implicit functions Find an equation of theline tangent to the curve x2 + xy - y3 = 7 at (3, 2).arrow_forwardFind equations for the horizontal tangent lines to the curve y = x3 - 3x - 2. Also find equations for the lines that are perpendicular to these tangent lines at the points of tangency.arrow_forwardA straight line perpendicular to and passing through the point of tangency of the tangent line is called the normal to the curve. Find an equation of the tangent line and the normal to the curve y = x3 − 3x + 2 at the point (3, 20). (Let x be the independent variable and y be the dependent variable.)arrow_forward
- equation of the tangent line to the curve at the given point. y = 2 + x3 , (−1, 1)arrow_forwardA straight line perpendicular to and passing through the point of tangency of the tangent line is called the normal to the curve. Find an equation of the tangent line and the normal to the curve y = x3 − 3x + 3 at the point (2, 5). (Let x be the independent variable and y be the dependent variable.) tangent liney= 24x−45 normal liney=arrow_forwardTangent lines Find an equation of the line tangent to the cycloidx = t - sin t, y = 1 - cos t at the points corresponding to t = π/6 and t = 2π/3.arrow_forward
- Find equations of the tangent lines to the curve y=(5ln(x)) / x at the points (1,0) and (e, 5/e) at the point (1,0) y= at the point (e, 5/e) y= Illustrate by graphing the curve and its tangent lines.arrow_forwardTangent and Normal Line equation: x^2+2y^2=9 TL perpendicular to the line x+3y=7arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageFunctions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage LearningTrigonometry (MindTap Course List)TrigonometryISBN:9781305652224Author:Charles P. McKeague, Mark D. TurnerPublisher:Cengage Learning
- Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning