Cornu Spiral Consider the cornu spiral given by
(a) Use a graphing utility to graph the spiral over the interval
(b) Show that the cornu spiral is symmetric with respect to the origin.
(c) Find the length of the comu spiral from
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Chapter 10 Solutions
Calculus
- A particle is traveling along a circualr path defined by x^2 + y^2 = 4, where x and y are measured in centimeters. If the particle starts at the point (2,0) and moves in a counterclockwise direction at a speed of 6cm/sec, what will be the coodeinate fo the particle's position after 10 seconds? (show detailed diagrams)arrow_forwardII. Determine the equation of the family of curves as described then find thedifferential equation by eliminating the arbitrary constants. Draw the figure showingthe family of curves. 1. Parabolas with axis parallel to the x-axis with focal distance “a” fixed.2. Circles tangent to the x-axis.3. Straight lines with sum of x and y intercept equal to a constant “k”.arrow_forwardArc length of polar curvesFind the arc length of the spiral r = ƒ(θ) = θ, for 0 ≤ θ ≤ 2πarrow_forward
- The curve (x - a) ^ 2 + z ^ 2 = r ^ 2 lying on the plane of XZ- in R ^ 3 region formed by rotating the around the Z- axis (where a and r are positive constants) Cartesian (closed) and parametric Find the equation.arrow_forwardII. Determine the equation of the family of curves as described then find thedifferential equation by eliminating the arbitrary constants. Draw the figure showingthe family of curves. 2. Circles tangent to the x-axis.arrow_forwardArc length of the curve from point P to Q. x^2=(y-4)^3, P(1,5), Q(27,13)arrow_forward
- A radial line is drawn from the origin to the spiral r = aθ (a > 0and θ ≥ 0). Find the area swept out during the second revolution of the radial line that was not swept out during the first revolution.arrow_forwardparametric 1 Determine the parametric equations of the path of a particle that travels the circle: (x−1)2 + (y−1)2=81 on a time interval of 0 ≤ t ≤ 2π: if the particle makes one full circle starting at the point ( 10 , 1 ) traveling counterclockwisex( t ) = y( t ) = if the particle makes one full circle starting at the point ( 1 , 10 ) traveling clockwise x( t ) = y( t ) = if the particle makes one half of a circle starting at the point ( 10 , 1 ) traveling clockwise x( t ) = y( t ) =arrow_forwarda. Show that when you express ds2 = dx2 + dy2 + dz2 in terms of cylindrical coordinates, you get ds2 = dr2 + r2 du2 + dz2. b. Interpret this result geometrically in terms of the edges and a diagonal of a box. Sketch the box. c. Use the result in part (a) to find the length of the curve r = eu, z = eu, 0 <=u<=ln 8.arrow_forward
- Walking on a surface Consider the following surfaces specified in the form z = ƒ(x, y) and the oriented curve C in the xy-plane. a.In each case, find z’(t). b.Imagine that you are walking on the surface directly above the curve C in the direction of positive orientation. Find the values of t for which you are walking uphill (that is, z is increasing). z = 4x2 - y2 + 1, C: x = cos t, y = sin t; 0 ≤ t ≤ 2πarrow_forwardHorizontal tangent planes Find the points at which the following surfaces have horizontal tangent planes. z = cos 2x sin y in the region -π ≤ x ≤ π, -π ≤ y ≤ πarrow_forwardWalking on a surface Consider the following surfaces specified in the form z = ƒ(x, y) and the oriented curve C in the xy-plane. a. In each case, find z'(t). b. Imagine that you are walking on the surface directly above the curve C in the direction of positive orientation. Find the values of t for which you are walking uphill (that is, z is increasing). z = x2 + 4y2 + 1, C: x = cos t, y = sin t; 0 ≤ t ≤ 2πarrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage