Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)
6th Edition
ISBN: 9781524908102
Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE
Publisher: Kendall Hunt Publishing
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Chapter 10.4, Problem 57PS
To determine
To calculate: the Frenet- Serret formula is to be proved.
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2. Consider the following vector function:
r(t) = a cos wti + a sin wtj + bwtk,
where a, b and w are nonzero constants. Compute the radius of curvature for this curve and show
that it is a constant number (independent of the parameter t).
b. The vector function r(t)=sin 2ti-cos 2tj+t√5k determines a curve C'in space.
i. Find the unit tangent vector I and the principal unit normal N
ii. Determine the curvature of Cat time t
iii. Determine the tangential and normal component of the acceleration vector.
Consider the helix r(t) = (cos(-4t), sin(-4t), 3t). Compute, at t = 픔
A. The unit tangent vector T = (
000
B. The unit normal vector N = (00
C. The unit binormal vector B=(
000
D. The curvature k =
E|O
Chapter 10 Solutions
Calculus: Special Edition: Chapters 1-5 (w/ WebAssign)
Ch. 10.1 - Prob. 1PSCh. 10.1 - Prob. 2PSCh. 10.1 - Prob. 3PSCh. 10.1 - Prob. 4PSCh. 10.1 - Prob. 5PSCh. 10.1 - Prob. 6PSCh. 10.1 - Prob. 7PSCh. 10.1 - Prob. 8PSCh. 10.1 - Prob. 9PSCh. 10.1 - Prob. 10PS
Ch. 10.1 - Prob. 11PSCh. 10.1 - Prob. 12PSCh. 10.1 - Prob. 13PSCh. 10.1 - Prob. 14PSCh. 10.1 - Prob. 15PSCh. 10.1 - Prob. 16PSCh. 10.1 - Prob. 17PSCh. 10.1 - Prob. 18PSCh. 10.1 - Prob. 19PSCh. 10.1 - Prob. 20PSCh. 10.1 - Prob. 21PSCh. 10.1 - Prob. 22PSCh. 10.1 - Prob. 23PSCh. 10.1 - Prob. 24PSCh. 10.1 - Prob. 25PSCh. 10.1 - Prob. 26PSCh. 10.1 - Prob. 27PSCh. 10.1 - Prob. 28PSCh. 10.1 - Prob. 29PSCh. 10.1 - Prob. 30PSCh. 10.1 - Prob. 31PSCh. 10.1 - Prob. 32PSCh. 10.1 - Prob. 33PSCh. 10.1 - Prob. 34PSCh. 10.1 - Prob. 35PSCh. 10.1 - Prob. 36PSCh. 10.1 - Prob. 37PSCh. 10.1 - Prob. 38PSCh. 10.1 - Prob. 39PSCh. 10.1 - Prob. 40PSCh. 10.1 - Prob. 41PSCh. 10.1 - Prob. 42PSCh. 10.1 - Prob. 43PSCh. 10.1 - Prob. 44PSCh. 10.1 - Prob. 45PSCh. 10.1 - Prob. 46PSCh. 10.1 - Prob. 47PSCh. 10.1 - Prob. 48PSCh. 10.1 - Prob. 49PSCh. 10.1 - Prob. 50PSCh. 10.1 - Prob. 51PSCh. 10.1 - Prob. 52PSCh. 10.1 - Prob. 53PSCh. 10.1 - Prob. 54PSCh. 10.1 - Prob. 55PSCh. 10.1 - Prob. 56PSCh. 10.1 - Prob. 57PSCh. 10.1 - Prob. 58PSCh. 10.1 - 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Prob. 37PSCh. 10.3 - Prob. 38PSCh. 10.3 - Prob. 39PSCh. 10.3 - Prob. 40PSCh. 10.3 - Prob. 41PSCh. 10.3 - Prob. 42PSCh. 10.3 - Prob. 43PSCh. 10.3 - Prob. 44PSCh. 10.3 - Prob. 45PSCh. 10.3 - Prob. 46PSCh. 10.3 - Prob. 47PSCh. 10.3 - Prob. 48PSCh. 10.3 - Prob. 49PSCh. 10.3 - Prob. 50PSCh. 10.3 - Prob. 51PSCh. 10.3 - Prob. 52PSCh. 10.3 - Prob. 53PSCh. 10.3 - Prob. 54PSCh. 10.3 - Prob. 55PSCh. 10.3 - Prob. 56PSCh. 10.3 - Prob. 57PSCh. 10.3 - Prob. 58PSCh. 10.3 - Prob. 59PSCh. 10.3 - Prob. 60PSCh. 10.4 - Prob. 1PSCh. 10.4 - Prob. 2PSCh. 10.4 - Prob. 3PSCh. 10.4 - Prob. 4PSCh. 10.4 - Prob. 5PSCh. 10.4 - Prob. 6PSCh. 10.4 - Prob. 7PSCh. 10.4 - Prob. 8PSCh. 10.4 - Prob. 9PSCh. 10.4 - Prob. 10PSCh. 10.4 - Prob. 11PSCh. 10.4 - Prob. 12PSCh. 10.4 - Prob. 13PSCh. 10.4 - Prob. 14PSCh. 10.4 - Prob. 15PSCh. 10.4 - Prob. 16PSCh. 10.4 - Prob. 17PSCh. 10.4 - Prob. 18PSCh. 10.4 - Prob. 19PSCh. 10.4 - Prob. 20PSCh. 10.4 - Prob. 21PSCh. 10.4 - Prob. 22PSCh. 10.4 - Prob. 23PSCh. 10.4 - Prob. 24PSCh. 10.4 - Prob. 25PSCh. 10.4 - 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Prob. 15PSCh. 10.5 - Prob. 16PSCh. 10.5 - Prob. 17PSCh. 10.5 - Prob. 18PSCh. 10.5 - Prob. 19PSCh. 10.5 - Prob. 20PSCh. 10.5 - Prob. 21PSCh. 10.5 - Prob. 22PSCh. 10.5 - Prob. 23PSCh. 10.5 - Prob. 24PSCh. 10.5 - Prob. 25PSCh. 10.5 - Prob. 26PSCh. 10.5 - Prob. 27PSCh. 10.5 - Prob. 28PSCh. 10.5 - Prob. 29PSCh. 10.5 - Prob. 30PSCh. 10.5 - Prob. 31PSCh. 10.5 - Prob. 32PSCh. 10.5 - Prob. 33PSCh. 10.5 - Prob. 34PSCh. 10.5 - Prob. 35PSCh. 10.5 - Prob. 36PSCh. 10.5 - Prob. 37PSCh. 10.5 - Prob. 38PSCh. 10.5 - Prob. 39PSCh. 10.5 - Prob. 40PSCh. 10.5 - Prob. 41PSCh. 10.5 - Prob. 42PSCh. 10.5 - Prob. 43PSCh. 10.5 - Prob. 44PSCh. 10.5 - Prob. 45PSCh. 10.5 - Prob. 46PSCh. 10.5 - Prob. 47PSCh. 10.5 - Prob. 48PSCh. 10.5 - Prob. 49PSCh. 10.5 - Prob. 50PSCh. 10.5 - Prob. 51PSCh. 10.5 - Prob. 52PSCh. 10.5 - Prob. 53PSCh. 10.5 - Prob. 54PSCh. 10.5 - Prob. 55PSCh. 10.5 - Prob. 56PSCh. 10.5 - Prob. 57PSCh. 10.5 - Prob. 58PSCh. 10.5 - Prob. 59PSCh. 10.5 - Prob. 60PSCh. 10 - Prob. 1PECh. 10 - Prob. 2PECh. 10 - Prob. 3PECh. 10 - Prob. 4PECh. 10 - Prob. 5PECh. 10 - Prob. 6PECh. 10 - Prob. 7PECh. 10 - Prob. 8PECh. 10 - Prob. 9PECh. 10 - Prob. 10PECh. 10 - Prob. 11PECh. 10 - Prob. 12PECh. 10 - Prob. 13PECh. 10 - Prob. 14PECh. 10 - Prob. 15PECh. 10 - Prob. 16PECh. 10 - Prob. 17PECh. 10 - Prob. 18PECh. 10 - Prob. 19PECh. 10 - Prob. 20PECh. 10 - Prob. 21PECh. 10 - Prob. 22PECh. 10 - Prob. 23PECh. 10 - Prob. 24PECh. 10 - Prob. 25PECh. 10 - Prob. 26PECh. 10 - Prob. 27PECh. 10 - Prob. 28PECh. 10 - Prob. 29PECh. 10 - Prob. 30PECh. 10 - Prob. 1SPCh. 10 - Prob. 2SPCh. 10 - Prob. 3SPCh. 10 - Prob. 4SPCh. 10 - Prob. 5SPCh. 10 - Prob. 6SPCh. 10 - Prob. 7SPCh. 10 - Prob. 8SPCh. 10 - Prob. 9SPCh. 10 - Prob. 10SPCh. 10 - Prob. 11SPCh. 10 - Prob. 12SPCh. 10 - Prob. 13SPCh. 10 - Prob. 14SPCh. 10 - Prob. 15SPCh. 10 - Prob. 16SPCh. 10 - Prob. 17SPCh. 10 - Prob. 18SPCh. 10 - Prob. 19SPCh. 10 - Prob. 20SPCh. 10 - Prob. 21SPCh. 10 - Prob. 22SPCh. 10 - Prob. 23SPCh. 10 - Prob. 24SPCh. 10 - Prob. 25SPCh. 10 - Prob. 26SPCh. 10 - 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- A curve C in the plane is defined by the parametric equations: x=t² i. Find the length of C from t=0 to 1-2 ii. Find the curvature of C at t = 1 b. The vector function r(t)=sin 2ti-cos2tj+t√√5k determines a curve C in space. i. Find the unit tangent vector T and the principal unit normal N ii. Determine the curvature of C at time t x=f² +4y=2²=1 iii. Determine the tangential and normal component of the acceleration vector. 5. Let f(x,y)=xln(x/y) + xy² a. Calculate f, and f b. Determine the directional derivative off at the point (2, 2) in the direction of the vector a=i-2j af Ət c. Suppose that x = ste' and y=2se. Calculate d. Determine an equation for the tangent plane to the surface == f(x, y) at the point (2, 2, 8) on the surface. 6. Let F(x, y, z)=2xy² +2y=²+2x²z. a. Determine the maximum directional derivative of F at the point (1, -1, 1). b. Find the directional derivative off at the point (-2, 1, -1) in the direction parallel to the line x=34t, y=2-t, z=3t. c. Determine…arrow_forward#3. Use the cross product formula to calculate the curvature of the following vector function. r(t) = (2t,-t², t³).arrow_forwardConsider the curve given by 7(t) = (etcost, et sint, 5et) Find the value of t that corresponds to the position vector with terminal point (1,0,5). a. b. Find the unit tangent vector 7, principle unit normal vector №, and curvature x at the time found in part a.arrow_forward
- r(t) = Find the curvature when t = 0.5. Enter your answer in decimal form below, rounded off to two decimal places. QUESTION 6 r(t) = (21, 12) Find the unit tangent vector, T(t), when t = 0.25. Enter the y coordinate of the vector in space below as a decimal, rounded off to two decimal places.arrow_forwardAnswer only one of the following: 1. Write the tangential and normal components of acceleration for r(t) = (tsint)i + (tcost)j + tk, at t-0 2. Calculate the curvature, unit binormal vector and torsion for r(t) = (e cost)i + (e'sint)j +5karrow_forwardShow all/each step of the problem.arrow_forward
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