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Problem 79

Zero curvature Prove that the curve r(t)=a+btp,c+dtp,e+ftp where a,b,c,d,e, and f are real numbers and p is a positive integer, has zero curvature. Give an explanation.

Problem 79

Determine whether the following statements are true and give an explanation or counterexample. a. The vectors r(t) and r(t) are parallel for all values of t in the domain. b. The curve described by the function r(t)=t,t22t,cosπt is smooth, for \(-\infty

Problem 79

Tilted ellipse Consider the curve r(t)=cost,sint,csint, for 0t2π, where c is a real number. Assuming the curve lies in a plane, prove that the curve is an ellipse in that plane.

Problem 80

Equal area property Consider the ellipse r(t)=acost,bsint for 0t2π. where a and b are real numbers. Let θ be the angle between the position vector and the x -axis. a. Show that tanθ=batant. b. Find θ(t). c. Recall that the area bounded by the polar curve r=f(θ) on the interval [0,θ] is A(θ)=120θ(f(u))2du. Letting f(θ(t))=|r(θ(t))|, show that A(t)=12ab. d. Conclude that as an object moves around the ellipse, it sweeps out equal areas in equal times.

Problem 80

Suppose the vector-valued function r(t)=f(t),g(t),h(t) is smooth on an interval containing the point t0 The line tangent to r(t) at t=t0 is the line parallel to the tangent vector r(t0) that passes through (f(t0),g(t0),h(t0)). For each of the following functions, find an equation of the line tangent to the curve at t=t0. Choose an orientation for the line that is the same as the direction of r. r(t)=et,e2t,e3t;t0=0

Problem 80

Practical formula for N Show that the definition of the principal unit normal vector N=dT/ds|dT/ds| implies the practical formula N=dT/dt|dT/dt|. Use the Chain Rule and recall that |v|=dsdt>0.

Problem 81

Suppose the vector-valued function r(t)=f(t),g(t),h(t) is smooth on an interval containing the point t0 The line tangent to r(t) at t=t0 is the line parallel to the tangent vector r(t0) that passes through (f(t0),g(t0),h(t0)). For each of the following functions, find an equation of the line tangent to the curve at t=t0. Choose an orientation for the line that is the same as the direction of r. r(t)=2+cost,3+sin2t,t;t0=π2

Problem 81

Another property of constant |r| motion Suppose an object moves on the surface of a sphere with |r(t)| constant for all t Show that r(t) and a(t)=r(t) satisfy r(t)a(t)=|v(t)|2.

Problem 81

Consider the "superparabolas" fn(x)=x2n where n is a positive integer. a. Find the curvature function of fn for n=1,2, and 3 b. Plot fn and their curvature functions, for n=1,2, and 3, and check for consistency. c. At what points does the maximum curvature occur, for n=1,2, and 3? d. Let the maximum curvature for fn occur at x=±zm Using either analytical methods or a calculator, determine lim zn Interpret your result.

Problem 82

Suppose the vector-valued function r(t)=f(t),g(t),h(t) is smooth on an interval containing the point t0 The line tangent to r(t) at t=t0 is the line parallel to the tangent vector r(t0) that passes through (f(t0),g(t0),h(t0)). For each of the following functions, find an equation of the line tangent to the curve at t=t0. Choose an orientation for the line that is the same as the direction of r. r(t)=2t+1,sinπt,4;t0=4

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