5.3E: Exercises

Exercise \(\PageIndex{1}\)

Find the arc length of the curve on the given interval.

a) \(\mathrm{r(t)=t^2 \mathbf{i}+14t \mathbf{j},0≤t≤7}\). This portion of the graph is shown here:

b) \(\mathrm{r(t)=t^2 \mathbf{i}+(2t^2+1)\mathbf{j},1≤t≤3}\)

c) \(\mathrm{r(t)=⟨2 \sin t,5t,2 \cos t⟩,0≤t≤π}\). This portion of the graph is shown here:

d) \(\mathrm{r(t)=⟨t^2+1,4t^3+3⟩,−1≤t≤0}\)

e) \(\mathrm{r(t)=⟨e^{−t \cos t},e^{−t \sin t}⟩}\) over the interval \(\mathrm{[0,\frac{π}{2}]}\). Here is the portion of the graph on the indicated interval:

e) Find the length of one turn of the helix given by \(\mathrm{r(t)= \frac{1}{2} \cos t \mathbf{i}+\frac{1}{2} \sin t \mathbf{j}+\sqrt{\frac{3}{4}}t \mathbf{k}.}\)

f) Find the arc length of the vector-valued function \(\mathrm{r(t)=−t \mathbf{i}+4t \mathbf{j}+3t \mathbf{k}}\) over \(\mathrm{[0,1]}\).

g) A particle travels in a circle with the equation of motion \(\mathrm{r(t)=3 \cos t \mathbf{i}+3 \sin t \mathbf{j} +0 \mathbf{k}}\). Find the distance traveled around the circle by the particle.

h) Set up an integral to find the circumference of the ellipse with the equation \(\mathrm{r(t)= \cos t \mathbf{i}+2 \sin t \mathbf{j}+0\mathbf{k}.}\)

i) Find the length of the curve \(\mathrm{r(t)=⟨\sqrt{2}t,e^t,e^{−t}⟩}\) over the interval \(\mathrm{0≤t≤1}\). The graph is shown here:

j) Find the length of the curve \(\mathrm{r(t)=⟨2 \sin t,5t,2 \cos t⟩}\) for \(\mathrm{t∈[−10,10]}\).

Answer

a) \(\mathrm{8\sqrt{5}}\)

d) \(\mathrm{\frac{1}{54}(37^{3/2}−1)}\)

e) Length \(\mathrm{=2π}\)

g) \(\mathrm{6π}\)

i) \(\mathrm{e−\frac{1}{e}}\)

Exercise \(\PageIndex{2}\)

a) The position function for a particle is \(\mathrm{r(t)=a \cos( ωt) \mathbf{i}+b \sin (ωt) \mathbf{j}}\). Find the unit tangent vector and the unit normal vector at \(\mathrm{t=0.}\)

b) Given \(\mathrm{r(t)=a \cos (ωt) \mathbf{i} +b \sin (ωt) \mathbf{j}}\), find the binormal vector \(\mathrm{B(0)}\).

c) Given \(\mathrm{r(t)=⟨2e^t,e^t \cos t,e^t \sin t⟩}\), determine the tangent vector \(\mathrm{T(t)}\).

d) Given \(\mathrm{r(t)=⟨2e^t,e^t \cos t,e^t \sin t⟩}\), determine the unit tangent vector \(\mathrm{T(t)}\) evaluated at \(\mathrm{t=0}\).

e) Given \(\mathrm{r(t)=⟨2e^t,e^t \cos t,e^t \sin t⟩}\), find the unit normal vector \(\mathrm{N(t)}\) evaluated at \(\mathrm{t=0}\), \(\mathrm{N(0)}\).

f) Given \(\mathrm{r(t)=⟨2e^t,e^t \cos t,e^t \sin t⟩}\), find the unit normal vector evaluated at \(\mathrm{t=0}\).

g) Given \(\mathrm{r(t)=t \mathbf{i}+t^2 \mathbf{j}+t\mathbf{k}}\), find the unit tangent vector \(\mathrm{T(t)}\). The graph is shown here:

h) Find the unit tangent vector \(\mathrm{T(t)}\) and unit normal vector \(\mathrm{N(t)}\) at \(\mathrm{t=0}\) for the plane curve \(\mathrm{r(t)=⟨t^3−4t,5t^2−2⟩}\). The graph is shown here:

i) Find the unit tangent vector \(\mathrm{T(t)}\) for \(\mathrm{r(t)=3t \mathbf{i}+5t^2 \mathbf{j}+2t\mathbf{k}}\)

j) Find the principal normal vector to the curve \(\mathrm{r(t)=⟨6 \cos t,6 \sin t⟩}\) at the point determined by \(\mathrm{t=π/3}\).

k) Find \(\mathrm{T(t)}\) for the curve \(\mathrm{r(t)=(t^3−4t) \mathbf{i}+(5t^2−2)\mathbf{j}}\).

l) Find \(\mathrm{N(t)}\) for the curve \(\mathrm{r(t)=(t^3−4t)\mathbf{i}+(5t^2−2)\mathbf{j}}\).

m) Find the unit normal vector \(\mathrm{N(t)}\) for \(\mathrm{r(t)=⟨2sint,5t,2cost⟩}\).

n) Find the unit tangent vector \(\mathrm{T(t)}\) for \(\mathrm{r(t)=⟨2 \sin t,5t,2 \cos t⟩}\).

Answer

a) \(\mathrm{T(0)= \mathbf{j}, N(0)=−\mathbf{i}}\)

c) \(\mathrm{T(t)=⟨2e^t,e^t \cos t−e^t \sin t,e^t \cos t+e^t \sin t⟩}\)

e) \(\mathrm{N(0)=⟨\frac{\sqrt{2}}{2},0,\frac{\sqrt{2}}{2}⟩}\)

g) \(\mathrm{T(t)=\frac{1}{\sqrt{4t^2+2}}<1,2t,1>}\)

i) \(\mathrm{T(t)=\frac{1}{\sqrt{100t^2+13}}(3\mathbf{i}+10t\mathbf{j}+2\mathbf{k})}\)

k) \(\mathrm{T(t)=\frac{1}{\sqrt{9t^4+76t^2+16}}([3t^2−4]\mathbf{i}+10t\mathbf{j})}\)

m) \(\mathrm{N(t)=⟨−sint,0,−cost⟩}\)

Exercise \(\PageIndex{3}\)

a) Find the arc-length function \(\mathrm{s(t)}\) for the line segment given by \(\mathrm{r(t)=⟨3−3t,4t⟩}\). Write r as a parameter of s.

b) Parameterize the helix \(\mathrm{r(t)= \cos t \mathbf{i}+ \sin t \mathbf{j}+t \mathbf{k}}\) using the arc-length parameter s, from \(\mathrm{t=0}\).

c) Parameterize the curve using the arc-length parameter s, at the point at which \(\mathrm{t=0}\) for \(\mathrm{r(t)=e^t \sin t \mathbf{i} + e^t \cos t \mathbf{j}}\)

d) Find the curvature of the curve \(\mathrm{r(t)=5 \cos t \mathbf{i}+4 \sin t \mathbf{j}}\) at \(\mathrm{t=π/3}\). (Note: The graph is an ellipse.)

e) Find the x-coordinate at which the curvature of the curve \(\mathrm{y=1/x}\) is a maximum value.

f) Find the curvature of the curve \(\mathrm{r(t)=5 \cos t \mathbf{i}+5 \sin t \mathbf{j}}\). Does the curvature depend upon the parameter t?

h) Find the curvature \(κ\) for the curve \(\mathrm{y=x−\frac{1}{4}x^2}\) at the point \(\mathrm{x=2}\).

i) Find the curvature \(κ\) for the curve \(\mathrm{y=\frac{1}{3}x^3}\) at the point \(\mathrm{x=1}\).

j) Find the curvature κκ of the curve \(\mathrm{r(t)=t \mathbf{i}+6t^2 \mathbf{j}+4t \mathbf{k}}\). The graph is shown here:

k) Find the curvature of \(\mathrm{r(t)=⟨2 \sin t,5t,2 \cos t⟩}\).

l) Find the curvature of \(\mathrm{r(t)=\sqrt{2}t \mathbf{i}+e^t \mathbf{j}+e^{−t} \mathbf{k}}\) at point \(\mathrm{P(0,1,1)}\).

m) At what point does the curve \(\mathrm{y=e^x}\) have maximum curvature?

n) What happens to the curvature as \(\mathrm{x→∞}\) for the curve \(\mathrm{y=e^x}\)?

o) Find the point of maximum curvature on the curve \(\mathrm{y=\ln x}\).

p) Find the equations of the normal plane and the osculating plane of the curve \(\mathrm{r(t)=⟨2 \sin (3t),t,2 \cos (3t)⟩}\) at point \(\mathrm{(0,π,−2)}\).

q) Find equations of the osculating circles of the ellipse \(\mathrm{4y^2+9x^2=36}\) at the points \(\mathrm{(2,0)}\) and \(\mathrm{(0,3)}\).

r) Find the equation for the osculating plane at point \(\mathrm{t=π/4}\) on the curve \(\mathrm{r(t)=\cos (2t) \mathbf{i}+ \sin (2t) \mathbf{j}+t}\).

Answer

a) Arc-length function: \(\mathrm{s(t)=5t}\); r as a parameter of s: \(\mathrm{r(s)=(3−\frac{3s}{5})\mathbf{i}+\frac{4s}{5}\mathbf{j}}\)

c) \(\mathrm{(s)=(1+\frac{s}{\sqrt{2}}) \sin ( \ln (1+ \frac{s}{\sqrt{2}})) \mathbf{i} +(1+ \frac{s}{\sqrt{2}}) \cos [ \ln (1+\frac{s}{\sqrt{2}})]\mathbf{j}}\)

e) The maximum value of the curvature occurs at \(\mathrm{x=\sqrt[4]{5}}\).

h) \(\mathrm{\frac{1}{2}}\)

j) \(\mathrm{κ≈\frac{49.477}{(17+144t^2)^{3/2}}}\)

l) \(\mathrm{\frac{1}{2\sqrt{2}}}\)

n) The curvature approaches zero.

p) \(\mathrm{y=6x+π}\) and \(\mathrm{x+6=6π}\)

r) \(\mathrm{x+2z=\frac{π}{2}}\)

Exercise \(\PageIndex{4}\)

a) Find the radius of curvature of \(\mathrm{6y=x^3}\) at the point \(\mathrm{(2,\frac{4}{3}).}\)

b) Find the curvature at each point \(\mathrm{(x,y)}\) on the hyperbola \(\mathrm{r(t)=⟨a \cosh( t),b \sinh (t)⟩}\).

c) Calculate the curvature of the circular helix \(\mathrm{r(t)=r \sin (t) \mathbf{i}+r \cos (t) \mathbf{j}+t \mathbf{k}}\).

d) Find the radius of curvature of \(\mathrm{y= \ln (x+1)}\) at point \(\mathrm{(2,\ln 3)}\).

e) Calculate the curvature of the circular helix \(\mathrm{r(t)=r \sin (t) \mathbf{i}+r \cos (t) \mathbf{j}+t \mathbf{k}}\).

f) Find the radius of curvature of \(\mathrm{y= \ln (x+1)}\) at point \(\mathrm{(2,\ln 3)}\).

g) Find the curvature of the plane curve at \(\mathrm{t=0,1,2}\).

h) Describe the curvature as t increases from \(\mathrm{t=0}\) to \(\mathrm{t=2}\).

Answer

a) \(\mathrm{\frac{a^4b^4}{(b^4x^2+a^4y^2)^{3/2}}}\)

d) \(\mathrm{\frac{10\sqrt{10}}{3}}\)

f) \(\mathrm{\frac{38}{3}}\)

h) The curvature is decreasing over this interval.

Exercise \(\PageIndex{5}\)

The surface of a large cup is formed by revolving the graph of the function \(\mathrm{y=0.25x^{1.6}}\) from \(\mathrm{x=0}\) to \(\mathrm{x=5}\) about the y-axis (measured in centimeters).

a) [T] Use technology to graph the surface.

b) Find the curvature \(κ\) of the generating curve as a function of x.

c) [T] Use technology to graph the curvature function.

Answer

b) \(\mathrm{κ=\frac{6}{x^{2/5}(25+4x^{6/5})}}\)