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5.3E: Exercises

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    26284
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    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:

    This figure is the graph of a curve beginning at the origin and increasing.

    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:

    This figure is the graph of a curve in 3 dimensions. It is inside of a box. The box represents an octant. The curve begins in the upper right corner of the box and bends through the box to the other side.

    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:

    This figure is the graph of a curve in the first quadrant. It begins approximately at 0.20 on the y axis and increases to approximately where x = 0.3. Then the curve decreases, meeting the x-axis at 1.0.

    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:

    This figure is the graph of a curve in 3 dimensions. It is inside of a box. The box represents an octant. The curve begins in the upper left corner of the box and bends through the box to the bottom of the other side.

    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:

    This figure is the graph of a curve in 3 dimensions. It is inside of a box. The box represents an octant. The curve begins in the bottom left corner of the box and bends through the box to the upper left side.

    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:

    This figure is the graph of a curve above the x-axis. The curve decreases in the second quadrant, passes through the y-axis at y=20. Then it intersects the origin. The curve loops at the origin, increasing back through y=20 into the first quadrant.

    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.)

    This figure is the graph of an ellipse. The ellipse is oval along the x-axis. It is centered at the origin and intersects the y-axis at -4 and 4.

    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:

    This figure is the graph of a curve in 3 dimensions. It is inside of a box. The box represents an octant. The curve has a parabolic shape in the middle of the box.

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


    This page titled 5.3E: Exercises is shared under a not declared license and was authored, remixed, and/or curated by Pamini Thangarajah.

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