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# 12.4E: Exercises for Section 12.4

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## Determining Arc Length

In questions 1 - 6, find the arc length of the curve on the given interval.

1) $$\vecs r(t)=t^2 \,\hat{\mathbf{i}}+(2t^2+1)\,\hat{\mathbf{j}}, \quad 1≤t≤3$$

$$8\sqrt{5}$$ units

2) $$\vecs r(t)=t^2 \,\hat{\mathbf{i}}+14t \,\hat{\mathbf{j}},\quad 0≤t≤7$$. This portion of the graph is shown here: 3) $$\vecs r(t)=⟨t^2+1,4t^3+3⟩, \quad −1≤t≤0$$

$$\frac{1}{54}(37^{3/2}−1)$$ units

4) $$\vecs r(t)=⟨2 \sin t,5t,2 \cos t⟩,\quad 0≤t≤π$$. This portion of the graph is shown here: 5) $$\vecs r(t)=⟨e^{−t \cos t},e^{−t \sin t}⟩$$ over the interval $$[0,\frac{π}{2}]$$. Here is the portion of the graph on the indicated interval: 6)

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

Length $$=2π$$ units

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

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

$$6π$$ units

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

11) Find the length of the curve $$\vecs r(t)=⟨\sqrt{2}t,e^t,e^{−t}⟩$$ over the interval $$0≤t≤1$$. The graph is shown here: $$\left(e−\frac{1}{e}\right)$$ units

12) Find the length of the curve $$\vecs r(t)=⟨2 \sin t,5t,2 \cos t⟩$$ for $$t∈[−10,10]$$.

## Unit Tangent Vectors and Unit Normal Vectors

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

$$\vecs T(0)= \hat{\mathbf{j}}, \quad \vecs N(0)=−\hat{\mathbf{i}}$$

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

15) Given $$\vecs r(t)=⟨2e^t,e^t \cos t,e^t \sin t⟩$$, determine the tangent vector $$\vecs T(t)$$.

$$\vecs T(t)=⟨2e^t,e^t \cos t−e^t \sin t,e^t \cos t+e^t \sin t⟩$$

16) Given $$\vecs r(t)=⟨2e^t,e^t \cos t,e^t \sin t⟩$$, determine the unit tangent vector $$\vecs T(t)$$ evaluated at $$t=0$$.

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

$$\vecs N(0)=⟨\frac{\sqrt{2}}{2},0,\frac{\sqrt{2}}{2}⟩$$

18) Given $$\vecs r(t)=⟨2e^t,e^t \cos t,e^t \sin t⟩$$, find the unit normal vector evaluated at $$t=0$$.

19) Given $$\vecs r(t)=t \,\hat{\mathbf{i}}+t^2 \,\hat{\mathbf{j}}+t \,\hat{\mathbf{k}}$$, find the unit tangent vector $$\vecs T(t)$$. The graph is shown here: $$\vecs T(t)=\frac{1}{\sqrt{4t^2+2}}<1,2t,1>$$

20) Find the unit tangent vector $$\vecs T(t)$$ and unit normal vector $$\vecs N(t)$$ at $$t=0$$ for the plane curve $$\vecs r(t)=⟨t^3−4t,5t^2−2⟩$$. The graph is shown here: 21) Find the unit tangent vector $$\vecs T(t)$$ for $$\vecs r(t)=3t \,\hat{\mathbf{i}}+5t^2 \,\hat{\mathbf{j}}+2t \,\hat{\mathbf{k}}$$.

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

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

23) Find $$\vecs T(t)$$ for the curve $$\vecs r(t)=(t^3−4t) \,\hat{\mathbf{i}}+(5t^2−2) \,\hat{\mathbf{j}}$$.

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

24) Find $$\vecs N(t)$$ for the curve $$\vecs r(t)=(t^3−4t)\,\hat{\mathbf{i}}+(5t^2−2)\,\hat{\mathbf{j}}$$.

25) Find the unit normal vector $$\vecs N(t)$$ for $$\vecs r(t)=⟨2sint,5t,2cost⟩$$.

$$\vecs N(t)=⟨−\sin t,0,−\cos t⟩$$

26) Find the unit tangent vector $$\vecs T(t)$$ for $$\vecs r(t)=⟨2 \sin t,5t,2 \cos t⟩$$.

## Arc Length Parameterizations

27) Find the arc-length function $$\vecs s(t)$$ for the line segment given by $$\vecs r(t)=⟨3−3t,4t⟩$$. Then write the arc-length parameterization of $$r$$ with $$s$$ as the parameter.

Arc-length function: $$s(t)=5t$$; The arc-length parameterization of $$\vecs r(t)$$: $$\vecs r(s)=(3−\frac{3s}{5})\mathbf{i}+\frac{4s}{5}\mathbf{j}$$

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

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

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

## Curvature and the Osculating Circle

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

The maximum value of the curvature occurs at $$\mathrm{x=\sqrt{5}}$$.

32) 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?

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

$$\mathrm{\frac{1}{2}}$$

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

35) Find the curvature $$κ$$ of the curve $$\mathrm{r(t)=t \mathbf{i}+6t^2 \mathbf{j}+4t \mathbf{k}}$$. The graph is shown here: $$\mathrm{κ≈\frac{49.477}{(17+144t^2)^{3/2}}}$$

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

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

$$\mathrm{\frac{1}{2\sqrt{2}}}$$

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

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

The curvature approaches zero.

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

41) 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 $$\mathrm{(0,π,−2)}$$.

$$\mathrm{y=6x+π}$$ and $$\mathrm{x+6=6π}$$

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

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

$$\mathrm{x+2z=\frac{π}{2}}$$

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

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

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

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

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

$$\mathrm{\frac{10\sqrt{10}}{3}}$$

48) Find the radius of curvature of the hyperbola $$\mathrm{xy=1}$$ at point $$\mathrm{(1,1)}$$.

A particle moves along the plane curve $$C$$ described by $$\vecs r(t)=t \,\hat{\mathbf{i}}+t^2 \,\hat{\mathbf{j}}$$. Use this parameterization to answer questions 49 - 51.

49) Find the length of the curve over the interval $$\mathrm{[0,2]}$$.

$$\mathrm{\frac{38}{3}}$$

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

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

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).
53) Find the curvature $$κ$$ of the generating curve as a function of $$x$$.
$$\mathrm{κ=\frac{6}{x^{2/5}(25+4x^{6/5})}}$$