Answer: Rounding the answers to four decimal places, use a

In Exercises 1 and 2, write in the form \(a=a_{T} T+a_{N} N\) without finding \(T\) and \(N\). \(r(t)=(a \cos t) i+(a \sin t) j+b t k\) Equation Transcription: Text Transcription: a=a_T T + a_N N T N r(t)=(a cos t)i + (a sin t)j + bt k

Problem 2E In Exercises 1 and 2, write a in the form without finding T and N.

In Exercises , write a in the form \(a=a_{T} T+a_{N} N\) at the given value of t without finding T and N. \(r(t)=(t+1) i+2 j+t^{2} k, t=1\) Equation Transcription: Text Transcription: a=a_T T + a_N N r(t)=(t+1)i +2j +t^2 k, t=1

Problem 4E In Exercises 3–6, write a in the form at the given value of t without finding T and N.

In Exercises , write a in the form \(a=a_{T} T+a_{N} N\) at the given value of t without finding T and N. \(r(t)=t^{2} i+\left(t+(1 / 3) t^{3}\right) j+\left(1-(1 / 3) t^{3}\right) k, t=0\) Equation Transcription: Text Transcription: a=a_T T + a_N N r(t)=t^2 i+ (t+1(1/3)t^3 )j+ (1-(1/3 )t^3 )k, t=0

Problem 6E In Exercises 3–6, write a in the form at the given value of t without finding T and N.

Finding the TNB Frame In Exercises 7 and 8, find \(r, T, N\), and \(B\) at the given value of \(t\). Then find equations for the osculating, normal, and rectifying planes at that value of \(t\). \(r(t)=(\cos t) i+(\sin t) j-k, t=\pi / 4\) Equation Transcription: Text Transcription: r, T, N B t t r(t)=(cos t)i + (sin t)j - k, t=pi/4

Finding the TNB Frame In Exercises 7 and 8, find \(r, T, N\), and \(B\) at the given value of \(t\). Then find equations for the osculating, normal, and rectifying planes at that value of \(t\). \(r(t)=(\cos t) i+(\sin t) j+t k, t=0\) Equation Transcription: Text Transcription: r, T, N B t t r(t)=(cos t)i + (sin t)j + t k, t=0

Problem 9E In Exercises 9–16 of Section 12.4, you found T, N, and . Now, in the following Exercises 9–16, find B and for these space curves.

Problem 10E In Exercises 9–16 of Section 12.4, you found T, N, and . Now, in the following Exercises 9–16, find B and for these space curves.

In Exercises of Section , you found \(T, N\), and \(\kappa\). Now, in the following Exercises , find \(B\) and \(\tau\) for these space curves. \(r(t)=\left(e^{t} \cos t\right) i+\left(e^{t} \sin t\right) j+2 k\) Equation Transcription: Text Transcription: T, N kappa B tau r(t)=(e^t cos t)i + (e^t sin t)j +2 k

Problem 12E In Exercises 9–16 of Section 12.4, you found T, N, and . Now, in the following Exercises 9–16, find B and for these space curves.

In Exercises of Section , you found \(T, N\), and \(\kappa\). Now, in the following Exercises , find \(B\) and \(\tau\) for these space curves. \(r(t)=\left(t^{3} / 3\right) i+\left(t^{2} / 2\right) j, t>0\) Equation Transcription: Text Transcription: T, N kappa B tau r(t)=(t^3 /3)i +(t^2 /2)j, t>0

Problem 14E In Exercises 9–16 of Section 12.4, you found T, N, and . Now, in the following Exercises 9–16, find B and for these space curves.

In Exercises of Section , you found \(T, N\), and \(\kappa\). Now, in the following Exercises , find \(B\) and \(\tau\) for these space curves. \(r(t)=t i+(a \cosh (t / a)) j, \quad a>0\) Equation Transcription: Text Transcription: T, N kappa B tau r(t)=t i + (a cosh (t/a))j, a>0

In Exercises of Section , you found \(T, N\), and \(\kappa\). Now, in the following Exercises , find \(B\) and \(\tau\) for these space curves. \(r(t)=(\cosh t) i-(\sinh t) j+t k\) Equation Transcription: Text Transcription: T, N kappa B tau r(t)=cosh t)i - (sinh t)j + t k

Problem 17E The speedometer on your car reads a steady 35 mph. Could you be accelerating? Explain.

Problem 18E Can anything be said about the acceleration of a particle that is moving at a constant speed? Give reasons for your answer.

Problem 19E Can anything be said about the speed of a particle whose acceleration is always orthogonal to its velocity? Give reasons for your answer.

Problem 20E An object of mass m travels along the parabola y = x2 with a constant speed of 10 units sec. What is the force on the object due to its acceleration at (0, 0)? At (21/2, 2)? Write your answers in terms of i and j. (Remember Newton’s law, F = ma.)

Vector formula for curvature For a smooth curve, use Equation (1) to derive the curvature formula \(\kappa=\frac{|v \times a|}{|v|^{3}}\) Equation Transcription: Text Transcription: kappa = | v times a | / | v |^3

Problem 22E Show that a moving particle will move in a straight line if the normal component of its acceleration is zero.

Problem 23E A sometime shortcut to curvature If you already know |aN| and |v|, then the formula aN = k |v|2 gives a convenient way to find the curvature. Use it to find the curvature and radius of curvature of the curve (Take aN and |v| from Example 1.)

Show that \(\kappa\) and \(\tau\) are both zero for the line \(r(t)=\left(x_{0}+A t\right) i+\left(y_{0}+B t\right) j+\left(z_{0}+C t\right) k\) Equation Transcription: Text Transcription: kappa tau r(t)=(x_0 + At)i +(y_0 +Bt)j + (z_0 +Ct)k

Problem 25E What can be said about the torsion of a smooth plane curve ? Give reasons for your answer.

The torsion of a helix Show that the torsion of the helix \(r(t)=(a \cos t) i\ +\ (a \sin t) j\ +\ b t k, \quad a, b \geq 0\) is \(\tau=b /\left(a^{2}\ +\ b^{2}\right)\). What is the largest value \(\tau\) can have for a given value of \({a}\)? Give reasons for your answer. Equation Transcription: Text Transcription: r(t) = (a cos t)i + (a sin t)j + btk, a, b geq 0 tau = b/(a^2 + b^2) tau a

Problem 27E Differentiable curves with zero torsion lie in planes That a sufficiently differentiable curve with zero torsion lies in a plane is a special case of the fact that a particle whose velocity remains perpendicular to a fixed vector C moves in a plane perpendicular to C. This, in turn, can be viewed as the following result. Suppose is twice differentiable for all t in an interval [a, b], that r = 0 when t = a, and that for all t in [a, b]. Show that h(t) = 0 for all t in [a, b]. (Hint: Start with a = d2r/dt2 and apply the initial conditions in reverse order.).

A formula that calculates \(\tau\) from B and v If we start with the definition \(\tau=-(d B / d s) \cdot N\) and apply the Chain Rule to rewrite \(d B / d s\) as \(\frac{d B}{d s}=\frac{d B}{d t} \frac{d t}{d s}=\frac{d B}{d t} \frac{1}{|v|}\) we arrive at the formula \(\tau=-\frac{1}{|v|}\left(\frac{d B}{d t} \cdot N\right)\) The advantage of this formula over Equation (5) is that it is easier to derive and state. The disadvantage is that it can take a lot of work to evaluate without a computer. Use the new formula to find the torsion of the helix in Exercise 26 . Equation Transcription: Text Transcription: tau tau=-(dB/ds) . N dB/ds dB/ds = dB/dt dt/ds = dB/dt 1/|v| tau=-1/|v| (dB/dt . N)

Problem 29CE Rounding the answers to four decimal places, use a CAS to find v, a, speed, T, N, B, and the tangential and normal components of acceleration for the curves in Exercises 29–32 at the given values of t.

Rounding the answers to four decimal places, use a CAS to find \({ v, a }\), speed, \({ T, } \ N, \ B, \ \kappa, \ \tau\), and the tangential and normal components of acceleration for the curves in Exercises 29-32 at the given values of \({t}\). \(r(t)=\left(e^{t} \cos t\right) i\ +\ \left(e^{t} \sin t\right) j\ +\ e^{t} k, \quad t=\ln 2\) Equation Transcription: Text Transcription: v, a T, N, B, kappa, tau t r(t) = (e^t cos t)i + (e^t sin t)j + e^t k, t = ln 2

Rounding the answers to four decimal places, use a CAS to find \({ v, a }\), speed, \({ T, } \ N, \ B, \ \kappa, \ \tau\), and the tangential and normal components of acceleration for the curves in Exercises 29-32 at the given values of \({t}\). \(r(t)=(t-\sin t) i\ +\ (1-\cos t) j\ +\ \sqrt{-t} \ k, \quad t=-3 \pi\) Equation Transcription: Text Transcription: v, a T, N, B, kappa, tau t r(t) = (t - sin t)i + (1 - cos t)j + sqrt -t k, t = -3pi

Problem 32CE Rounding the answers to four decimal places, use a CAS to find v, a, speed, T, N, B, and the tangential and normal components of acceleration for the curves in Exercises 29–32 at the given values of t.