?The helix \(\mathbf{r}_{1}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+t \mathbf{k}\)

What is a vector function? How do you find its derivative and its integral?

What is the connection between vector functions and space curves?

How do you find the tangent vector to a smooth curve at a point? How do you find the tangent line? The unit tangent vector?

If \(\mathbf{u}\) and \(\mathbf{v}\) are differentiable vector functions, c is a scalar, and f is a real-valued function, write the rules for differentiating the following vector functions. (a) \(\mathbf{u}(t)+\mathbf{v}(t)\) (b) \(\operatorname{cu}(t)\) (c) \(f(t) \mathbf{u}(t)\) (d) \(\mathbf{u}(t) \cdot \mathbf{v}(t)\) (e) \(\mathbf{u}(t) \times \mathbf{v}(t)\) (f) \(\mathbf{u}(f(t))\)

How do you find the length of a space curve given by a vector function r(t)?

(a) What is the definition of curvature? (b) Write a formula for curvature in terms of r’(t) and T’(t). (c) Write a formula for curvature in terms of r’(t) and r’’(t). (d) Write a formula for the curvature of a plane curve with equation y = f(x).

(a) Write formulas for the unit normal and binormal vectors of a smooth space curve r(t). (b) What is the normal plane of a curve at a point? What is the osculating plane? What is the osculating circle?

(a) How do you find the velocity, speed, and acceleration of a particle that moves along a space curve? (b) Write the acceleration in terms of its tangential and normal components.

State Kepler’s Laws.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The curve with vector equation r(t) = t3 i + 2t3 j + 3t2 k is a line.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The curve r(t) = (0, t2, 4t) is a parabola.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The curve r(t) = (2t, 3 - t, 0) is a line that passes through the origin.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The derivative of a vector function is obtained by differentiating each component function.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(\mathbf{u}(t)\) and \(\mathbf{v}(t)\) are differentiable vector functions, then \(\frac{d}{d t}[\mathbf{u}(t) \times \mathbf{v}(t)]=\mathbf{u}^{\prime}(t) \times \mathbf{v}^{\prime}(t)\)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If r(t) is a differentiable vector function, then

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If T(t) is the unit tangent vector of a smooth curve, then the curvature is k = | dT/dt |.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The binormal vector is \(\mathrm{B}(\mathrm{t})=\mathrm{N}(\mathrm{t}) \times \mathrm{T}(\mathrm{t})\)

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. Suppose f is twice continuously differentiable. At an inflection point of the curve y = f(x), the curvature is 0.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If k(t) = 0 for all t, the curve is a straight line.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If | r(t) | = 1 for all t, then | r’(t) | is a constant.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If \(|\mathbf{r}(t)|=1\) for all \(t\), then \(\mathbf{r}^{\prime}(t)\) is orthogonal to \(\mathbf{r}(t)\) for all \(t\).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The osculating circle of a curve C at a point has the same tangent vector, normal vector, and curvature as C at that point.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. Different parameterizations of the same curve result in identical tangent vectors at a given point on the curve.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The projection of the curve r(t) = (cos 2t, t, sin 2t) onto the xz-plane is a circle.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The vector equations r(t) = (t, 2t, t + 1) and r(t) = (t - 1, 2t - 2, t) are parametrizations of the same line.

(a) Sketch the curve with vector function \(\mathbf{r}(t)=t \mathbf{i}+\cos \pi t \mathbf{j}+\sin \pi t \mathbf{k} \quad t \geqslant 0\) (b) Find \(\mathbf{r}^{\prime}(t)\) and \(\mathbf{r}^{\prime \prime}(t)\)

Let r(t) = [, (et - 1)/t, ln(t + 1) ]. (a) Find the domain of r. (b) Find . (c) Find r’(t).

Find a vector function that represents the curve of intersection of the cylinder \(x^2 + y^2 = 16\) and the plane x + z = 5.

Find parametric equations for the tangent line to the curve x=2 sin t, y=2 sin 2t, z=2 sin 3t at the point \((1, \sqrt{3}, 2)\). Graph the curve and the tangent line on a common screen.

If \(\mathbf{r}(t)=t^2 \mathbf{i}+t \cos \pi t \mathbf{j}+\sin \pi t \mathbf{k}\), evaluate \(\int_0^1 \mathbf{r}(t) d t\)

Let \(C\) be the curve with equations \(x=2-t^{3}, y=2 t-1\), \(z=\ln t . Find (a) the point where \(C\) intersects the \(x z\)-plane, (b) parametric equations of the tangent line at \((1,1,0)\), and (c) an equation of the normal plane to \(C\) at \((1,1,0)\). Equation Transcription: Text Transcription: x=2-t^3, y = 2t - 1, z =ln t C xz (1, 1, 0)

Use Simpson's Rule with \(n=6\) to estimate the length of the \(\operatorname{arc}\) of the curve with equations \(x=t^{2}, y=t^{3}, z=t^{4}\), \(0 \leqslant t \leqslant 3\). Equation Transcription: ? ? Text Transcription: n=6 x = t^2, y = t^3, z = t^4, 0 less than or equal slant t less than or equal slant 3

Find the length of the curve \(\mathbf{r}(t)=\left\langle 2 t^{3 / 2}, \cos 2 t, \sin 2 t\right\rangle\), \(0 \leqslant t \leqslant 1\). Equation Transcription: ? t ? 1 Text Transcription: r(t) = (2t^3/2, cos 2t, sin 2t), 0 less than or equal slant t less than or equal slant 1

The helix \(\mathbf{r}_{1}(t)=\cos t \mathbf{i}+\sin t \mathbf{j}+t \mathbf{k}\) intersects the curve \(\mathbf{r}_{2}(t)=(1+t) \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k}\) at the point \((1,0,0)\). Find the angle of intersection of these curves. Equation Transcription: Text Transcription: r_1(t) = cos t i + sin t j + t k r_2(t) = (1 + t)i + t^2 j + t^3 (1, 0, 0)

Reparametrize the curve \(\mathbf{r}(t)=e^t \mathbf{i}+e^t \sin t \mathbf{j}+e^t \cos t \mathbf{k}\) with respect to arc length measured from the point (1, 0, 1) in the direction of increasing t.

For the curve given by\(\mathbf{r}(t)=\left\langle\sin ^{3} t, \cos ^{3} t, \sin ^{2} t\right\rangle\), \(0 \leqslant t \leqslant \pi / 2\), find (a) the unit tangent vector. (b) the unit normal vector. (c) the unit binormal vector. (d) the curvature. (e) the torsion. Equation Transcription: ? t ? ????/2 Text Transcription: r(t) = ( sin^3 t, cos^3 t, sin^2 t), 0 less than or equal slant t less than or equal slant pi/2

Find the curvature of the ellipse \(x=3 \cos t, y=4 \sin t\) at the points \((3,0)\) and \((0,4)\). Equation Transcription: Text Transcription: x = 3 cos t, y = 4 sin t (3, 0) (0, 4)

Find the curvature of the curve \(y=x^{4}\) at the point \((1, 1)\). Equation Transcription: Text Transcription: y = x^4 (1, 1)

Find an equation of the osculating circle of the curve \(y=x^{4}-x^{2}\) at the origin. Graph both the curve and its osculating circle. Equation Transcription: Text Transcription: y = x^4-x^2

Find an equation of the osculating plane of the curve \(x=\sin 2 t, y=t, z=\cos 2 t\) at the point \((0, \pi, 1)\). Equation Transcription: Text Transcription: x = sin 2t, y = t, z = cos 2t (0, pi, 1)

The figure shows the curve \(C\) traced by a particle with position vector \9\mathbf{r}(t)\) at time \(t\). (a) Draw a vector that represents the average velocity of the particle over the time interval \(3 \leqslant t \leqslant 3.2\). (b) Write an expression for the velocity \(\mathbf{v}(3)\). (c) Write an expression for the unit tangent vector \(\mathbf{T}(3)\) and draw it. Equation Transcription: 3 ? t ? 3.2 Text Transcription: C r(t) t 3 less than or equal slant t less than or equal slant 3.2 v(3) T(3)

A particle moves with position function \(\mathbf{r}(t)=t \ln t \mathbf{i}+t \mathbf{j}+e^{-t} \mathbf{k}\). Find the velocity, speed, and acceleration of the particle. Equation Transcription: Text Transcription: r(t) = t ln t i + t j + e^-t k

Find the velocity, speed, and acceleration of a particle moving with position function \(\mathbf{r}(t)=\left(2 t^{2}-3\right) \mathbf{i}+2 t \mathbf{j}\). Sketch the path of the particle and draw the position, velocity, and acceleration vectors for \(t=1\). Equation Transcription: Text Transcription: r(t) = (2t^2-3) i + 2t j t=1

A particle starts at the origin with initial velocity \(\mathbf{i}-\mathbf{j}+3 \mathbf{k}\). Its acceleration is \(\mathbf{a}(t)-6 t \mathbf{i}+12 t^{2} \mathbf{j}-6 t \mathbf{k}\). Find its position function. Equation Transcription: Text Transcription: i - j + 3k a(t) = 6t i + 12t^2 j - 6t k

An athlete throws a shot at an angle of \(45^{\circ}\) to the horizontal at an initial speed of 43 ft/s. It leaves the athlete's hand 7 ft above the ground. (a) Where is the shot 2 seconds later? (b) How high does the shot go? (c) Where does the shot land?

A projectile is launched with an initial speed of \(40 \mathrm{~m} / \mathrm{s}\) from the floor of a tunnel whose height is \(30 \mathrm{~m}\). What angle of elevation should be used to achieve the maximum possible horizontal range of the projectile? What is the maximum range? Equation Transcription: Text Transcription: 40 m/s 30 m

Find the tangential and normal components of the acceleration vector of a particle with position function \(\mathbf{r}(t)=t \ \mathbf{i}+2 t \ \mathbf{j}+t^{2} \ \mathbf{k}\)

A disk of radius 1 is rotating in the counterclockwise direction at a constant angular speed \(\omega\). A particle starts at the center of the disk and moves toward the edge along a fixed radius so that its position at time \(t, \ t \geqslant 0\), is given by \(\mathbf{r}(t)=t \ \mathbf{R}(t)\), where \(\mathbf{R}(t)=\cos \omega t \ \mathbf{i}+\sin \omega t \ \mathbf{j}\) (a) Show that the velocity \(\mathbf{v}\) of the particle is \(\mathbf{v}=\cos \omega t \ \mathbf{i}+\sin \omega \ t \mathbf{j}+t \mathbf{v}_{d}\) where \(\mathbf{v}_{d}=\mathbf{R}^{\prime}(t)\) is the velocity of a point on the edge of the disk. (b) Show that the acceleration a of the particle is \(\mathbf{a}=2 \mathbf{v}_{d}+t \mathbf{a}_{d}\) where \(\mathbf{a}_{d}=\mathbf{R}^{\prime \prime}(t)\) is the acceleration of a point on the edge of the disk. The extra term \(2 \mathbf{v}_{d}\) is called the Coriolis acceleration; it is the result of the interaction of the rotation of the disk and the motion of the particle. One can obtain a physical demonstration of this acceleration by walking toward the edge of a moving merry-go-round. (c) Determine the Coriolis acceleration of a particle that moves on a rotating disk according to the equation \(\mathbf{r}(t)=e^{-t} \cos \omega t \ \mathbf{i}+e^{-t} \sin \omega t \ \mathbf{j}\)

In designing transfer curves to connect sections of straight railroad tracks, it's important to realize that the acceleration of the train should be continuous so that the reactive force exerted by the train on the track is also continuous. Because of the formulas for the components of acceleration in Section 13.4, this will be the case if the curvature varies continuously. (a) A logical candidate for a transfer curve to join existing tracks given by \(y=1\) for \(x \leqslant 0\) and \(y=\sqrt{2}-x\) for \(x \geqslant 1 / \sqrt{2}\) might be the function \(f(x)=\sqrt{1-x^{2}}\) \(0<x<1 / \sqrt{2}\), whose graph is the arc of the circle shown in the figure. It looks reasonable at first glance. Show that the function \(F(x)= \begin{cases}1 & \text { if } x \leqslant 0 \\ \sqrt{1-x^{2}} & \text { if } 0<x<1 / \sqrt{2} \\ \sqrt{2}-x & \text { if } x \geqslant 1 / \sqrt{2}\end{cases}\) is continuous and has continuous slope, but does not have continuous curvature. Therefore \(f\) is not an appropriate transfer curve. (b) Find a fifth-degree polynomial to serve as a transfer curve between the following straight line segments: \(y=0\) for \(x \leqslant 0\) and \(y=x\) for \(x \geqslant 1\). Could this be done with a fourth-degree polynomial? Use a graphing calculator or computer to sketch the graph of the "connected" function and check to see that it looks like the one in the figure. Equation Transcription: x ? 0 x ? 1 { Text Transcription: y = 1 x leqslant 0 y = sqrt 2- x x geqslant 1 2 f(x) = sqrt 1 - x^2, 0 < x < 1/ sqrt 2 F(x)= 1 & \text { if } x \leqslant 0 \\ \sqrt{1-x^{2}} & \text { if } 0<x<1 / \sqrt{2} \\ \sqrt{2}-x & \text { if } x \geqslant 1 / \sqrt{2}

A particle \(P\) moves with constant angular speed \(\omega\) around a circle whose center is at the origin and whose radius is \(R\). The particle is said to be in uniform circular motion. Assume that the motion is counterclockwise and that the particle is at the point \((R, 0)\) when \(t=0\) . The position vector at time \(t \geqslant 0\) is \(\mathbf{r}(t)=R \cos \omega t \mathbf{i}+R \sin \omega t \mathbf{j}\). (a) Find the velocity vector \(\mathbf{v}\) and show that \(\mathbf{v} \cdot \mathbf{r}=0\). Conclude that \(\mathbf{v}\) is tangent to the circle and points in the direction of the motion. (b) Show that the speed \(|\mathbf{v}|\) of the particle is the constant \(\omega R\). The period \(T\) of the particle is the time required for one complete revolution. Conclude that \(T=\frac{2 \pi R}{|\mathbf{v}|}=\frac{2 \pi}{\omega}\) (c) Find the acceleration vector a. Show that it is proportional to \(\mathbf{r}\) and that it points toward the origin. An acceleration with this property is called a centripetal acceleration. Show that the magnitude of the acceleration vector is \(|\mathbf{a}|=R \omega^{2}\). (d) Suppose that the particle has mass \(m\). Show that the magnitude of the force \(\mathbf{F}\) that is required to produce this motion, called a centripetal force, is \(|\mathbf{F}|=\frac{m|\mathbf{v}|^{2}}{R}\) Equation Transcription: t ? 0 T = Text Transcription: P R (R, 0) t = 0 t geqslant 0 r(t) = R cos omega t i + R sin omega t j v r = 0 T = 2 pi R/| v |=2 pi/omega |a| = R omega^2 | F | = m | v |^2/R

A circular curve of radius \(R\) on a highway is banked at an angle \(\theta\) so that a car can safely traverse the curve without skidding when there is no friction between the road and the tires. The loss of friction could occur, for example, if the road is covered with a film of water or ice. The rated speed \(v_{R}\) of the curve is the maximum speed that a car can attain without skidding. Suppose a car of mass \(m\) is traversing the curve at the rated speed \(v_{R}\). Two forces are acting on the car: the vertical force, \(m g\), due to the weight of the car, and a force \(\mathbf{F}\) exerted by, and normal to, the road (see the figure). The vertical component of \(\mathbf{F}\) balances the weight of the car, so that \(|\mathbf{F}| \cos \theta=m g\). The horizontal component of \(\mathbf{F}\) produces a centripetal force on the car so that, by Newton's Second Law and part (d) of Problem 1, \(|\mathbf{F}| \sin \theta=\frac{m v_{R}^{2}}{R}\) (a) Show that \(v_{R}^{2}=R g \tan \theta\). (b) Find the rated speed of a circular curve with radius \(400 \mathrm{ft}\) that is banked at an angle of \(12^{\circ}\). (c) Suppose the design engineers want to keep the banking at \(12^{\circ}\), but wish to increase the rated speed by \(50 \%\). What should the radius of the curve be?

A projectile is fired from the origin with angle of elevation \(\alpha\) and initial speed \(v_0\). Assuming that air resistance is negligible and that the only force acting on the projectile is gravity, g, we showed in Example 13.4.5 that the position vector of the projectile is \(\mathbf{r}(t)=\left(v_0 \cos \alpha\right) t \mathbf{i}+\left[\left(v_0 \sin \alpha\right) t-\frac{1}{2} g t^2\right] \mathbf{j}\) We also showed that the maximum horizontal distance of the projectile is achieved when \(\alpha=45^{\circ}\) and in this case the range is \(R=v_0^2 / g\). (a) At what angle should the projectile be fired to achieve maximum height and what is the maximum height? (b) Fix the initial speed \(v_0\) and consider the parabola \(x^2+2 R y-R^2=0\), whose graph is shown in the figure at the left. Show that the projectile can hit any target inside or on the boundary of the region bounded by the parabola and the x-axis, and it can't hit any target outside this region. (c) Suppose that the gun is elevated to an angle of inclination \(\alpha\) in order to aim at a target that is suspended at a height h directly over a point D units downrange (see the following figure). The target is released at the instant the gun is fired. Show that the projectile always hits the target, regardless of the value \(v_0\), provided the projectile does not hit the ground "before" D.

(a) A projectile is fired from the origin down an inclined plane that makes an angle \(\theta\) with the horizontal. The angle of elevation of the gun and the initial speed of the projectile are \(\alpha\) and \(v_{0}\), respectively. Find the position vector of the projectile and the parametric equations of the path of the projectile as functions of the time \(t\). (Ignore air resistance.) (b) Show that the angle of elevation \(\alpha\) that will maximize the downhill range is the angle halfway between the plane and the vertical. (c) Suppose the projectile is fired up an inclined plane whose angle of inclination is \(\theta\). Show that, in order to maximize the (uphill) range, the projectile should be fired in the direction halfway between the plane and the vertical. (d) In a paper presented in 1686, Edmond Halley summarized the laws of gravity and projectile motion and applied them to gunnery. One problem he posed involved firing a projectile to hit a target a distance \(R\) up an inclined plane. Show that the angle at which the projectile should be fired to hit the target but use the least amount of energy is the same as the angle in part (c). (Use the fact that the energy needed to fire the projectile is proportional to the square of the initial speed, so minimizing the energy is equivalent to minimizing the initial speed.) Equation Transcription: Text Transcription: theta v_0 R t

A ball rolls off a table with a speed of \(2 \mathrm{ft} / \mathrm{s}\). The table is \(3.5 \mathrm{ft}\) high. (a) Determine the point at which the ball hits the floor and find its speed at the instant of impact. (b) Find the angle \(\theta\) between the path of the ball and the vertical line drawn through the point of impact (see the figure). (c) Suppose the ball rebounds from the floor at the same angle with which it hits the floor, but loses \(20 \%\) of its speed due to energy absorbed by the ball on impact. Where does the ball strike the floor on the second bounce?

Find the curvature of the curve with parametric equations \(x=\int_{0}^{t} \sin \left(\frac{1}{2} \pi \theta^{2}\right) d \theta \quad y=\int_{0}^{t} \cos \left(\frac{1}{2} \pi \theta^{2}\right) d \theta\) Equation Transcription: Text Transcription: x = integral_0^1 sin(1/2 pi theta)d theta y = integral_0^1 cos(1/2 pi theta^2)d theta

If a projectile is fired with angle of elevation \(\alpha\) and initial speed \(v\), then parametric equations for its trajectory are \(x=(v \cos \alpha) t \quad y=(v \sin \alpha) t-\frac{1}{2} g t^{2}\) (See Example 13.4.5.) We know that the range (horizontal distance traveled) is maximized when \(\alpha=45^{\circ}\). What value of \(\alpha\) maximizes the total distance traveled by the projectile? (State your answer correct to the nearest degree.) Equation Transcription: ???? = 45° Text Transcription: x = (v cos alpha)t y = (v sin alpha)t - 1/2 gt^2 alpha = 45 degree

A cable has radius \(r\) and length \(L\) and is wound around a spool with radius \(R\) without overlapping. What is the shortest length along the spool that is covered by the cable?

Show that the curve with vector equation \(\mathbf{r}(t)=\left\langle a_1 t^2+b_1 t+c_1, a_2 t^2+b_2 t+c_2, a_3 t^2+b_3 t+c_3\right\rangle\) lies in a plane and find an equation of the plane.