Chapter 13

Find an equation for the circle of curvature of the curve \(\mathbf{r}(t)=\) \((2 \ln t) \mathbf{i}-[t+(1 / t)] \mathbf{j}, e^{-2} \leq t \leq e^{2},\) at the point \((0,-2)\) where \(t=1 .\)

Throwing a baseball A baseball is thrown from the stands 32 \(\mathrm{ft}\) above the field at an angle of \(30^{\circ}\) up from the horizontal. When and how far away will the ball strike the ground if its initial speed is 32 \(\mathrm{ft} / \mathrm{sec}\) ?

Use Simpon's Rule with \(n=10\) to approximate the length of arc of \(\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k}\) from the origin to the point \((2,4,8) .\)

As mentioned in the text, the tangent line to a smooth curve \(\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}\) at \(t=t_{0}\) is the line that passes through the point \(\left(f\left(t_{0}\right), g\left(t_{0}\right), h\left(t_{0}\right)\right)\) parallel to \(\mathbf{v}\left(t_{0}\right),\) the curve's velocity vector at \(t_{0}\) Find parametric equations for the line that is tangent to the given curve at the given parameter value \(t=t_{0}\) . \begin{equation} \mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}+(\sin 2 t) \mathbf{k}, \quad t_{0}=\frac{\pi}{2} \end{equation}

Firing golf balls A spring gun at ground level fires a golf ball at an angle of \(45^{\circ} .\) The ball lands 10 m away. a. What was the ball's initial speed? b. For the same initial speed, find the two firing angles that make the range 6 \(\mathrm{m} .\)

Motion along a circle Each of the following equations in parts (a) \((\) e) describes the motion of a particle having the same path, namely the unit circle \(x^{2}+y^{2}=1 .\) Although the path of each particle in parts \((a)-(e)\) is the same, the behavior, or "dynamics," of each particle is different. For each particle, answer the following questions. \(\begin{array}{l}{\text { 1) Does the particle have constant speed? If so, what is its contant }} \\ {\text {speed? }} \\ {\text { ii) Is the particle's acceleration vector always orthogonal to its }} \\ {\text { velocity vector? }}\end{array}\) \(\begin{array}{l}{\text { iii) Does the particle move clockwise or counterclockwise }} \\ {\text { around the circle? }} \\ {\text { iv) Does the particle begin at the point }(1,0) ?}\end{array}\) \(\begin{array}{l}{\text { a. } \mathbf{r}(t)=(\cos t) \mathbf{i}+(\sin t) \mathbf{j}, \quad t \geqq 0} \\ {\text { b. } \mathbf{r}(t)=\cos (2 t) \mathbf{i}+\sin (2 t) \mathbf{j}, \quad t \geq 0} \\ {\text { c. } \mathbf{r}(t)=\cos (t-\pi / 2) \mathbf{i}+\sin (t-\pi / 2) \mathbf{j}, \quad t \geq 0} \\ {\text { d. } \mathbf{r}(t)=(\cos t) \mathbf{i}-(\sin t) \mathbf{j}, \quad t \geq 0} \\ {\text { e. } \mathbf{r}(t)=\cos \left(t^{2}\right) \mathbf{i}+\sin \left(t^{2}\right) \mathbf{j}, \quad t \geq 0}\end{array}\)

The formula $$ \kappa(x)=\frac{\left|f^{\prime \prime}(x)\right|}{\left[1+\left(f^{\prime}(x)\right)^{2}\right]^{3 / 2}} $$ derived in Exercise \(5,\) expresses the curvature \(\kappa(x)\) of a twice- differentiable plane curve \(y=f(x)\) as a function of \(x\) . Find the curvature function of each of the curves. Then graph \(f(x)\) together with \(\kappa(x)\) over the given interval. You will find some surprises. $$ y=x^{2}, \quad-2 \leq x \leq 2 $$

What can be said about the torsion of a smooth plane curve \(\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j} ?\) Give reasons for your answer.

The formula $$ \kappa(x)=\frac{\left|f^{\prime \prime}(x)\right|}{\left[1+\left(f^{\prime}(x)\right)^{2}\right]^{3 / 2}} $$ derived in Exercise \(5,\) expresses the curvature \(\kappa(x)\) of a twice- differentiable plane curve \(y=f(x)\) as a function of \(x\) . Find the curvature function of each of the curves. Then graph \(f(x)\) together with \(\kappa(x)\) over the given interval. You will find some surprises. $$ y=x^{4} / 4, \quad-2 \leq x \leq 2 $$

Beaming electrons An electron in a TV tube is beamed hori- zontally at a speed of \(5 \times 10^{6} \mathrm{m} / \mathrm{sec}\) toward the face of the tube 40 \(\mathrm{cm}\) away, About how far will the electron drop before it hits?

Motion along a circle Show that the vector-valued function \(\mathbf{r}(t)=(2 \mathbf{i}+2 \mathbf{j}+\mathbf{k})\) \begin{equation} +\cos t\left(\frac{1}{\sqrt{2}} \mathbf{i}-\frac{1}{\sqrt{2}} \mathbf{j}\right)+\sin t\left(\frac{1}{\sqrt{3}} \mathbf{i}+\frac{1}{\sqrt{3}} \mathbf{j}+\frac{1}{\sqrt{3}} \mathbf{k}\right) \end{equation} describes the motion of a particle moving in the circle of radius 1 centered at the point \((2,2,1)\) and lying in the plane \(x+y-2 z=2\) .

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 \(\mathbf{C}\) moves in a plane perpendicular to \(\mathbf{C} .\) This, in turn, can be viewed as the following result. Suppose \(\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}\) is twice differentiable for all \(t\) in an interval \([a, b],\) that \(\mathbf{r}=0\) when \(t=a\) , and that \(\mathbf{v} \cdot \mathbf{k}=0\) for all \(t\) in \([a, b] .\) Show that \(h(t)=0\) for all \(t\) in \([a, b] .\) (Hint: Start with \(\mathbf{a}=d \mathbf{r} / d t^{2}\) and apply the initial conditions in reverse order.)

The formula $$ \kappa(x)=\frac{\left|f^{\prime \prime}(x)\right|}{\left[1+\left(f^{\prime}(x)\right)^{2}\right]^{3 / 2}} $$ derived in Exercise \(5,\) expresses the curvature \(\kappa(x)\) of a twice- differentiable plane curve \(y=f(x)\) as a function of \(x\) . Find the curvature function of each of the curves. Then graph \(f(x)\) together with \(\kappa(x)\) over the given interval. You will find some surprises. $$ y=\sin x, \quad 0 \leq x \leq 2 \pi $$

Equal-range firing angles What two angles of elevation will enable a projectile to reach a target 16 \(\mathrm{km}\) downrange on the same level as the gun if the projectile's initial speed is 400 \(\mathrm{m} / \mathrm{sec}\) ?

Motion along a parabola A particle moves along the top of the parabola \(y^{2}=2 x\) from left to right at a constant speed of 5 units per second. Find the velocity of the particle as it moves through the point \((2,2) .\)

The formula $$ \kappa(x)=\frac{\left|f^{\prime \prime}(x)\right|}{\left[1+\left(f^{\prime}(x)\right)^{2}\right]^{3 / 2}} $$ derived in Exercise \(5,\) expresses the curvature \(\kappa(x)\) of a twice- differentiable plane curve \(y=f(x)\) as a function of \(x\) . Find the curvature function of each of the curves. Then graph \(f(x)\) together with \(\kappa(x)\) over the given interval. You will find some surprises. $$ y=e^{x},-1 \leq x \leq 2 $$

Motion along a cycloid A particle moves in the \(x y\) -plane in such a way that its position at time \(t\) is \begin{equation} \mathbf{r}(t)=(t-\sin t) \mathbf{i}+(1-\cos t) \mathbf{j} \end{equation} \(\begin{array}{l}{\text { a. Graph } \mathbf{r}(t) . \text { The resulting curve is a cycloid. }} \\ {\text { b. Find the maximum and minimum values of }|\mathbf{v}| \text { and }|\mathbf{a}| \text { . }} \\ {\text { (Hint: Find the extreme values of }|\mathbf{v}|^{2} \text { and }|\mathbf{a}|^{2} \text { first and take }} \\ {\text { square roots later.) }}\end{array}\)

Osculating circle Show that the center of the osculating circle for the parabola \(y=x^{2}\) at the point \(\left(a, a^{2}\right)\) is located at $$ \left(-4 a^{3}, 3 a^{2}+\frac{1}{2}\right) $$

Let \(\mathbf{r}\) be a differentiable vector function of \(t .\) Show that if \(\mathbf{r} \cdot(d \mathbf{r} / d t)=0\) for all \(t,\) then \(|\mathbf{r}|\) is constant.

Rounding the answers to four decimal places, use a CAS to find v, a, speed, \(\mathbf{T}, \mathbf{N}, \mathbf{B}, \kappa, \tau,\) and the tangential and normal components of acceleration for the curves in Exercises \(27-30\) at the given values of \(t\) \(\mathbf{r}(t)=(t \cos t) \mathbf{i}+(t \sin t) \mathbf{j}+t \mathbf{k}, \quad t=\sqrt{3}\)

Rounding the answers to four decimal places, use a CAS to find v, a, speed, \(\mathbf{T}, \mathbf{N}, \mathbf{B}, \kappa, \tau,\) and the tangential and normal components of acceleration for the curves in Exercises \(27-30\) at the given values of \(t.\) \(\mathbf{r}(t)=\left(e^{t} \cos t\right) \mathbf{i}+\left(e^{t} \sin t\right) \mathbf{j}+e^{t} \mathbf{k}, \quad t=\ln 2\)

Derivatives of triple scalar products \begin{equation} \begin{array}{l}{\text { a. Show that if } \mathbf{u}, \mathbf{v}, \text { and w are differentiable vector functions of }} \\ {t, \text { then }}\end{array} \end{equation} $$\begin{array}{l}{\frac{d}{d t}(\mathbf{u} \cdot \mathbf{v} \times \mathbf{w})=\frac{d \mathbf{u}}{d t} \cdot \mathbf{v} \times \mathbf{w}+\mathbf{u} \cdot \frac{d \mathbf{v}}{d t} \times \mathbf{w}+\mathbf{u} \cdot \mathbf{v} \times \frac{d \mathbf{w}}{d t}} \\\ {\text { b. Show that }}\end{array}$$ $$\frac{d}{d t}\left(\mathbf{r} \cdot \frac{d \mathbf{r}}{d t} \times \frac{d^{2} \mathbf{r}}{d t^{2}}\right)=\mathbf{r} \cdot\left(\frac{d \mathbf{r}}{d t} \times \frac{d^{3} \mathbf{r}}{d t^{3}}\right)$$ (Hint: Differentiate on the left and look for vectors whose products are zero.)

Colliding marbles The accompanying figure shows an experiment with two marbles. Marble \(A\) was launched toward marble \(B\) with launch angle \(\alpha\) and initial speed \(v_{0 .}\) At the same instant, marble \(B\) was released to fall from rest at \(R \tan \alpha\) units directly above a spot \(R\) units downrange from \(A .\) The marbles were found to collide regardless of the value of \(v_{0}\) . Was this mere coincidence, or must this happen? Give reasons for your answer.

You will use a CAS to explore the osculating circle at a point \(P\) on a plane curve where \(\kappa \neq 0 .\) Use a CAS to perform the following steps: a. Plot the plane curve given in parametric or function form over the specified interval to see what it looks like. b. Calculate the curvature \(\kappa\) of the curve at the given value \(t_{0}\) using the appropriate formula from Exercise 5 or \(6 .\) Use the parametrization \(x=t\) and \(y=f(t)\) if the curve is given as a function \(y=f(x) .\) c. Find the unit normal vector \(N\) at \(t_{0}\) . Notice that the signs of the components of \(N\) depend on whether the unit tangent vector \(T\) is turning clockwise or counterclockwise at \(t=t_{0}\) . (See Exercise 7\()\) d. If \(\mathbf{C}=a \mathbf{i}+b \mathbf{j}\) is the vector from the origin to the center \((a, b)\) of the osculating circle, find the center \(\mathbf{C}\) from the vector equation $$ \mathbf{C}=\mathbf{r}\left(t_{0}\right)+\frac{1}{\kappa\left(t_{0}\right)} \mathbf{N}\left(t_{0}\right) $$ The point \(P\left(x_{0}, y_{0}\right)\) on the curve is given by the position vector \(\mathbf{r}\left(t_{0}\right) .\) e. Plot implicitly the equation \((x-a)^{2}+(y-b)^{2}=1 / \kappa^{2}\) of the osculating circle. Then plot the curve and osculating circle together. You may need to experiment with the size of the viewing window, but be sure the axes are equally scaled. $$ \mathbf{r}(t)=(3 \cos t) \mathbf{i}+(5 \sin t) \mathbf{j}, \quad 0 \leq t \leq 2 \pi, \quad t_{0}=\pi / 4 $$

Firing from \(\left(x_{0}, y_{0}\right)\) Derive the equations $$\begin{array}{l}{x=x_{0}+\left(v_{0} \cos \alpha\right) t} \\\ {y=y_{0}+\left(v_{0} \sin \alpha\right) t-\frac{1}{2} g t^{2}}\end{array}$$ (see Equation \((7)\) in the text) by solving the following initial value problem for a vector \(r\) in the plane. $$\text{Differential equation:}\quad \frac{d^{2} \mathbf{r}}{d t^{2}}=-g \mathbf{j}$$ $$\text{Initial conditions:}\begin{array}{l}{\mathbf{r}(0)=x_{0} \mathbf{i}+y_{0} \mathbf{j}} \\ {\frac{d \mathbf{r}}{d t}(0)=\left(v_{0} \cos \alpha\right) \mathbf{i}+\left(v_{0} \sin \alpha\right) \mathbf{j}}\end{array}$$

You will use a CAS to explore the osculating circle at a point \(P\) on a plane curve where \(\kappa \neq 0 .\) Use a CAS to perform the following steps: a. Plot the plane curve given in parametric or function form over the specified interval to see what it looks like. b. Calculate the curvature \(\kappa\) of the curve at the given value \(t_{0}\) using the appropriate formula from Exercise 5 or \(6 .\) Use the parametrization \(x=t\) and \(y=f(t)\) if the curve is given as a function \(y=f(x) .\) c. Find the unit normal vector \(N\) at \(t_{0}\) . Notice that the signs of the components of \(N\) depend on whether the unit tangent vector \(T\) is turning clockwise or counterclockwise at \(t=t_{0}\) . (See Exercise 7\()\) d. If \(\mathbf{C}=a \mathbf{i}+b \mathbf{j}\) is the vector from the origin to the center \((a, b)\) of the osculating circle, find the center \(\mathbf{C}\) from the vector equation $$ \mathbf{C}=\mathbf{r}\left(t_{0}\right)+\frac{1}{\kappa\left(t_{0}\right)} \mathbf{N}\left(t_{0}\right) $$ The point \(P\left(x_{0}, y_{0}\right)\) on the curve is given by the position vector \(\mathbf{r}\left(t_{0}\right) .\) e. Plot implicitly the equation \((x-a)^{2}+(y-b)^{2}=1 / \kappa^{2}\) of the osculating circle. Then plot the curve and osculating circle together. You may need to experiment with the size of the viewing window, but be sure the axes are equally scaled. $$ \mathbf{r}(t)=\left(\cos ^{3} t\right) \mathbf{i}+\left(\sin ^{3} t\right) \mathbf{j}, \quad 0 \leq t \leq 2 \pi, \quad t_{0}=\pi / 4 $$

Where trajectories erest For a projectile fired from the ground at launch angle \(\alpha\) with initial speed \(v_{0},\) consider \(\alpha\) as a variable and \(v_{0}\) as a fixed constant. For each \(\alpha, 0<\alpha<\pi / 2,\) we obtable a parabolic trajectory as shown in the accompanying figure. Show that the points in the plane that give the maximum heights of these parabolic trajectories all lie on the ellipse $$x^{2}+4\left(y-\frac{v_{0}^{2}}{4 g}\right)^{2}=\frac{v_{0}^{4}}{4 g^{2}}$$ where \(x \geqq 0\)

You will use a CAS to explore the osculating circle at a point \(P\) on a plane curve where \(\kappa \neq 0 .\) Use a CAS to perform the following steps: a. Plot the plane curve given in parametric or function form over the specified interval to see what it looks like. b. Calculate the curvature \(\kappa\) of the curve at the given value \(t_{0}\) using the appropriate formula from Exercise 5 or \(6 .\) Use the parametrization \(x=t\) and \(y=f(t)\) if the curve is given as a function \(y=f(x) .\) c. Find the unit normal vector \(N\) at \(t_{0}\) . Notice that the signs of the components of \(N\) depend on whether the unit tangent vector \(T\) is turning clockwise or counterclockwise at \(t=t_{0}\) . (See Exercise 7\()\) d. If \(\mathbf{C}=a \mathbf{i}+b \mathbf{j}\) is the vector from the origin to the center \((a, b)\) of the osculating circle, find the center \(\mathbf{C}\) from the vector equation $$ \mathbf{C}=\mathbf{r}\left(t_{0}\right)+\frac{1}{\kappa\left(t_{0}\right)} \mathbf{N}\left(t_{0}\right) $$ The point \(P\left(x_{0}, y_{0}\right)\) on the curve is given by the position vector \(\mathbf{r}\left(t_{0}\right) .\) e. Plot implicitly the equation \((x-a)^{2}+(y-b)^{2}=1 / \kappa^{2}\) of the osculating circle. Then plot the curve and osculating circle together. You may need to experiment with the size of the viewing window, but be sure the axes are equally scaled. $$ \mathbf{r}(t)=t^{2} \mathbf{i}+\left(t^{3}-3 t\right) \mathbf{j}, \quad-4 \leq t \leq 4, \quad t_{0}=3 / 5 $$

Component test for continuity at a point Show that the vector function \(\mathbf{r}\) defined by \(\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}\) is continuous at \(t=t_{0}\) if and only if \(f, g,\) and \(h\) are continuous at \(t_{0}\)

Limits of cross products of vector functions Suppose that \(\mathbf{r}_{1}(t)=f_{1}(t) \mathbf{i}+f_{2}(t) \mathbf{j}+f_{3}(t) \mathbf{k}, \mathbf{r}_{2}(t)=g_{1}(t) \mathbf{i}+g_{2}(t) \mathbf{j}+g_{3}(t) \mathbf{k},\) \(\lim _{t \rightarrow t_{0}} \mathbf{r}_{1}(t)=\mathbf{A},\) and \(\lim _{t \rightarrow t_{0}} \mathbf{r}_{2}(t)=\mathbf{B} .\) Use the determinant for- mula for cross products and the Limit Product Rule for scalar functions to show that \begin{equation} \lim _{t \rightarrow t_{0}}\left(\mathbf{r}_{1}(t) \times \mathbf{r}_{2}(t)\right)=\mathbf{A} \times \mathbf{B} \end{equation}

Volleyball A volleyball is hit when it is 4 ft above the ground and 12 ft from a 6 -ft-high net. It leaves the point of impact with an initial velocity of 35 \(\mathrm{ft} / \mathrm{sec}\) at an angle of \(27^{\circ}\) and slips by the opposing team untouched. a. Find a vector equation for the path of the volleyball. b. How high does the volleyball go, and when does it reach maximum height? c. Find its range and flight time. d. When is the volleyball 7 ft above the ground? How far e. Suppose that the net is raised to 8 8 ft. Does this change things? Explain.

Differentiable vector functions are continuous Show that if \(\mathbf{r}(t)=f(t) \mathbf{i}+g(t) \mathbf{j}+h(t) \mathbf{k}\) is differentiable at \(t=t_{0},\) then it is continuous at \(t_{0}\) as well.

You will use a CAS to explore the osculating circle at a point \(P\) on a plane curve where \(\kappa \neq 0 .\) Use a CAS to perform the following steps: a. Plot the plane curve given in parametric or function form over the specified interval to see what it looks like. b. Calculate the curvature \(\kappa\) of the curve at the given value \(t_{0}\) using the appropriate formula from Exercise 5 or \(6 .\) Use the parametrization \(x=t\) and \(y=f(t)\) if the curve is given as a function \(y=f(x) .\) c. Find the unit normal vector \(N\) at \(t_{0}\) . Notice that the signs of the components of \(N\) depend on whether the unit tangent vector \(T\) is turning clockwise or counterclockwise at \(t=t_{0}\) . (See Exercise 7\()\) d. If \(\mathbf{C}=a \mathbf{i}+b \mathbf{j}\) is the vector from the origin to the center \((a, b)\) of the osculating circle, find the center \(\mathbf{C}\) from the vector equation $$ \mathbf{C}=\mathbf{r}\left(t_{0}\right)+\frac{1}{\kappa\left(t_{0}\right)} \mathbf{N}\left(t_{0}\right) $$ The point \(P\left(x_{0}, y_{0}\right)\) on the curve is given by the position vector \(\mathbf{r}\left(t_{0}\right) .\) e. Plot implicitly the equation \((x-a)^{2}+(y-b)^{2}=1 / \kappa^{2}\) of the osculating circle. Then plot the curve and osculating circle together. You may need to experiment with the size of the viewing window, but be sure the axes are equally scaled. $$ \begin{array}{l}{\mathbf{r}(t)=(2 t-\sin t) \mathbf{i}+(2-2 \cos t) \mathbf{j}, \quad 0 \leq t \leq 3 \pi} \\ {t_{0}=3 \pi / 2}\end{array} $$

You will use a CAS to explore the osculating circle at a point \(P\) on a plane curve where \(\kappa \neq 0 .\) Use a CAS to perform the following steps: a. Plot the plane curve given in parametric or function form over the specified interval to see what it looks like. b. Calculate the curvature \(\kappa\) of the curve at the given value \(t_{0}\) using the appropriate formula from Exercise 5 or \(6 .\) Use the parametrization \(x=t\) and \(y=f(t)\) if the curve is given as a function \(y=f(x) .\) c. Find the unit normal vector \(N\) at \(t_{0}\) . Notice that the signs of the components of \(N\) depend on whether the unit tangent vector \(T\) is turning clockwise or counterclockwise at \(t=t_{0}\) . (See Exercise 7\()\) d. If \(\mathbf{C}=a \mathbf{i}+b \mathbf{j}\) is the vector from the origin to the center \((a, b)\) of the osculating circle, find the center \(\mathbf{C}\) from the vector equation $$ \mathbf{C}=\mathbf{r}\left(t_{0}\right)+\frac{1}{\kappa\left(t_{0}\right)} \mathbf{N}\left(t_{0}\right) $$ The point \(P\left(x_{0}, y_{0}\right)\) on the curve is given by the position vector \(\mathbf{r}\left(t_{0}\right) .\) e. Plot implicitly the equation \((x-a)^{2}+(y-b)^{2}=1 / \kappa^{2}\) of the osculating circle. Then plot the curve and osculating circle together. You may need to experiment with the size of the viewing window, but be sure the axes are equally scaled. $$ \mathbf{r}(t)=\left(e^{-t} \cos t\right) \mathbf{i}+\left(e^{-t} \sin t\right) \mathbf{j}, \quad 0 \leq t \leq 6 \pi, \quad t_{0}=\pi / 4 $$

Constant Function Rule Prove that if \(u\) is the vector function with the constant value \(C,\) then \(d \mathbf{u} / d t=0 .\)

Use a CAS to perform the following steps in Exercises \(35-38\) \begin{equation} \begin{array}{l}{\text { a. Plot the space curve traced out by the position vector } \mathbf{r} \text { . }} \\ {\text { b. Find the components of the velocity vector } d \mathbf{r} / d t \text { . }} \\ {\text { c. Evaluate } d \mathbf{r} / d t \text { at the given point } t_{0} \text { and determine the equation of }} \\ {\text { the tangent line to the curve at } \mathbf{r}\left(t_{0}\right) .} \\ {\text { d. Plot the tangent line together with the curve over the given interval. }}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\mathbf{r}(t)=(\sin t-t \cos t) \mathbf{i}+(\cos t+t \sin t) \mathbf{j}+t^{2} \mathbf{k}} \\ {0 \leq t \leq 6 \pi, \quad t_{0}=3 \pi / 2}\end{array} \end{equation}

Hitting a baseball under a wind gust A baseball is hit when it is 2.5 \(\mathrm{ft}\) above the ground. It leaves the bat with an initial velocity of 145 \(\mathrm{ft} / \mathrm{sec}\) at a launch angle of \(23^{\circ} .\) At the instant the ball is hit, an instantaneous gust of wind blows against the ball, adding a component of \(-14 \mathrm{i}(\mathrm{ft} / \mathrm{sec})\) to the ball's initial velocity. \(\mathrm{A} 15\) -ft- high fence lies 300 \(\mathrm{ft}\) from home plate in the direction of the flight. a. Find a vector equation for the path of the baseball. b. How high does the baseball go, and when does it reach maximum height? c. Find the range and flight time of the baseball, assuming that the ball is not caught. d. When is the baseball 20 ft high? How far (ground distance) is e. Has the batter hit a home run? Explain.

Use a CAS to perform the following steps. \(\begin{array}{l}{\text { a. Plot the space curve traced out by the position vector } \mathbf{r} \text { . }} \\ {\text { b. Find the components of the velocity vector } d \mathbf{r} / d t \text { . }} \\ {\text { c. Evaluate } d \mathbf{r} / d t \text { at the given point } t_{0} \text { and determine the equation of }} \\ {\text { the tangent line to the curve at } \mathbf{r}\left(t_{0}\right) .} \\ {\text { d. Plot the tangent line together with the curve over the given interval. }}\end{array}$$ \mathbf{r}(t)=\sqrt{2} t \mathbf{i}+e^{t} \mathbf{j}+e^{-i} \mathbf{k}, \quad-2 \leq t \leq 3, \quad t_{0}=1\)

Linear drag Derive the equations $$\begin{aligned} x &=\frac{v_{0}}{k}\left(1-e^{-k l}\right) \cos \alpha \\ y &=\frac{v_{0}}{k}\left(1-e^{-k t}\right)(\sin \alpha)+\frac{g}{k^{2}}\left(1-k t-e^{-k t}\right) \end{aligned}$$ by solving the following initial value problem for a vector \(r\) in the plane. $$ \text{Differential equation:}\frac{d^{2} \mathbf{r}}{d t^{2}}=-g \mathbf{j}-k \mathbf{v}=-g \mathbf{j}-k \frac{d \mathbf{r}}{d t}$$ $$\text{Initial conditions:}\begin{aligned} \mathbf{r}(0) &=\mathbf{0} \\\\\left.\frac{d \mathbf{r}}{d t}\right|_{t=0} &=\mathbf{v}_{0}=\left(v_{0} \cos \alpha\right) \mathbf{i}+\left(\boldsymbol{v}_{0} \sin \alpha\right) \mathbf{j} \end{aligned}$$ The drag coefficient \(k\) is a positive constant representing resistance due to air density, \(v_{0}\) and \(\alpha\) are the projectile's initial speed and launch angle, and \(g\) is the acceleration of gravity.

Use a CAS to perform the following steps. \begin{equation} \begin{array}{l}{\text { a. Plot the space curve traced out by the position vector } \mathbf{r} \text { . }} \\ {\text { b. Find the components of the velocity vector } d \mathbf{r} / d t \text { . }} \\ {\text { c. Evaluate } d \mathbf{r} / d t \text { at the given point } t_{0} \text { and determine the equation of }} \\ {\text { the tangent line to the curve at } \mathbf{r}\left(t_{0}\right) .} \\ {\text { d. Plot the tangent line together with the curve over the given interval. }}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\mathbf{r}(t)=(\sin 2 t) \mathbf{i}+(\ln (1+t)) \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 4 \pi} \\ {t_{0}=\pi / 4}\end{array} \end{equation}

Use a CAS to perform the following steps. \begin{equation} \begin{array}{l}{\text { a. Plot the space curve traced out by the position vector } \mathbf{r} \text { . }} \\ {\text { b. Find the components of the velocity vector } d \mathbf{r} / d t \text { . }} \\ {\text { c. Evaluate } d \mathbf{r} / d t \text { at the given point } t_{0} \text { and determine the equation of }} \\ {\text { the tangent line to the curve at } \mathbf{r}\left(t_{0}\right) .} \\ {\text { d. Plot the tangent line together with the curve over the given interval. }}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\mathbf{r}(t)=\left(\ln \left(t^{2}+2\right) \mathbf{i}+\left(\tan ^{-1} 3 t\right) \mathbf{j}+\sqrt{t^{2}+1} \mathbf{k}\right.} \\ {-3 \leq t \leq 5, \quad t_{0}=3}\end{array} \end{equation}

Establish the following properties of integrable vector functions. a. The Constant Scalar Multiple Rule: $$\int_{a}^{b} k \mathbf{r}(t) d t=k \int_{a}^{b} \mathbf{r}(t) d t \quad(\text { any scalar } k)$$ The Rule for Negatives, $$\int_{a}^{b}(-\mathbf{r}(t)) d t=-\int_{a}^{b} \mathbf{r}(t) d t$$ is obtained by taking \(k=-1\) b. The Sum and Difference Rules: $$\int_{a}^{b}\left(\mathbf{r}_{1}(t) \pm \mathbf{r}_{2}(t)\right) d t=\int_{a}^{b} \mathbf{r}_{1}(t) d t \pm \int_{a}^{b} \mathbf{r}_{2}(t) d t$$ c. The Constant Vector Multiple Rules: $$\int_{a}^{b} \mathbf{C} \cdot \mathbf{r}(t) d t=\mathbf{C} \cdot \int_{a}^{b} \mathbf{r}(t) d t \quad \text { (any constant vector } \mathbf{C} )$$ and $$\int_{a}^{b} \mathbf{C} \times \mathbf{r}(t) d t=\mathbf{C} \times \int_{a}^{b} \mathbf{r}(t) d t \quad(\text { any constant vector } \mathbf{C})$$

In Exercises 39 and \(40,\) you will explore graphically the behavior of the helix \begin{equation} \mathbf{r}(t)=(\cos a t) \mathbf{i}+(\sin a t) \mathbf{j}+b t \mathbf{k} \end{equation} as you change the values of the constants \(a\) and \(b .\) Use a CAS to perform the steps in each exercise. \begin{equation} \begin{array}{l}{\text { Set } b=1 . \text { Plot the helix } \mathbf{r}(t) \text { together with the tangent line to the }} \\ {\text { curve at } t=3 \pi / 2 \text { for } a=1,2,4, \text { and } 6 \text { over the interval }} \\\ {0 \leq t \leq 4 \pi . \text { Describe in your own words what happens to the }} \\ {\text { graph of the helix and the position of the tangent line as a }} \\ {\text { increases through these positive values. }}\end{array} \end{equation}

Products of scalar and vector functions Suppose that the scalar function \(u(t)\) and the vector function \(\mathbf{r}(t)\) are both defined for \(a \leq t \leq b .\) a. Show that \(u \mathbf{r}\) is continuous on \([a, b]\) if \(u\) and \(\mathbf{r}\) are continuous on \([a, b] .\) b. If \(u\) and \(\mathbf{r}\) are both differentiable on \([a, b],\) show that \(u \mathbf{r}\) is differentiable on \([a, b]\) and that $$\frac{d}{d t}(u \mathbf{r})=u \frac{d \mathbf{r}}{d t}+\mathbf{r} \frac{d u}{d t}$$

You will explore graphically the behavior of the helix \begin{equation} \mathbf{r}(t)=(\cos a t) \mathbf{i}+(\sin a t) \mathbf{j}+b t \mathbf{k} \end{equation} as you change the values of the constants \(a\) and \(b .\) Use a CAS to perform the steps in each exercise. \begin{equation} \begin{array}{l}{\text { Set } a=1 . \text { Plot the helix } \mathbf{r}(t) \text { together with the tangent line to the }} \\ {\text { curve at } t=3 \pi / 2 \text { for } b=1 / 4,1 / 2,2, \text { and } 4 \text { over the interval }} \\ {0 \leq t \leq 4 \pi . \text { Describe in your own words what happens to the }} \\ {\text { graph of the helix and the position of the tangent line as } b} \\ {\text { increases through these positive values. }}\end{array} \end{equation}

Antiderivatives of vector functions a. Use Corollary 2 of the Mean Value Theorem for scalar functions to show that if two vector functions \(\mathbf{R}_{1}(t)\) and \(\mathbf{R}_{2}(t)\) have identical derivatives on an interval \(I\) , then the functions differ by a constant vector value throughout I. b. Use the result in part (a) to show that if \(\mathbf{R}(t)\) is any antiderivative of \(\mathbf{r}(t)\) on \(I\) , then any other antiderivative of \(\mathbf{r}\) on \(I\) equals \(\mathbf{R}(t)+\mathbf{C}\) for some constant vector \(\mathbf{C}\) .