Chapter 1

An airplane traveling horizontally at \(100 \mathrm{~m} / \mathrm{s}\) over flat ground at an elevation of 4000 meters must drop an emergency package on a target on the ground. The \(\begin{array}{lllll}\text { trajectory of the package } & \text { is } & \text { given } & \text { by }\end{array}\) \(x=100 t, y=-4.9 t^{2}+4000, t \geq 0\) where the origin is the point on the ground directly beneath the plane at the moment of release. How many horizontal meters before the target should the package be released in order to hit the target?

The trajectory of a bullet is given by \(x=v_{0}(\cos \alpha) t y=v_{0}(\sin \alpha) t-\frac{1}{2} g t^{2}\) where \(v_{0}=500 \mathrm{~m} / \mathrm{s}, \quad g=9.8=9.8 \mathrm{~m} / \mathrm{s}^{2}, \quad\) and \(\alpha=30\) degrees. When will the bullet hit the ground? How far from the gun will the bullet hit the ground?

Use technology to sketch the curve represented by \(x=\sin (4 t), y=\sin (3 t), 0 \leq t \leq 2 \pi\).

Sketch the curve known as an epitrochoid, which gives the path of a point on a circle of radius \(b\) as it rolls on the outside of a circle of radius \(a\). The equations are $$ \begin{array}{l} x=(a+b) \cos t-c \cdot \cos \left[\frac{(a+b) t}{b}\right] \\ y=(a+b) \sin t-c \cdot \sin \left[\frac{(a+b) t}{b}\right] \end{array} $$ Let \(a=1, b=2, c=1\).

Use technology to sketch the spiral curve given by \(x=t \cos (t), y=t \sin (t)\) from \(-2 \pi \leq t \leq 2 \pi .\)

Use technology to graph the curve given by the \(\begin{array}{l}\text { parametric } & \text { equations } \\ x=2 \cot (t), y=1-\cos (2 t),-\pi / 2 \leq t \leq \pi / 2 . & \text { This }\end{array}\) curve is known as the witch of Agnesi.

Sketch the curve given by parametric equations \(x=\cosh (t)\) \(y=\sinh (t)\) where \(-2 \leq t \leq 2\).

Each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line. $$ x=3+t, \quad y=1-t $$

Each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line. $$ x=4-3 t, \quad y=-2+6 t $$

Each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line. $$ x=-5 t+7, \quad y=3 t-1 $$

Determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. $$ x=3 \sin t, \quad y=3 \cos t, \quad t=\frac{\pi}{4} $$

Determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. $$ x=\cos t, \quad y=8 \sin t, t=\frac{\pi}{2} $$

Determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. $$ x=2 t, \quad y=t^{3}, \quad t=-1 $$

Determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. $$ x=t+\frac{1}{t}, \quad y=t-\frac{1}{t}, \quad t=1 $$

Determine the slope of the tangent line, then find the equation of the tangent line at the given value of the parameter. $$ x=\sqrt{t}, \quad y=2 t, \quad t=4 $$

Find all points on the curve that have the given slope. $$ x=4 \cos t, \quad y=4 \sin t, \text { slope }=0.5 $$

Find all points on the curve that have the given slope. $$ x=2 \cos t, \quad y=8 \sin t, \text { slope }=-1 $$

Find all points on the curve that have the given slope. $$ x=t+\frac{1}{t}, \quad y=t-\frac{1}{t}, \text { slope }=1 $$

Find all points on the curve that have the given slope. $$ x=2+\sqrt{t}, \quad y=2-4 t, \text { slope }=0 $$

Write the equation of the tangent line in Cartesian coordinates for the given parameter \(t\). $$ x=e^{\sqrt{t}}, \quad y=1-\ln t^{2}, \quad t=1 $$

Write the equation of the tangent line in Cartesian coordinates for the given parameter \(t\). $$ x=e^{t}, \quad y=(t-1)^{2}, \quad \text { at }(1,1) $$

For \(x=\sin (2 t), y=2 \sin t\) where \(0 \leq t<2 \pi\). Find all values of \(t\) at which a horizontal tangent line exists.

For \(x=\sin (2 t), y=2 \sin t\) where \(0 \leq t<2 \pi\). Find all values of \(t\) at which a vertical tangent line exists.

Find all points on the curve \(x=4 \cos (t), y=4 \sin (t)\) that have the slope of \(\frac{1}{2}\).

Find \(\frac{d y}{d x}\) for \(x=\sin (t), y=\cos (t)\).

Find the equation of the tangent line to \(x=\sin (t), y=\cos (t)\) at \(t=\frac{\pi}{4}\).

For the curve \(x=4 t, y=3 t-2,\) find the slope and concavity of the curve at \(t=3\).

For the parametric curve whose equation is \(x=4 \cos \theta, y=4 \sin \theta, \quad\) find the slope and concavity of the curve at \(\theta=\frac{\pi}{4}\).

Find all points on the curve \(x=t+4, y=t^{3}-3 t\) at which there are vertical and horizontal tangents.

Find all points on the curve \(x=\sec \theta, y=\tan \theta\) at which horizontal and vertical tangents exist.

Find \(d^{2} y / d x^{2}\). $$ x=t^{4}-1, \quad y=t-t^{2} $$

Find \(d^{2} y / d x^{2}\). $$ x=\sin (\pi t), \quad y=\cos (\pi t) $$

Find \(d^{2} y / d x^{2}\). $$ x=e^{-t}, \quad y=t e^{2 t} $$

Find points on the curve at which tangent line is horizontal or vertical. $$ x=t\left(t^{2}-3\right), \quad y=3\left(t^{2}-3\right) $$

Find points on the curve at which tangent line is horizontal or vertical. $$ x=\frac{3 t}{1+t^{3}}, \quad y=\frac{3 t^{2}}{1+t^{3}} $$

Find \(d y / d x\) at the value of the parameter. $$ x=\cos t, \quad y=\sin t, \quad t=\frac{3 \pi}{4} $$

Find \(d y / d x\) at the value of the parameter. $$ x=\sqrt{t}, \quad y=2 t+4, \quad t=9 $$

Find \(d y / d x\) at the value of the parameter. $$ x=4 \cos (2 \pi s), \quad y=3 \sin (2 \pi s), \quad s=-\frac{1}{4} $$

Find \(d^{2} y / d x^{2}\) at the given point without eliminating the parameter. $$ x=\frac{1}{2} t^{2}, \quad y=\frac{1}{3} t^{3}, \quad t=2 $$

Find \(d^{2} y / d x^{2}\) at the given point without eliminating the parameter. $$ x=\sqrt{t}, \quad y=2 t+4, \quad t=1 $$

Find \(t\) intervals on which the curve \(x=3 t^{2}, y=t^{3}-t\) is concave up as well as concave down.

Determine the concavity of the curve \(x=2 t+\ln t, y=2 t-\ln t\).

Sketch and find the area under one arch of the cycloid \(x=r(\theta-\sin \theta), y=r(1-\cos \theta)\).

Find the area bounded by the curve \(x=\cos t, y=e^{t}, 0 \leq t \leq \frac{\pi}{2}\) and the lines \(y=1\) and \(x=0\).

Find the area enclosed by the ellipse \(x=a \cos \theta, y=b \sin \theta, 0 \leq \theta<2 \pi\).

Find the area of the region bounded by \(x=2 \sin ^{2} \theta, y=2 \sin ^{2} \theta \tan \theta,\) for \(0 \leq \theta \leq \frac{\pi}{2}\).

Find the area of the regions bounded by the parametric curves and the indicated values of the parameter. $$ x=2 \cot \theta, y=2 \sin ^{2} \theta, 0 \leq \theta \leq \pi $$

Find the area of the regions bounded by the parametric curves and the indicated values of the parameter. $$ x=2 a \cos t-a \cos (2 t), y=2 a \sin t-a \sin (2 t), 0 \leq t<2 \pi $$

Find the area of the regions bounded by the parametric curves and the indicated values of the parameter. $$ \begin{aligned} &x=2 a \cos t-a \sin (2 t), y=b \sin t, 0 \leq t<2 \pi \quad \text { (the }\\\ &\text { "teardrop") } \end{aligned} $$

Find the arc length of the curve on the indicated interval of the parameter. $$ x=4 t+3, \quad y=3 t-2, \quad 0 \leq t \leq 2 $$