Chapter 1

Convert \(\pi, 3 \pi,-\pi / 4\) to degrees and \(60^{\circ}, 90^{\circ}, 270^{\circ}\) to radians. What angles between 0 and \(2 \pi\) correspond to \(\theta=480^{\circ}\) and \(\theta=-1^{\circ} ?\)

Draw graphs of \(\tan \theta\) and \(\cot \theta\) from 0 to \(2 \pi .\) What is their (shortest) period?

For the same \(f(t)=t^{2}+t,\) find the average speed between (a) \(t=0\) and 1 (b) \(t=0\) and \(\frac{1}{2}\) (c) \(t=0\) and \(h\).

Show that \(\cos 2 \theta\) and \(\cos ^{2} \theta\) have period \(\pi\) and draw them on the same graph.

Problems \(5-11\) are about linear functions and constant slopes. Write down the slopes of these linear functions: (a) \(f(t)=1.1 t\) (b) \(f(t)=1-2 t\) (c) \(f(t)=4+5(t-6)\) Compute \(f(6)\) and \(f(7)\) for each function and confirm that \(f(7)-f(6)\) equals the slope.

At \(\theta=3 \pi / 2\) compute the six basic functions and check \(\cos ^{2} \theta+\sin ^{2} \theta, \sec ^{2} \theta-\tan ^{2} \theta, \csc ^{2} \theta-\cot ^{2} \theta\).

About linear functions and constant slopes. If \(f(t)=5+3(t-1)\) and \(g(t)=1.5+2.5(t-1)\) what is \(h(t)=f(t)-g(t) ?\) Find the slopes of \(f, g,\) and \(h .\)

Prepare a table showing the values of the six basic functions at \(\theta=0, \pi / 4, \pi / 3, \pi / 2, \pi\).

About linear functions and constant slopes. Suppose \(v(t)=2\) for \(t<5\) and \(v(t)=3\) for \(t>5\) (a) If \(f(0)=0\) find a two-part formula for \(f(t)\). (b) Check that \(f(10)\) equals the area under the graph of \(v(t)\) (two rectangles) up to \(t=10\)

Draw the graph of \(v(t)=1+2 t\). From geometry find the area under it from 0 to \(t .\) Find the slope of that area function \(\int(t)\).

The area of a circle is \(\pi r^{2}\). What is the area of the sector that has angle \(\theta\) ? It is a fraction _______ of the whole area.

About linear functions and constant slopes. Suppose \(v(t)=10\) for \(t<1 / 10, v(t)=0\) for \(t>1 / 10\). Starting from \(f(0)=1\) find \(f(t)\) in two pieces.

Find the distance from (1,0) to (0,1) along (a) a straight line (b) a quarter- circle (c) a semicircle centered at \(\left(\frac{1}{2}, \frac{1}{2}\right)\)

Draw graphs of \(f(t)=\cos 3 t\) and \(\cos 2 \pi t\) and \(2 \pi \cos t,\) marking the time axes. How long until each \(f\) repeats?

About linear functions and constant slopes. Suppose \(g(t)=2 t+1\) and \(f(t)=4 t .\) Find \(g(3)\) and \(f(g(3))\) and \(f(g(t))\). How is the slope of \(f(g(t))\) related to the slopes of \(f\) and \(g ?\)

Find the distance \(d\) from (1,0) to \(\left(\frac{1}{2}, \sqrt{3} / 2\right)\) and show on a circle why \(6 d\) is less than \(2 \pi\).

If \(f(t)=t^{2}\) what is the average velocity between \(t=9\) and \(t=1.1 ?\) What is the average between \(t-h\) and \(t+h\) ?

Decide whether these equations are true or false: (a) \(\frac{\sin \theta}{1-\cos \theta}=\frac{1+\cos \theta}{\sin \theta}\) (b) \(\frac{\sec \theta+\csc \theta}{\tan \theta+\cot \theta}=\sin \theta+\cos \theta\) (c) \(\cos \theta-\sec \theta=\sin \theta \tan \theta\) (d) \(\sin (2 \pi-\theta)=\sin \theta\)

The earth's population is growing at \(v=100\) million a year, starting from \(f=5.2\) billion in \(1990 .\) Graph \(f(t)\) and find \(f(2000)\)

Simplify \(\sin (\pi-\theta), \cos (\pi-\theta), \sin (\pi / 2+\theta), \cos (\pi / 2+\theta)\).

From the formula for \(\cos (2 t+t)\) find \(\cos 3 t\) in temss of \(\cos t\).

From the formula for \(\sin (2 t+t)\) find \(\sin 3 t\) in terms of \(\sin t\).

Find the ares under \(v=\cos t\) from the change in \(f=\sin t:\) $$ \text { from } t=0 \text { to } t=\pi $$

When you jump up and fall back your height is \(y=2 t-t^{2}\) in the right units. (a) Graph this parabola and its slope. (b) Find the time in the air and maximum height. (c) Prove: Half the time you are above \(y=\frac{3}{4}\). Basketball players "hang" in the air partly because of (c).

Find the ares under \(v=\cos t\) from the change in \(f=\sin t:\) $$ \text { from } t=0 \text { to } t=\pi / 6 $$

Show that \((\cos t+i \sin t)^{2}=\cos 2 t+i \sin 2 t,\) if \(i^{2}=-1\).

Find the ares under \(v=\cos t\) from the change in \(f=\sin t:\) $$ \text { from } t=0 \text { to } t=2 \pi $$

Draw \(\cos \theta\) and \(\sec \theta\) on the same graph. Find all points where \(\cos \theta=\sec \theta\).

Find the ares under \(v=\cos t\) from the change in \(f=\sin t:\) $$ \text { from } t=\pi / 2 \text { to } t=3 \pi / 2 . $$

Find all angles \(s\) and \(t\) between 0 and \(2 \pi\) where \(\sin (s+t)=\) \(\sin s+\sin t\).

The distance curve \(f=\sin 4 t\) yields the velocity curve \(v=4 \cos 4 t\). Explain both 4 's.

Draw rough graphs of \(y=\sqrt{x}\) and \(y=\sqrt{x-4}\) and \(y=\sqrt{x}-4\). They are "half-parabolas" with infinite slope at the start.

Involve numbers \(\int_{0}, f_{1}, f_{2}, \ldots\) and their differences \(v_{j}=f_{j}-f_{j-1}\), They give practice with subscripts \(0, \ldots, j\). Find \(f_{1}, f_{2}, f_{3}\) and a formula for \(f_{j}\) with \(f_{0}=0:\) (a) \(v=1,2,4,8, \ldots\) (b) \(v=-1,1,-1,1, \ldots\)

The distance curve \(f=2 \cos 3 t\) yields the velocity curve \(v=-6 \sin 3 t .\) Explain the -6

The velocity curve \(v=\cos 4 t\) yields the distance curve \(f=\frac{1}{4} \sin 4 t .\) Explain the \(\frac{1}{4}\).

Draw the graph of \(f(t)=\left|1-t^{2}\right|\) for \(0 \leqslant t \leqslant 2\). Find a three-part formula for \(v(t)\).

Involve numbers \(\int_{0}, f_{1}, f_{2}, \ldots\) and their differences \(v_{j}=f_{j}-f_{j-1}\), They give practice with subscripts \(0, \ldots, j\). From the area under the staircase (by rectangles and then by triangles) show that the first \(j\) whole numbers 1 to \(j\) add up to \(\frac{1}{2} j^{2}+\frac{1}{2} j\). Find \(1+2+\cdots+100\).

Find the slope of the sine curve at \(t=\pi / 3\) from \(v=\cos t\) Then find an average slope by dividing \(\sin \pi / 2-\sin \pi / 3\) by the time difference \(\pi / 2-\pi / 3\)

Problems 23-28 involve linear functions \(f(t)=o t+C\). Find the constants \(v\) and \(C\). What linear function has \(f(0)=3\) and \(f(2)=-11 ?\)

When does \(f(t)=t^{2}-3 t\) reach \(10 ?\) Find the average velocity up to that time and the instantaneous velocity at that time.

Find every \(\theta\) that satisfies the equation. $$ \sin \theta=-1 $$

Problems 23-28 involve linear functions \(f(t)=o t+C\). Find the constants \(v\) and \(C\). Find two linear functions whose domain is \(0 \leqslant t \leqslant 2\) and whose range is \(1 \leqslant f(t) \leqslant 9\)

If \(f(t)=\frac{1}{2} a t^{2}+b t+c,\) what is \(v(t) ?\) What is the slope of \(v(t) ?\) When does \(f(t)\) equal \(41,\) if \(a=b=c=1 ?\)

Find every \(\theta\) that satisfies the equation. $$ \sec \theta=-2 $$

The ball at \(x=\cos t, y=\sin t\) circles (1) connterclockwise (2) with radius 1 (3) starting from \(x=1, y=0\) (4) at speed 1. Find (1)(2)(3)(4) for the motions \(25-30\). $$ x=\cos 3 t, y=-\sin 3 t $$

Problems 23-28 involve linear functions \(f(t)=o t+C\). Find the constants \(v\) and \(C\). Find the linear function with \(f(1)=4\) and slope 6 .

If \(f(t)=t^{2}\) then \(v(t)=2 t .\) Does the speeded-up function \(f(4 t)\) have velocity \(v(4 t)\) or \(4 v(t)\) or \(4 v(4 t) ?\)

Find every \(\theta\) that satisfies the equation. $$ \sin \theta=\cos \theta $$

Involve numbers \(\int_{0}, f_{1}, f_{2}, \ldots\) and their differences \(v_{j}=f_{j}-f_{j-1}\), They give practice with subscripts \(0, \ldots, j\). If \(f(t)=t^{2}+t,\) compute \(f(99)\) and \(f(101)\). Between those times, what is the increase in \(f\) divided by the increase in \(t\) ?

If \(f(t)=t-t^{2}\) find \(v(t)\) and \(f(3 t)\). Does the slope of \(f(3 t)\) equal \(v(3 t)\) or \(3 v(t)\) or \(3 v(3 t) ?\)