1. Simplify the following expressions:
   • 1.
   • 2.
   • 3. (2⁴)²
   • 4. (2³)³
   • 5. 8^-4
   • 6.
   • 7.
   • 8. 2⁽³³⁾

   • Express the functions in the form y=x^p for some power p:
     • 1.
     • 2.
     • 3.
     • 4.
     • 5.
     • 6.

     • Express the functions in fractional and radical form, e.g., , etc...
       • 1. y=-x^-1
       • 2.
       • 3. y=x^-2
       • 4.
       • 5.
       • 6.

         You can check your work on these exercises with the program ExpRules in the Chapter 28 folder.

       • Find the surface area of a sphere as a function of its volume as in Exercise CD-28.3.2, and express your answer as a power function.

In a right triangle with acute angle as shown and with sides of lengths o opposite , a adjacent to and hypotenuse h, we have

Example CD-28.3 Derive the Functional Equation

The Pythagorean Theorem says

Section CD-28.9 reviews identities beyond the basic trig identities. Besides SOH-CAH-TOA, there are just three main identities that you must know. You do need to remember how to use these three, and the background review should help if you are rusty on that. The Mathematical Background chapter on Functional Identities studies identities more abstractly.

The radian measure of an angle A is defined to be the length of the circular arc, centered at its apex, that the two sides of the angle cut off. This is shown Figure CD-28.1.

Because the length of the unit circle , the distance all the way around a unit circle is . The distance half way around, corresponding to an angle of is . This simple relationship gives us a proportionality that we can use to convert degrees to radians or radians to degrees:

Figure CD-28.1: Radian measure

For example, can be converted to radian measure by

1. Conversion
   
   Find the radian measures of angles with degree measures
   , , , .
   Find the degree measures of angles with radian measures
   , , , , , 1.

2. Sine and Cosine as a Parametric Point
   
   Find the (x,y)-point on the unit circle that lies a distance of units counterclockwise along the unit circle measured from (1,0).
   The line L makes an angle of with the positive x axis, measured counterclockwise. Find the radian measure of this angle. Find the (x,y)-point in the second quadrant where the line L crosses the unit circle.

3. Odd and Even Functions
   
   Draw a unit circle and an angle with radian measure measured counterclockwise from the x-axis. Draw the negative angle measured clockwise. Observe that the point where the angle crosses the unit circle lies vertically above the point where the negative angle crosses, so the components of the point for the negative angle are (x,-y) when the components for the positive angle are (x,y).
   Prove that cosine is an even function, and that sine is an odd function, .

4. A plane passes overhead traveling at 600 mph in a straight horizontal line. At elapsed time 0 it is 5,280 ft. directly above you. Express the angle from the vertical that you look at the plane as a function of time. (Careful with units.)

5. The courtesy light in front of my house is 8 ft. tall and 4 ft. back from the sidewalk. My daughter is 5 ft. tall and walks down the sidewalk. Express the length of the shadow she casts as a function of the distance she is down the sidewalk from the point perpendicular to the lamp.

6. You stand in the middle of railroad tracks with a train approaching you at 100 ft/sec. The train is 15 ft. wide and is 1,000 ft. away at elapsed time 0. Express the angle of view subtended by the train as a function of time. (We do not recommend simulation. How long do you have to get off the tracks?)

Problem CD-28.1

Describe the motion of the piston shown in Figure CD-28.3 when the crankshaft turns 2000 revolutions per minute.

Figure CD-28.3: Crank and piston

Because the widespread use of inexpensive calculators, base 10 logs are no longer of much interest. Natural base or base "e" logs and exponentials are still very important, as we will see throughout the course. The fundamental functions' names are

Of course, it still makes perfectly good sense to use other bases, particularly for exponentials. However, calculus becomes much simpler when logs and exponentials are expressed in the "natural" base e. The following example uses base 2 and an exercise in Chapter 8 asks you to express this base 2 function in terms of base e. The reexpression does not seem advantageous now, but it will be once we have calculus, because the derivative

1. Explain the difference between an exponential function and a power function.

   Use a calculator or the program LogGth to compute the natural logs in the next exercise.

2. Logarithmic Growth
   A super duper computer can add ten billion terms of the form 1+1/2+1/3+1/4+1/5+... per second with perfect accuracy. The size of the sum 1+1/2+...+1/n is approximately . How many centuries would it take this computer to add enough terms to get a sum over 100?

Properties of exponents play an important role in exponential functions. Let's return to the number of mold cells in my basement

Use the program ExpGth in the High School Review folder for help with the algae growth computations in the next problem.

Problem CD-28.2 Growth of Algae

On a warm summer day with plenty of nutrients, supplied by runoff, the number of algae cells in a formerly clear pond doubles every 6 hours. This is the time it takes a cell to divide; but, naturally, the cells do not all divide simultaneously, so a population of many cells grows almost continuously with time. Suppose you begin with N₀ cells at elapsed time 0. Express the number of cells as a function of time in hours. How many cells are there after 3 hours? What do fractional values of N mean? How many are there after 1 day? How many in 1 week? If one cell has mass 1 mg. and we start with 1000 cells, what is the mass of the algae cells after 1 month? What is the approximate mass of all the water in Lake Michigan in these units? Suppose the lake fills at time t_f. How much algae is there 6 hours later? Can algae continue to double every six hours?

Symbolic calculus first gives rules for the basic kinds of functions and then gives rules for functions built up from basic functions by sums, products, and compositions. In order to understand the rules of calculus, you need to be familiar with function notation for high school functions.

We can think of this in terms of a function, y=f[x]=e^x, replacing the input x with the expression ,

You should also be familiar with the function notation for chains. If

Example CD-28.4 The Expression

In Chapter 1 of the main text we used the expression above to compute the slope of a secant line. Let's break down the meaning of this notation in the case

The expression is obtained by subtraction of the two results,

Finally, the expression

1. Show the following:
   • 1. If f[x]=x², then
   • 2. If f[x]=x⁴, then
   • 3. If f[x]=x⁵, then

   • Common Denominators
     Show that

     if , then

     Note that and that

   • Algebraic Conjugates
     Show that
     
     and that
     
     Conclude that

     if , then

   • Combinations of Functions
     
     • 1. Let h[t]=t²+1, q[t]=7e^t, . Write a single expression for:
       • (a) h[t]+q[t]
       • (b)
       • (c)
       • 2. Let V[r]=6r², A[r]=2^r+1. Write a single expression for:
         • (a) V[r+1]
         • (b) A[r+3]
         • (c)
         • (d)
         • (e) V[A[r]]
         • (f) A[V[r]]

The program Functions contains part of the solution to these exercises. We can substitute symbolic expressions in Mathematicaand Maplefunctions and make compositions.

Problem CD-28.3

Let

Compute f[-1], f[3]. Compute other values of f[x] and sketch the graph of f[x]. The graph consists of separate "pieces," and f[x] is called a "piecewise-defined" function.

Compute f[x+1] (it will also be a piecewise defined function).

Figure CD-28.4: Variation of the b parameter in

Figure CD-28.5: Variation of the m parameter in

In short, a parameter is another letter or unknown in our formulas. Roughly speaking, it is called a parameter if it is treated as an unknown constant as far as the independent variables are concerned. We will ask you to work with parameters often. If the extra letters confuse you in a problem, choose a special case or two and work through the problem with specific numbers instead of parameters. Then, generalize your work to a letter instead of your specific numbers.

1. • 1. What is the formula in terms of parameters for the two solutions of the equation in the variable x,
     
     where a, b and c are parameters? (You learned the "quadratic formula" in high school.)
   • 2. What are the specific solutions to this equation if a=2, b=3 and c=-2?

The next problem is easy once you understand the geometric meaning of the algebra. This is what we want you to learn (or review). You can check your work with the program SlideSquash.

Problem CD-28.4 Animation of Parameters

In high school, you learned that the graph of every quadratic polynomial is a parabola. A geometrically convenient way to write the parameters is

for , , (unknown) constants.

1) Let and plot the family of curves

for .

2) Let and plot the family of curves

for .

3) Let and plot the family of curves

for .

4) Can every quadratic of the form

also be written in the form

and vice versa? Why? What are the restrictions on a and ?

5) Verify your work using the program SlideSquash from the Chapter 28 folder. The first animation there corresponds to part (1) above but plots 41 graphs for

The animation uses the graphs to make a computer "movie" of the graph sliding across the screen. The parts of the program left for you to work correspond to parts (2) and (3) above.

Double-angle formulas are special cases of the addition formulas. Take

1. Use the basic trig identities to prove the following identities:
   
   

Example CD-28.5 Conversion of Base

Here is the way to convert b^x to the natural base. Solve

For example, if b=3, then and

The natural base logs and exponentials are important in calculus. The reason they are called "natural" is because they have the simplest derivatives, whereas other bases have a "less natural" calculus. This is why we convert to base e.

Example CD-28.6 Compound Interest

Compound interest on money is given by an exponential function. If you invest a "principal" P for a time t (in years) at the annual rate r (as a decimal), compounded n times per year, then your balance is

Quarterly compounded interest at a 10% annual rate for one year would give

Example CD-28.7 Double Your Money

Logarithms enter finances when you wish to answer questions such as "How long does it take to double an investment at 10% quarterly compounded interest?" We want our balance B to equal 2P after an unknown time t,

1. You are given that . Express the function f[x]=2^x in terms of the natural base exponential e=2.7182818284590452356028747..., f[x]=2^x=e^??

2. You are given the constant . Express the function f[x]=e^x+k in the form for some constant C. (HINT: Use the (ExpSum) identity.)

3. Triple Your Money
   How long would it take at 10% interest compounded monthly to triple your investment? Solve for t.