Chapter Summary

This chapter reviews the ideas ofindependentanddependentvariables andparameters. We do this in the context of some down-to-earth applications. We want to help you to develop careful working habits to use in calculus. We need you to understand function notation in order to communicate ideas.

The idea of a function is useful on a very general level.
We say that a quantity *A* (the answer) is a function of *I* (the input) over a certain domain of permissible values of *I* if each value of the input determines a unique answer.
For example, the computer plot command **Plot[ I ]** is a function whose answer is a graph and whose input

Definition:A real quantityyIs a Real Valued Function ofx,y=f[x]y(the dependent variable) is a function of another real quantityxover a certain domain of values ofx(the independent variable) if given an input value ofxin the domain, there is a unique output value associated to it. We denote this input-output relationship byy=f[x].

Additional function notation is reviewed in Section CD-28.7 below.

We want to understand the abstract definition above in concrete terms and develop common terminology. This Chapter helps you answer the

REVIEW QUESTIONS:

- 1. What are dependent and independent variables or functions? How did you study functions and variables in high school?
- 2. What is the difference between a variable and a parameter? What are some examples from high school?
- 3. How do variables and parameters arise in explicit formulas of science and mathematics? (Or, "What good are they?")
- 4.
What are the numerical, symbolic, and graphical aspects of functions with parameters?
The basic high school functions defined by explicit formulas are:

*Note:*Natural log is sometimes denoted ln*y)*in high school texts, where log*y)*denotes the base 10 log. We do not need base 10 logs.All of these functions are reviewed in this chapter.

The slope-intercept formula for a linear function,

expresses the "output" as the variable*y*in terms of the "input" variable*x*. In this case,*y*is the dependent variable, and*x*is the independent variable. The letters*m*and*b*are "parameters" that stand for constants (that do not change as*x*varies). All values of*x*are permitted as input, so the "domain" of this function consists of all real numbers.The exercises in Chapter 1 of the main text on linear functions are important building blocks for calculus. You should work them even if your high school math skills are sharp.

An advanced topic on linear functional identities is studied in the Math Background material on CD and in the Project book.

The scientific project on CO

uses linear functions to try to predict the increase of "greenhouse gases."_{2}Polynomials are functions of the form

sums of constants times integer powers of the independent variable*x*. For example,

In this case, the independent variable is*x*, the dependent variable is*y*, and the letters , and*n*are parameters. The parameters do not change as*x*varies. Again, the domain consists of all real values of*x*.Many basic geometrical formulas are single-term polynomials (monomials), for example,

**Exercise set CD-28.1**

*Simplify the following polynomial expressions by doing the indicated algebra.*- 1.
*(6x*^{2}+3x+7)+(10x^{2}-2x+12) - 2.
*(-7x*^{3}+x^{2}+5x+1)-(3x^{2}+2x-4) - 3.
*(2x+1)(x+6)* - 4.
*(x+7)(x-7)* - 5.
*(3x-4)(x+8)* *6.**(x+1)(x-1)(x+2)*Check your work with the computer using the

**Polynomials**program from the High School Review Folder in the courseware for either*Mathematica*or*Maple*.

- 1.

The area of a rectangle is a function of its length and width

In this case, area is a function of two independent variables

where the "domains" or permissible input values of*l*and*w*satisfy

*A*is a function of*l*and*w*in that once their values are specified, then*A*is determined. For example, if and , then .**Example CD-28.1***The area of a Square is a Function of the Length of its Side*

with domain

*A*is a function of a single independent variable. The formula*A=s*makes perfectly good mathematical sense when^{2}*s*is negative, but the geometrical meaning as length does not. Geometrically, the function*A=A[s]*has a domain restricted to .**Procedure**Format for Homework

In homework problems involving applications, we want you to (1) Explicitly list your variables with units and sketch a figure if appropriate. (2) Translate the information stated in the problem into formulas in your variables. (If this translation is difficult, you may not have chosen the best variables.) Often, it is helpful to balance units on both sides of an equation. (3) Formulate the question in terms of your variables and solve. (4) Explicitly interpret your solution.

Be sure to restate the question in terms of your variables. We will say more about this as the exercises become more difficult.

**Example CD-28.2***Express the Circumference of a Circle as a Function of its Area.*Solution:Step 1: We use the following variables:

Step 2: We know the formulas for circumference and area of a circle:

This translation step just translates implicit knowledge of circles.Step 3: The question asks us to find circumference as a function of area as the independent variable: Find

*C=C[A]*. Notice that we have introduced an extra variable, the radius*r*, to help us solve the problem.Begin by solving the area equation for

*r*:

This gives the solution

Step 4: The interpretation of the solution is simply that this formula gives circumference as a function of area.Notice that the circumference function is not a polynomial because it involves a square root. Square roots are special power functions studied in the next section.

**Exercise set CD-28.2**

*Find the perimeter of a square as a function of its area.**Find the surface area of a sphere as a function of its volume. (HINT: The volume of a sphere in terms of radius is and the surface area in terms of radius is .)*

Power functions are functions of the form

for constants*a*and*p*. The formula for the circumference of a circle in terms of its area (derived in the previous section) can be expressed as a power,

where*c*is the constant . In this case, the domain of the function consists of only positive values of the independent variable*A*, because zero areas do not make sense. The formula does make sense with*A=0*, but even the formula is not real valued if*A*is negative. (It can be given as a complex number, of course.)The formula for the surface area of a cube as a function of its volume may be written

because*A=6s*and^{2}*V=s*, where^{3}*s*is the length of the cube's edge, so and substituting we obtain

You may need to brush up on rules of exponents from high school. You need to be proficient at the use of these rules in order to take advantage of the powerful symbolics in calculus.**28.4.1**Rules of Exponents - 1.
Addition of Exponents:

for example, , . - 2.
Repeated Exponents:

for example, , . - 3.
Negative Exponents:

for example, , . - 4.
Fractional Exponents:

for example, , .**Exercise set CD-28.3**

*Simplify the following expressions:*- 1.
- 2.
- 3.
*(2*^{4})^{2} - 4.
*(2*^{3})^{3} - 5.
*8*^{-4} - 6.
- 7.
*8.**2*^{(33)}*Express the functions in the form**y=x*for some power^{p}*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.*

- 1.

**28.5**You may need to review some simple facts of high school trigonometry such as "SOH-CAH-TOA" and the Pythagorean Theorem.Trig Functions

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

SOH-CAH-TOA gives

If we divide the Pythagorean identity by*h*we have^{2}

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.

Trig functions also have associated inverse functions. We may know the ratio**28.5.1**Arctangent *o/a*, which is the tangent of an angle and want the angle itself. This is given by the arctangent function, which is built into most calculators.

for . For example, if*o*is fixed at 7.5, then as a function of*a*is

This formula is part of the speeding train problem (Exercise CD-28.5.6).

The notion of radian measure is one of the essential things you need to remember from high school. The reason that radian measure is important in calculus is because it relates a distance measure of angle to the distance measure of sine and cosine that locates a point on the unit circle.**28.5.2**Radian Measure 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 measureFor example, can be converted to radian measure by

An angle of has radian measure . An angle of radian measure has its degree measure computed as follows:

An angle of radian measure has degree measure . Suppose we measure an angle on the unit circle centered at the origin of the standard x-y-plane. We use the x-axis as one side and measure counterclockwise around the circle a distance , the radian measure. The*(x,y)*coordinates of the point on the unit circle where the side of the angle crosses are

**Figure CD-28.2: Sine and cosine in radian measure.**

We have and because the hypotenuse of the triangle is the unit radius of the circle. Using SOH-CAH-TOA,

These equations can even be taken as the "definition" of sine and cosine because this SOH-CAH-TOA argument can be reversed if we start from the unit hypotenuse and extend to a larger triangle.**Exercise set CD-28.4**

- Conversion
*1*. - Sine and Cosine as a Parametric Point
*(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. - Odd and Even Functions
*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, . *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.)*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.**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 pistonNotice the difference between an exponential function

and a power function

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

and the inverse

Note that some books use*ln[y]*for the natural logarithm, and a few still use log*[y]*for base 10 logarithm.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

**Exercise set CD-28.5**

*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.- 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?

Suppose the mold in my basement doubles the number of cells every hour. At midnight, there are 56 cells. At 1:00 am, there are cells. At 2:00 am, there are cells and . At 3:00 am there are cells. You can see that at integer hours, , there are**28.6.1**Exponential Growth: A First Look

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

How many mold cells should there be in hour? If we substitute , the answer is

Is this right? If so, how many in the next half hour? We should use the same rule, so

Rules of exponents say that this agrees with the integer formula

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 AlgaeOn 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

*N*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_{0}*t*. How much algae is there 6 hours later? Can algae continue to double every six hours?_{f}

for constants*a*and*b*. This expresses the intensity*I*of radiation at (angular) frequency for a body at absolute temperature*T*. The specific details are not important yet. We want to point out that*I*is built up using addition, multiplication, and division from a polynomial power formula

substituting for*x*, subtracting 1, multiplying, and dividing. When*T*is fixed, we want to use symbolic formulas of calculus to find . (The maximum is called Wien's Law of radiation.) By the end of the semester, you will think of Planck's formula as a straightforward combination of simple functions. (The way Planck found the formula is not simple. He got a Nobel Prize for the connection with science.)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.

In Planck's formula, we substitute into the natural exponential function. In old-fashioned notation, we might write**28.7.1**Composition in Function Notation

In a computer program, if we first entered both expressions and then asked for the symbolic value of*y*, we would get the answer, . Linking variables together in a chain like this is an important symbolic construction.We can think of this in terms of a function,

*y=f[x]=e*, replacing the input^{x}*x*with the expression ,

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

then the substitution of variables is the function

This notation is useful in some contexts.**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

This function rule means "Take any input value*x*and cube it." We could express this by leaving a blank space,

When we put*x*in the blank space, we get*x*. If we put the expression into the blank space, we get^{3}

The expression is obtained by subtraction of the two results,

This expression can be expanded, but that is another matter.Finally, the expression

Expansion and simplification of this expression gives

**Exercise set CD-28.6**

*Show the following:*- 1.
If
*f[x]=x*, then^{2} - 2.
If
*f[x]=x*, then^{4} *3. If**f[x]=x*, then^{5}- 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]=t*,^{2}+1*q[t]=7e*, . Write a single expression for:^{t}- (a)
*h[t]+q[t]* - (b)
- (c)
*2. Let**V[r]=6r*,^{2}*A[r]=2*. Write a single expression for:^{r}+1- (a)
*V[r+1]* - (b)
*A[r+3]* - (c)
- (d)
- (e)
*V[A[r]]* *(f)**A[V[r]]*

- (a)

- (a)

- 1.
If

The program

**Functions**contains part of the solution to these exercises. We can substitute symbolic expressions in*Mathematica*and*Maple*functions 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).In the slope-intercept formula

we "hold*m*and*b*fixed" while we vary*x*. Only varying*x*gives a single, straight-line graph. We can also plot a family of lines as we change a parameter. Here are two examples.

Figure CD-28.4: Variation of the*b*parameter in

Figure CD-28.5: Variation of the*m*parameter inIn 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.

**Exercise set CD-28.7**

- 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*?

- 1.
What is the formula in terms of parameters for the two solutions of the equation in the variable

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 ParametersIn 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.All of the identities you need to recall from high school are

but you must be able to*use*these identities. Some practice exercises using these basic identities are given in the next two sections.

The seven identities above can be used to find other identities. For example, to prove that**28.9.1**Review of High School Trig Identities

we need to use the (CircleIden), and the relationship between the trig functions, and . Simply divide the basic identity by ,

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

Half-angle formulas can be found by a similar change of variables, .**Exercise set CD-28.8**

*Use the basic trig identities to prove the following identities:*

In addition to the four high school identities,**28.9.2**Review of High School Log and Exponential Identities

you need to remember that the natural log and exponential are inverse functions, that is,

We may write the inverse relationship more operationally as

Using Log we can convert any other base*b*to an expression in base^{x}*e*as follows:**Example CD-28.5***Conversion of Base*Here is the way to convert

*b*to the natural base. Solve^{x}

The Law of Repeated Exponents gives,

for this value of the constant*k*.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

For example, suppose you invest $500 at 10% (*r*= 0.1) compounded each 1/2 year. In six months your balance is

If this amount all now earns interest, after six more months, you have

This is two compoundings in a year,

Notice that this gives $1.25 more interest than simple interest for a year, because the six month interest earned interest for the second half of the year.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*,

Take the log of both sides and use the (LogPower) identity to solve for*t*,

Simple noncompounded interest at 10% would take 10 years to double, whereas it takes only 7 years with compound interest. In 10 years compounded quarterly, we have 2.685 times our original investment.**Exercise set CD-28.9**

*You are given that . Express the function**f[x]=2*in terms of the natural base exponential^{x}*e=2.7182818284590452356028747...*,*f[x]=2*^{x}=e^{??}*You are given the constant . Express the function**f[x]=e*in the form for some constant^{x+k}*C*. (HINT: Use the (ExpSum) identity.)- Triple Your Money
*How long would it take at 10% interest compounded monthly to triple your investment? Solve for**t*.

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