# Chapter 14 – Pre-Calculus and Calculus

Beginning with a conference at Tulane University in January 1986, there developed in the mathematics community a sense that calculus was not being taught in a way befitting a subject that was at once the culmination of the secondary mathematics curriculum and the gateway to collegiate science and mathematics.

Pre-calculus can be considered an extension of Algebra II and trigonometry. It fills in the gaps and reviews students in preparation for calculus. It can be approached from a function and/or graphic perspective. Due to the advances in technology, a combined approach is a more feasible option when you consider the dynamics of the tools available to teachers and students. When it comes to calculus, the sad truth is that calculus is not a realization of secondary school preparation and an exciting beginning to future mathematical study. Instead, calculus continues to serve as an exit from the study of mathematics and related subject areas for many students

## Exercises

### Exercise 14.1

- It is said that you learn something best when you teach it. That is true{em}but does that give you license to use a class as guinea pigs? How much should you know about a topic before embarking on the study of it with a class?
- You undoubtedly have had some logic as a part of your undergraduate program. Out of that information, what could be inserted into a precalculus course and why? If you have not had logic beyond basic truth tables, research the subject to determine what should be included in the precalculus class. As a part of your research, you should include a description of how much time it will take you to learn the material well enough to teach it.
- Part 2 of this exercise mentions learning material prior to teaching it. Does this imply you will be lecturing? Is lecturing more acceptable in an advanced course, because these are more capable students and there is so much information to cover? Why or why not?

### Exercise 14.2

Using a graphing calculator or software, do the following:

- Graph one of each type of the functions:

*f*(*x*) = Constant

*f*(*x*) = Linear

*f*(*x*) = Quadratic

*f*(*x*) = Polynomial

*f*(*x*) = Rational

*f*(*x*) = Exponential

*f*(*x*) = Logarithmic

Select any three of these functions and describe their similarities and differences. List the main points you would bring out to students if you were comparing and contrasting the selected three in a precalculus class.

- Graph at least four trigonometric functions:

*f*(*x*) = sin(*x*)

*f(x)* = cos(*x*)

*f(x)* = tan(*x*)

*f(x)* = csc(*x*)

*f(x)* = sec(*x*)

*f(x)* = cot(*x*)

It is rather common to have an elaborate explanation of the development of sine using a unit circle. Use the unit circle to explain why one of the other trigonometric functions behaves as it does. Build your discussion in the form of a lesson plan. You should incorporate technology.

- Graph one of each type of the functions:

*f*(*x*) = Power

*f*(*x*) = Reciprocal

*f(x)* = Absolute value

*f(x)* = Trigonometric inverse

*f(x)* = Greatest-integer

*f(x)* = Piecewise

Which of these functions were you exposed to prior to a precalculus or calculus course? Is it reasonable to expect similar backgrounds from students taking the course at the time you are reading this question? Why or why not?

- Which of the graphs in parts 1–3 of this exercise accepts a vertical or horizontal shift?
- Which of the graphs in parts 1–3 of this exercise can be compressed or stretched? Describe an example of one of the compressions or stretches in the format that would be appropriate for students in a precalculus class.
- What change or rotation would make one graph from each of parts 1–3 of this exercise no longer fit the definition of a function? Develop a lesson plan for each of the three examples you select.
- Do you think a beginning teacher should be assigned to teach a precalculus class? Why or why not?

### Exercise 14.3

- The text discusses dealing with
*f*(*g*(*x*)) when*f*(*x*) = 2*x*and*g*(*x*) = 3*x*– 4. Describe the advantages and disadvantages of approaching this by using a graphing calculator or function plotting software. - When
*f*(*x*) =*x*^{2}+ 5*x*– 6 and*g*(*x*) =*x*^{2}+ 7, what is*f*(*g*(*x*))? What is*g*(*f*(*x*))? Does*f*(*g*(*x*)) =*g*(*f*(*x*)) in general? When does*g*(*f*(*x*)) =*f*(*g*(*x*)), if ever? Develop a lesson plan for this problem. It should include technology, and you should assume the students have the appropriate skills and background with the calculator or software selected. - Describe how you would determine a class background on compound functions.
- How extensive should the treatment of compound functions be in a precalculus class? Defend your position.

### Exercise 14.4

- Find a different derivation of the quadratic formula. Compare and contrast it with the one presented here. Are the differences significant or mostly cosmetic and author preference? How should secondary students react to these different avenues to arrive at the same destination? Why?
- Can the quadratic formula be introduced to students prior to the traditional algebra class? Defend your position.
- Are there other derivations like that of the quadratic formula given here that students should have seen in prerequisite courses for college algebra? If you say yes, list at least three and discuss their value to the precalculus course. If you say no, defend your position, part of which should include a rationalization for why the particular derivation should or should not be included in the precalculus course.

### Exercise 14.5

- Create a lesson designed to teach a class about a quadratic function (parabola) that is symmetric and opens downward.
- Using part 1 of this exercise, modify the lesson so the function shows the other two possible cases of roots. Do you think this is too much to cover in one day? Why or why not?

### Exercise 14.6

- Summarize the situations in the lives of Newton and Leibniz that impacted their association with each other.
- Find two historical texts on mathematics by different authors. Who is credited with the discovery of calculus in each text?
- Name two other individuals who are credited with making significant contributions to the development of calculus.

### Exercise 14.7

- Describe how technology has been used as a part of your learning calculus. List the strengths and weaknesses of your learning process in calculus. What would you suggest to make the course better through the use of technology?
- Define your position on the use of technology in the calculus class at the secondary level. Advanced Placement (AP) exams are now written assuming technology is available for the student. How much emphasis should technology receive in the learning of calculus?
- Should tools like the Casio CFX-9850, TI-84 Plus, and WolframAlpha be permitted in high school calculus classes? Why or why not? (It is likely that ETS will permit their use within a few years of the publication of this text.)
- Should calculus class time be used to instruct students on the use of technology? Why or why not?
- The AP examinations now permit the use of graphing calculators, but exclude the use of computers and software. Is that a reasonable position in light of the power of some graphing calculators? Why or why not?
- How would you respond to a member of a college mathematics department who was criticizing you for using graphing calculators, the Casio CFX-9850, TI-84 Plus, and WolframAlpha, in your high school calculus class because the students are not permitted to use those tools in their college calculus class?
- Describe a calculus situation that would be negatively impacted by a student’s assumption that two items are equal when they are actually only approximately equal.
- Does $$\frac{22}{7}$$ equal ? $$\pi $$ Is it a close approximation of $$\pi $$ Justify your answer.

### Exercise 14.8

- Draw a curve on a sheet of paper. Select two points on it. Let one point be fixed and let the other “move” along the curve toward the fixed one. Fold the paper several times between the fixed and moving points to depict the secant line approaching the tangent line.
- Create a lesson plan designed to have students perform the activity described in part 1 of this exercise.
- Are activities like the one in part 1 of this exercise appropriate for a secondary calculus class? Why or why not?

### Exercise 14.9

- Use software to create an animated representation of Figure 14.12 in the text. Describe the benefits of your creation as contrasted with pictures such as those in Figures 14.13 and 14.14 in the text.

### Exercise 14.10

- Prepare a lesson plan designed to have students learn how the secant line approaches the tangent line.

### Exercise 14.11

- Create a dynamic demonstration of Figure 14.15 in the text where the tangent line is horizontal and the secant line approaches the tangent line.
- Create a lesson plan in which the dynamic demonstration you built in Exercise 1 of this section is the foundation of your explanation of the basics of Rolle’s theorem.

## Problem Solving Challenges

- What would be the units digit of 3
^{9999}?

Hint: Look for a pattern and begin with 3^{0}, 3^{1}, 3^{2}, 3^{3}, etc

## Answer

Look for a pattern by making a table of smaller values.

Remainder when power is divided by 4 | |

3^{0} = 1 | $$\frac{0}{4}=0$$ remainder 0 |

3^{1} = 3 | $$\frac{1}{4}=0$$ remainder 1 |

3^{2} = 9 | $$\frac{2}{4}=0$$ remainder 2 |

3^{3} = 27 | $$\frac{3}{4}=0$$ remainder 3 |

3^{4} = 81 | $$\frac{4}{4}=1$$ remainder 0 |

3^{5} = 243 | $$\frac{5}{4}=1$$ remainder 1 |

3^{6} = 729 | $$\frac{6}{4}=1$$ remainder 2 |

3^{7} = 2187 | $$\frac{7}{4}=1$$ remainder 3 |

3^{8} = 6561 | $$\frac{8}{4}=2$$ remainder 0 |

3^{9} = 19683 | $$\frac{9}{4}=2$$ remainder 1 |

Notice that the units digits are either 1, 3, 9, or 7. Therefore, you know that the answer must be one of 4 digits. Divide 9999 by 4 and the remainder yields the clue to this puzzle. The possible remainders when dividing by 4 are 0, 1, 2, or 3. When dividing the power by 4 and the remainder is 0, the units digit is 1. When dividing the power by 4 and the remainder is 1, the units digit is 3. When dividing the power by 4 and the remainder is 2, the units digit is 9. When dividing the power by 4 and the remainder is 3, the units digit is 7. Since 9999/4 = 2499 remainder 3, the units digit will be 7.

- There is a hill that is two miles from the base to the top on the north side and one mile from the top to the bottom on the south side. Jack has an old car that can only go up the hill at an average speed of 40 miles per hour, but he can race down the hill as fast as he desires. What will Jack’s average speed have to be going down the south side of the hill to average 60 miles per hour over the entire hill?

Hint: It might be easier to think of the hill as 80 miles on the north side and 40 miles on the south side for a total of 120 miles.

## Answer

Answer/solution: It is not possible!

It might be easier to think of the hill as 80 miles on the north side and 40 miles on the south side for a total of 120 miles. Averaging 60 miles an hour means the trip can be made in 2 hours. Jack’s car can do only 40 mph uphill so Jack uses 2 hours to get to the top of the hill. He has used all the allowed time and still has to go down the south side, meaning it is an impossible situation.

- You have three circles of radii 6, 7, and 8 units. Each is tangent to the other two. There is a circle inscribed in the central region created by the three larger circles. This little circle is tangent to the other three as well. What is the radius of this little inscribed circle?

Hint: Drawing a picture on a coordinate plane can help. Call the radius of the inner circle r. The center is 8 + r away from the center of the circle with radius 8, and 7 + r and 6 + r from the other two respectively.

## Answer

Answer/solution:

$$\frac{168}{157}$$We will call the radius of the inner circle *r*. The center is 8 + *r* away from the center of the circle with radius 8, and 7 + *r* and 6 + *r *from the other two respectively. Connect the centers of the three larger circles to get a triangle, which happens to be a 13, 14, 15 triangle. Put this triangle on the coordinate plane, with the side of length 14 on the x-axis and the altitude to this side, including the vertex on the y-axis. Put the 15 side to the left of the x-axis and use the fact that the altitude on the y-axis is 12, the three vertices are (–9,0), (5,0), and (0,12). From the earlier idea that the center of the inner circle is a certain distance away, basically, the center is 8 + r away from (–9,0), 7 + *r* away from (0,12), and 6 + *r* away from (0,5). Use these points and centers to get circle equations. The one spot where the three circles intersect is the center of the inner circle.

(*x* + 9)^{2 }+ *y*^{2 }= (8 + *r*)^{2} and

(*x* – 5)^{2 }+ *y*^{2} = (6 + *r*)^{2} and

*x*^{2 }+ (*y* – 12)^{2 }= (7 +* r*)^{2}.

Subtract the second equation from the first to get 14(2*x* + 4) = 2(14 + 2*r*), which simplifies to r = 7*x* + 7 or $$x=\frac{r-7}{7}$$. Substitute into the first and third and get (*x* + 9)^{2 }+ *y*^{2} = (15 + 7*x*)^{2} and *x*^{2 }+ (*y* – 12)^{2 }= (14 + 7*x*)^{2}. Expanding and moving around variables, these simplify to *y*^{2} = 48*x*^{2 }+ 192*x* + 144 and *y*^{2 }– 24*y* = 48*x*^{2 }+ 196*x* + 52. Subtracting, get –24*y* + 144 = 4*x* + 52, which simplifies to $$y=\frac{23-x}{6}$$. Substitute $$x=\frac{r-7}{7}$$ to get $$y=\frac{168-r}{42}$$. Substitute these x and y values in terms of *r* into (*x* + 9)^{2 }+ *y*^{2} = (8 + *r*)^{2} to get $${(\frac{r-7}{7}+9)}^{2}+{\left(\frac{168-r}{42}\right)}^{2}$$ = (8 + r)^{2}. After algebra, this becomes $$\frac{1727}{1764}{r}^{2}+\frac{292}{21}r-16=0$$. The only positive solution of this is 168/157.

- Observe the 400-digit number:

1234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890123456789012345678901234567890

First, eliminate all digits that are in odd-numbered places, starting at the left-most place. Repeat this process with the remaining 200-digit number. Continue this process until all digits are gone. What was the last number to be eliminated?

Hint: Try a much smaller problem using 12345678901234567890

## Answer

Answer/solution: 6

After the first set of numbers is crossed off, the remaining 200-digit number is 2468023468…24680. After the second set is eliminated the remaining 100-digit number is 4826048260…48260. Next would leave the 50-digit number 8642086420…86420 followed by the 25-digit number 6284062840..62840 then the 12-digit number 246802468024 and the 6-digit number 482604 then 864 and lastly 6.

## Additional Learning Activities

There is no Additional Learning Activities for this chapter.