Chapter 14: Exercises
Exercise 14.1
- 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 14.2
- 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
equal π? Is it a close approximation of π ? Justify your answer.
Exercise 14.3
- 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 14.4
- Use software to create an animated representation of Figure 14.4. Describe the benefits of your creation as contrasted with pictures such as those in Figures 14.3 and 14.4.
Exercise 14.5
- Prepare a lesson plan designed to have students learn how the secant line approaches the tangent line.
Exercise 14.6
- Create a dynamic demonstration of Figure 14.4 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.
Chapter 14: Problem Solving Challenges
Question 1
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?
Answer [Click to reveal...]
Answer: 
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.
Subtract the second equation from the first to get 14(2x + 4) = 2(14 + 2r), which simplifies to
Substitute into the first and third and get (x + 9)² + y² = (15 + 7x)² and x² + (y - 12)² = (14 + 7x)². Expanding and moving around variables, these simplify to y² = 48x² + 192x + 144 and y² - 24y = 48x2 + 196x + 52. Subtracting, get –24y + 144 = 4x + 52, which simplifies to 
. Substitute 
to get 
Substitute these 
. After algebra, this becomes 
The only positive solution of this is 168/157.
Question 2
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?
Answer [Click to reveal...]
Answer: 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.
Chaper 14: Videos