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CLEAR Calculus

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Lab 4: Different types of limits

Lab Preparation: Answer the following questions individually and bring your write-up to class. For easy comparisons between the two contexts, create two columns for your write-up where the answers to A. are in the left column and the answers to similar questions for B. are on the right.

  1. In Lab 3, you approximated the height of a hole in a graph above the \(x\)-axis. Suppose the hole occurred at \(x = 1\), and the function, \(f\), is decreasing.

                                                                                  

  1. In class, we investigated the motion of a bolt fired from a crossbow straight up into the air with an initial velocity of 49 m/s. Accounting for wind resistance proportional to the speed of the bolt, its height above the ground is given by the equation \(h(t) = 7350 - 245t - 7350 e^{-t/25}\) meters, (with \(t\) measured in seconds).

  1. You were approximating the height of the hole above the \(x\)-axis. Why couldn't you compute that height directly from the formula for \(f\)?

  2. How did you find an underestimate? an overestimate?

  3. How did you determine an error bound for these approximations?

  1. We were approximating the speed of the bolt when \(t = 2\) seconds. Why couldn't we determine the speed directly using \(\Delta h/ \Delta t\)?

  2. How did we find an underestimate? an overestimate?

  3. How did we determine an error bound for these approximations?

Lab (read carefully): Throughout this course, we will be dealing with many different types of limits. The basic structure will be the same: a quantity being approximated, approximations, errors, and error bounds. But the mathematical objects used (and the quantities they represent) in that basic structure will be different. The key to understanding the entire course is

  1. Developing a strong understanding of the basic limit/approximation structure.

  2. Developing an ability identify the relevant quantities and interpret their meanings

This lab will help lay out this basic structure and distinguish its application in different types of limits.

For the following, work with your group on ALL problems. We encourage you to collaborate both in and out of class, but you must write up your responses to ALL problems individually .

Answer the questions in numerical order. For easy comparisons between the two contexts, create two columns for your write-up where the answers to the height of the hole questions are in the left column and the answers to similar questions for the speed context are on the right.

Limits/Approximations for the height of a hole in a graph

Limits/Approximations for instantaneous speed of a moving object

In the Lab Preparation, you answered some basic questions about approximating the height of a hole in a graph. We did not specify what representation to use (numerical, graphical, algebraic, or descriptive). So now be intentional about answering the questions using each representation. Suppose the hole occurred at \(x = 1\), and the function, \(f\), is decreasing.                                                                                             

In class, we investigated the motion of a bolt fired from a crossbow straight up into the air with an initial velocity of 49 m/s. Accounting for wind resistance proportional to the speed of the bolt, its height above the ground is given by the equation \(h(t) = 7350 - 245t - 7350 e^{-t/25}\) meters (with t measured in seconds). In class, we approximated the speed of the bolt when \(t = 2\) seconds.

  1.  Answer the following briefly using only words (no numbers except 1, algebraic expressions, etc.).

    1. What was being approximated?
    2. How did you find an underestimate? an overestimate?
    3. What is the error for your underestimate?
    4. How did you determine an error bound for these approximations?

  1.  Answer the following briefly using only words (no numbers except 2, algebraic expressions, etc.).

    1. What was being approximated?
    2. How did you find an underestimate? an overestimate?
    3. How did you determine an error bound for these approximations?

  1. Answer the following using the table of values for \(f\).

     \(x\)   0.8  0.95 1  1.03 1.1
     \(f(x)\)  4.216  4.107 dne  4.104  3.9
    1. Can you write down an exact number for the height of the hole?
    2. What is an underestimate? an overestimate?
    3. Can you write down numerical values of the errors for your two approximations? Why or why not?
    4. What is an error bound for your approximations?

  1. Answer the following using the table of values for \(h\).

    \(t\) 1 1.5 2 2.4 2.6
    \(h(t)\) 43.198 60.531 75.095 84.789 88.994
    1. Can you write down an exact number for the speed of the bolt at \(t =2\) seconds? Why or why not?
    2. What is an underestimate? an overestimate?
    3. Can you write down numerical values of the errors for your two approximations? Why or why not?
    4. What is an error bound for your approximations?

  1.  Answer the following using only algebraic expressions. The only number you can use is 1. You can use the variables \(a\) and \(b\) where \(a < 1 < b\).

    1.  What are you approximating? Hint: it’s not \(f(1)\). Use limit notation to answer the question, then choose a single letter to represent this value for subsequent questions.

    2. What is an underestimate? an overestimate?

    3. What is an error for your underestimate?

    4. What is an error bound for your approximations?

  1. Answer the following using 

    only algebraic expressions. The only number you can use is 2. You can use the variables \(t_1\) and \(t_2\) where \(t_1 < 2 < t_2\).

    1. What are you approximating? Use limit notation to answer the question, then choose a single letter to represent this value for subsequent questions.

    2. What is an underestimate? an overestimate?

    3. What is an error for your underestimate?

    4. What is an error bound for your approximations?

  1. Draw a large graph on half of a page that could represent \( y = f(x)\). Be sure to include the hole, label your axes, and mark \(x = 1\). Answer the following questions by identifying a line segment in the graph whose length is the requested value. Draw and label your answers very clearly!

    1. What are you approximating?

    2. What is an underestimate? an overestimate?

    3. What is the error for your underestimate?

    4. What is an error bound for your approximations?

  1. Draw a large graph on the other half the page of \(h\) vs. \(t\). Be sure to label your axes, and mark \(t=2\). Answer the following questions by identifying a line in the graph whose slope is the requested value OR by identifying two lines where the difference between the slopes is the requested value. Draw and label your answers very clearly!

    1. What are you approximating?

    2. What is an underestimate? an overestimate?

    3. What is the error for your underestimate?

    4. What is an error bound for your approximations?

  1. Suppose you had a formula for \(f\) and needed an approximation for the height of the hole above the \(x\)-axis accurate to within 0.001? Explain what you would do.

  1. Suppose you needed an approximation for the speed of the bolt at \(t=2\) accurate to within one m/s. Explain what to do.