Calculus I, Pre-Lab 5: The Derivative as a Function

Name:

The Pre-lab is intended to encourage you to prepare for the Lab, so answer these questions in your

own words, and hand this sheet in at the beginning of lab.

  1. Activity #1 from Lab 5 asks you to read pages 115-118 in the text. Take a close look at the two graphs on page 116. Suppose that for the graph of the function f (the blue graph) the y-value represents the height of a roller-coaster ride at any time x between 0 and 5. (Use your imagination to see a little roller-coaster car rolling along the graph of f.) Now the graph of the derivative f' (the orange graph) would represent the rate of change in height with respect to time, or the verticalvelocityof the roller-coaster car. Explain what the following facts represent in this roller-coaster situation.
    i.) the value of the derivative is zero at A', B', and C'…
    ii.) the value of the derivative is 1 when x=1 (between A' and B') …
    iii.) the value of the derivative is positive between A' and B' …
    iv.) the value of the derivative is negative between B' and C' …
  1. Read activity #1 of Lab 5 and look at EXAMPLE 6 p. 116 -117. Do you think you could have done the algebra by hand that was required in order to find the limit of SecantSlope and get the derivative?
    Following EXAMPLE 6, complete exercise #19 on p. 121. Show all the algebra involved in your work.
    For exercise #19 on p. 121, EXAMPLE 6and EXAMPLE 7 on p. 116-117, complete the following table

Domain of / Domain of
EXAMPLE 6
EXAMPLE 7
Exercise #19
  1. As you see in activity #1 and #2 from Lab 5, we often use different notations to represent the derivative of a function. Look through section 2.2 p. 124-131 and notice the summarizing table of differentiation formulas on page 131. Fill in the following table using the different notations to write the differentiation rules: (the symbol is used for multiplication as in MapleV)

Constant Multiple Rule / Product Rule / Quotient Rule

notation / / /
notation / / /
notation / / /

  1. In Lab 5 Activity #3, you are asked to compute derivatives for some piecewise linear functions. Take a moment to think about what the derivative of a linear function means.
  2. We have considered problems called the tangent line problem, the instantaneous velocity problem, heat flow, and a host of problems in section 2.3, pages 134-143 that involve a common approach. Let us take a crack at summarizing this approach. These problems all involve a rate of change, and our physical reasoning leads us to construct an average rate of change (you know the slope of the secant line). Yet, we are asked to find rates at a single point, so we take an appropriate limit to pass from an average rate of change to an instantaneous rate of change. Since this process appears so often we have decided to give it a name, the derivative. Now look at one goal stated on the syllabus:

Gain knowledge about the infinite processes of limit and differentiation.

Write a paragraph explaining how differentiation is an infinite process. Use your own words and write in sentences that make sense.