Lab this week will be the first time we encounter the need to use calculus concepts, so let's review that a bit here so that it's fresh.

To be clear: you will never need to perform differentiation or integration in this class. Derivatives will always be given to you directly, and if they aren't, you can ask your TA to provide any you may need.

But you do need to understand, for example, what a derivative is and how to use it. Let's refresh on that.

One way to think about a derivative that will be especially useful in lab is that it represents how a variable y that depends on another variable x will change if x changes just a little bit. For example, let's say we have:

And we start at the point . What if we change x just a little bit to x=2.1? Of course we can solve directly:

But we could also expand the function:

And notice that since the change in x is very small, we can probably neglect the final term on the right as very very small. Then since this simplifies to:

This is very close to correct! The quantity 2x is the derivative of y with respect to x, written or .

This relation is generally true in that a good approximation for a change in y when x is small is:

Question

Lab this week will be the first time we encounter the need to use calculus concepts, so let's review that a bit here so that it's fresh.

To be clear: you will never need to perform differentiation or integration in this class. Derivatives will always be given to you directly, and if they aren't, you can ask your TA to provide any you may need.

But you do need to understand, for example, what a derivative is and how to use it. Let's refresh on that.

One way to think about a derivative that will be especially useful in lab is that it represents how a variable y that depends on another variable x will change if x changes just a little bit. For example, let's say we have:

And we start at the point . What if we change x just a little bit to x=2.1? Of course we can solve directly:

But we could also expand the function:

And notice that since the change in x is very small, we can probably neglect the final term on the right as very very small. Then since  this simplifies to:

This is very close to correct! The quantity 2x is the derivative of y with respect to x, written or .

This relation is generally true in that a good approximation for a change in y when x is small is:

BlackTom AI · Accepted Answer

The prompt presents a short exchange about calculus concepts, specifically derivatives and their interpretation. We are given two possible responses to the material: option 1 is "Got it" and option 2 is "What?".
Option 1: "Got it". This reply indicates comprehension or acknowledgment of the material just reviewed. In the context of a lab or teaching scenario, signaling that you understand the core idea (that a derivative describes how y changes with a small change in x, and that dx small allows a linear approximation) is appropriate and aligned with the goal of confirming learning.
Option 2: "What?". This response signals confusion or a need for clarification. While asking for clarification can be appropriate if concepts are unclear, in a typical setup where the goal is to review and reassure understanding, this option suggests misinterpretation or insufficient engagement with the presented material. It may reflect a barrier to proceeding with the lab tasks or indicate that the student did not follow the explanation.
To evaluate the options, we consider what each choice implies about understanding:
- If a student says Got it, they are affirming comprehension of the derivative concept and the idea of using a small change in x to approximate the change in y, which matches the instructional intent.
- If a student says What?, they may be expressing confusion about what has been said, such as the meaning of the derivative, the approximation, or the steps shown, and would warrant clarification or further explanation.
Without additional context, option 1 is the more appropriate acknowledgment of understanding the reviewed calculus concept, whereas option 2 introduces ambiguity about comprehension. However, it is important in learning to feel comfortable asking for clarification when something is not clear.
Overall, option 1 aligns with a student who has grasped the key takeaway about derivatives and small-change approximations, while option 2 points to a need for further explanation or different instructional approach to ensure comprehension.

PHY 105M MASTER Prelab 3

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