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Calculus I MATH 131 Fall 2025

Section Week 2 (9/2 - 9/5)

This week we cover sections 1.3 and some of 1.4 of the textbook. This introduces the derivative at a point, and the derivative function.
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Handout Tuesday, 9/2

Today we will start by discussing any questions about the practice problems due tonight. Then some discussion section 1.3 on the derivative at a point. Finally, we will do the first Learning Target Quiz.
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  • Look at question 2 from last Friday’s activity. We want to evaluate that limit. Notice that it is a very particular type of limit. It is a limit of an average velocity between two points. What are they?
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  • The points are \((a, f(a))\) and \((a+h, f(a+h))\text{,}\) where \(f(x) = x^2 + 5x\text{.}\)
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  • Look at the preview activity for 1.3. Go over this carefully.
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  • We will want to evaluate these sorts of limits a bunch. But what does \(\d\frac{f(a+h) - f(a)}{h}\) look like for different functions? Try this out with a bunch of different examples.
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  • The limit of this difference quotient is called the derivative of \(f\) at \(a\), and written \(f'(a)\text{.}\) Here we are thinking of \(a\) as a specific number. So examples would be \(f'(2)\) or \(f'(0)\text{.}\)
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Handout Wednesday, 9/3

Learning target quizzes discussion. Reminder: you try this again so that you can demonstrate mastery. Your next opportunity is next Tuesday. Let’s talk a little bit about instantaneous velocity.
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Let’s continue investigating the derivative of a function at a point.
  • Look at the interactive applet the textbook links to. Compare this to the definition of the derivative we discussed in class and the idea of the instantaneous velocity.
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  • We can always approximate a derivative at a point by approximating the limit of the average rate of change. In other words, approximating the difference quotient at that point.
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  • Sometimes, we can find the derivative exactly by computing that limit exactly.
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  • Let’s think of this as two distinct steps: First, we want to find an expression whose limit is the derivative at a point. Second, we want to evaluate that limit.
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  • Here are some examples. For each function below, write a limit that represents the derivative at the given point.
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  1. Let \(f(x) = \sqrt{x+1}\text{.}\) Find an expression for \(f'(3)\text{,}\) the derivative at \(a = 3\text{.}\) Then repeat for \(f'(8)\text{.}\)
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  2. Let \(f(x) = x^2 + 5x\text{.}\) Find an expression for \(f'(2)\text{,}\) the derivative at \(a = 2\text{.}\)
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  3. Let \(f(x) = x^3 - 2x + 1\text{.}\) Find an expression for \(f'(1)\text{,}\) the derivative at \(a = 1\text{.}\)
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  4. Let \(f(x) = \sin(x) + \cos(x)\text{.}\) Find an expression for \(f'(0)\text{,}\) the derivative at \(a = 0\text{.}\)
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Pay attention to how the substitution of the function and the difference quotient interact.
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Now, evaluate some of the limits we found. Note that the last one... no clue what to do there.
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Handout Friday, 9/5

Today we start looking at section 1.4, thinking about the derivative as a function.
  • Start with an example. Let \(f(x) = x^2 + 1\text{.}\) Find \(f'(3)\text{.}\) Then, what is \(f'(4)\text{?}\) Can we also find \(f'(42)\text{?}\)
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  • In fact, we can find \(f'(a)\) for any number \(a\text{.}\) One way to think of this is that we are creating a rule to help us find a lot of different derivatives. We are giving a recipe. No matter what \(a\) is, we can always find \(f'(a)\) by simply...
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  • But really we are doing more. At any point in the domain of \(f\text{,}\) we can find a derivative. That derivative changes when \(a\) changes. So there is actually a function of derivatives.
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  • We can describe a function in multiple ways. A formula. A table. A graph.
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  • Look at a graph of a function. At any input, we can ask: what is the slope of the line tangent to the graph? We can then collect these answers in a table. That is describing the derivative function! We can also graph these values to get a graph of \(f'(x)\)
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  • Let’s repeat this with a few functions.
    1. a piecewise linear function.
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    2. A parabola.
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    3. \(\displaystyle y=x^3\)
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    4. \(\displaystyle f(x) = \sin(x)\)
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    5. \(\displaystyle f(x) = e^x\)
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