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Calculus I MATH 131 Fall 2025

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Handout Monday 9/8

We are still working on 1.4 today. Thinking about the derivative as a function of slopes or of instantaneous rates of change. So instead of finding \(f'(a)\text{,}\) we are finding a function \(f'(x)\text{.}\)
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Here are two things to try:
  • Sketch the graph of \(f(x) = 4 + 3x - x^2\) and guess its derivative \(f'(x)\text{.}\)
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  • Let \(f(x) = 4 + 3x - x^2\text{.}\) Find \(f'(x)\) using the limit definition.
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  • Interesting, we get the equation of a line for the derivative. This line is NOT the tangent line: it is a line that gives slopes of tangent lines at any given point.
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  • Let’s try the same thing with \(f(x) = x^3 + 1\text{.}\)
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  • Graphically, what do we guess the derivative function will look like?
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  • Using the limit definition, find \(f'(x)\text{.}\) We will need to multiply out \((x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3\text{.}\)
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