1.9.1: Resources and Key Concepts

Math 400: Calculus I - Differential Calculus · LibreTexts

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    From LibreTexts, under CC BY-SA, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    Key Concepts

    Definitions

    • Difference Quotient:
      • Given a function \(f\) defined on an interval \(I\) containing \(a\), if \(x \neq a\) is in \(I\), then \(Q = \frac{f(x)-f(a)}{x-a}\) is a difference quotient.
      • If \(h \neq 0\) is chosen so that \(a+h\) is in \(I\), then \(Q = \frac{f(a+h)-f(a)}{h}\) is a difference quotient with increment \(h\).
    • Tangent Line: Let \(f(x)\) be a function defined in an open interval containing \(a\). The tangent line to \(f(x)\) at \(a\) is the line passing through the point \((a, f(a))\) having slope:
      • \(m_{tan} = \displaystyle \lim_{x \to a} \frac{f(x)-f(a)}{x-a}\)
      • or equivalently, \(m_{tan} = \displaystyle \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}\)
      provided these limits exist.
    • Derivative: Let \(f(x)\) be a function defined in an open interval containing \(a\). The derivative of the function \(f(x)\) at \(a\), denoted by \(f'(a)\), is defined by:
      • \(f'(a) = \displaystyle \lim_{x \to a} \frac{f(x)-f(a)}{x-a}\)
      • or alternatively, \(f'(a) = \displaystyle \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}\)
      provided this limit exists.
    • Differentiation: The process of finding a derivative.
    • Average Velocity: If \(s(t)\) is the position of an object moving along a coordinate axis, the average velocity of the object over a time interval \([a,t]\) (or \([t,a]\)) is given by the difference quotient \(v_{avg} = \frac{s(t)-s(a)}{t-a}\).
    • Instantaneous Velocity: As the values of \(t\) approach \(a\), the values of \(v_{avg}\) approach the value we call the instantaneous velocity at \(a\). Denoted \(v(a)\), it is given by \(v(a) = s'(a) = \displaystyle \lim_{t \to a} \frac{s(t)-s(a)}{t-a}\).
    • Instantaneous Rate of Change: The instantaneous rate of change of a function \(f(x)\) at a value \(a\) is its derivative \(f'(a)\).

    This page titled 1.9.1: Resources and Key Concepts was last modified on Mon, 15 Sep 2025 11:39:01 GMT and is shared under a CC BY-SA license and was authored, remixed, and/or curated by Roy Simpson.