diff --git a/docs/calc.html b/docs/calc.html index f1b95ed..1d86661 100644 --- a/docs/calc.html +++ b/docs/calc.html @@ -472,12 +472,18 @@

Colbyn’s Calculus Notes

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  • Solutions + +
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    Colbyn’s Calculus Notes

    name: 'Calculus - Definitions', path: 'calc/definitions.html' }, + { + name: 'Calculus - Solutions', + path: 'calc/solutions.html' + }, ]; let ul_node = document.querySelector("header#root > ul"); console.assert(ul_node); @@ -1007,7 +1017,51 @@

    Tangent

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    Intermediate Value Theorem

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    The Limit of a Function (§1.5)

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    Vertical Asymptote (Definition §1.5.6)
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    The vertical line \(x = a\) is called the vertical asymptote of the curve \(y = f(x)\) if at least one of the following statements is true:

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    +\[\begin{equation} +\begin{split} + \lim_{x \to a} f(x) &= \infty \\ + \lim_{x \to a} f(x) &= -\infty \\ + \lim_{x \to a^{-}} f(x) &= \infty \\ + \lim_{x \to a^{-}} f(x) &= -\infty \\ + \lim_{x \to a^{+}} f(x) &= \infty \\ + \lim_{x \to a^{+}} f(x) &= -\infty +\end{split} +\end{equation}\] +
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    Continuity (§1.5)

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    The Intermediate Value Theorem (IVT)
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    Suppose that \(f\) is continuous on the closed interval \(\lbrack a, b \rbrack\), and let N be any number between \(f(a)\) and \(f(b)\), where \(f(a) \neq f(b)\), then there exists a number \(c\) in \((a, b)\) such that \(f(c) = N\).

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    Or alternatively, suppose \(f\) is a function that is continuous at every point in the interval \(\lbrack a, b \rbrack\):

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      \(f\) will take on every value between \(f(a)\) and \(f(b)\) over the interval.

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      For any L between the values \(f(a)\) and \(f(b)\), there exists a number \(c\) in \(\lbrack a, b \rbrack\) for which \(f(c) = L\).

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    Note

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    • The intermediate value theorem states that if a continuous function attains two values, it must also attain all values in between these two values.

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      The Intermediate Value Theorem states that a continuous function takes on every intermediate value between the function values \(f(a)\) and \(f(b)\).

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      The statement of the theorem has multiple requirements, all of which are necessary for the conclusion to hold.

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    Tip
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    The statement of the theorem has multiple requirements, all of which are necessary for the conclusion to hold.

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    The IVT can be used to show that a root exists, if given some interval between \(a\) and \(b\):

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      \(f(a)\) is negative

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      \(f(b)\) is positive

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    or

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      \(f(a)\) is positive

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      \(f(b)\) is negative

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    then therefore, if the function is continuous, then there must be a value for which some \(x\) in \(f(x) = 0\) exists. I.e. that a root exists. Which should make sense.

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    More generally, this can be extended to any value between some interval. For instance, if:

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      \(f(a) = 3\)

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      \(f(b) = 9\)

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    Then there must exist some value of \(x\) for which:

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      \(f(x) = 6\)

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    Maximum and Minimum Values

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    Maximum and Minimum Values (§3.1)

    The extreme Value Theorem

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    Critica

    A critical number of a function \(f\) is a number \(c\) in the domain of \(f\) such that either \(f^\prime(c) = 0\) or \(f^\prime(c)\) does not exist.

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    To find an absolute maximum or minimum of a continuous function on a closed interval, we note that either it is local [in which case it occurs at a critical number by (7)] or it occurs at an endpoint of the interval, as we see from the examples in Figure 8. Thus the following three-step procedure always works. See The Closed Interval Method.

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    Critica
    Note

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      Critical numbers of \(f\) occur when \(f^\prime(c) = 0\), or when \(f^\prime(c) = \mathrm{undefined}\).

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    • Not every critical number gives rise to a maximum or a minimum.

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    To find an absolute maximum or minimum of a continuous function on a closed interval, we note that either it is local [in which case it occurs at a critical number by (7)] or it occurs at an endpoint of the interval, as we see from the examples in Figure 8. Thus the following three-step procedure always works. See The Closed Interval Method.

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    The Closed Interval Method

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      I.e. critical numbers of \(f\) occur when \(f^\prime(c) = 0\),

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      I.e. critical numbers of \(f\) occur when \(f^\prime(c) = 0\), or when \(f^\prime(c) = \mathrm{undefined}\)

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    The Mean Value Theorem

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    The Mean Value Theorem (§3.2)

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      Miscellaneous

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      Theorem (5)
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      Theorem (§3.2.5)

      If \(f^\prime(x) = 0\) for all x in an interval \((a, b)\), then f is constant on \((a, b)\).

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      Miscellaneo

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      Corollary (7)
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      Corollary (§3.2.7)

      if \(f^\prime(x) = g^\prime(x)\) for all \(x\) in an interval \((a, b)\), then \(f - g\) is constant on \((a, b)\); that is, \(f(x) = g(x) + c\) where \(c\) is a constant.

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      Solutions

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      Derivative of Absolute Value Functions (\({\scriptscriptstyle \frac{\mathrm{d}}{\mathrm{d}x}}\; \lvert x \rvert\))

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      +\[\begin{equation} +\begin{split} + \lvert x \rvert &= \sqrt{x^2} \neq x \\ + {\scriptscriptstyle \frac{\mathrm{d}}{\mathrm{d}x}}\; \lvert x \rvert &= \frac{x}{\lvert x \rvert} = \frac{\lvert x \rvert}{x} +\end{split} +\end{equation}\] +
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    Derivative of Absolute Value Functions (\({\scriptscriptstyle \frac{\mathrm{d}}{\mathrm{d}x}}\; \lvert x \rvert\))

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    +\[\begin{equation} +\begin{split} + \lvert x \rvert &= \sqrt{x^2} \neq x \\ + {\scriptscriptstyle \frac{\mathrm{d}}{\mathrm{d}x}}\; \lvert x \rvert &= \frac{x}{\lvert x \rvert} = \frac{\lvert x \rvert}{x} +\end{split} +\end{equation}\] +
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    + + + + + \ No newline at end of file diff --git a/docs/index.html b/docs/index.html index a3e562e..1b269d7 100644 --- a/docs/index.html +++ b/docs/index.html @@ -547,6 +547,10 @@

    Colbyn’s Math Notes

    name: 'Calculus - Definitions', path: 'calc/definitions.html' }, + { + name: 'Calculus - Solutions', + path: 'calc/solutions.html' + }, ]; let ul_node = document.querySelector("header#root > ul"); console.assert(ul_node); diff --git a/docs/trig.html b/docs/trig.html index efdd2e0..07dda8e 100644 --- a/docs/trig.html +++ b/docs/trig.html @@ -562,6 +562,10 @@

    Colbyn’s Trigonometry Notes

    name: 'Calculus - Definitions', path: 'calc/definitions.html' }, + { + name: 'Calculus - Solutions', + path: 'calc/solutions.html' + }, ]; let ul_node = document.querySelector("header#root > ul"); console.assert(ul_node);