Fix prose typos and grammatical errors across 26 .qmd files
Scan of all .qmd files under quarto/ found 43 genuine prose errors in 26 files. Changes by category: Duplicate words removed: - alternatives/makie_plotting.qmd: 'can can' -> 'can'; 'the the' -> 'the' - basics/vectors.qmd: 'the the' -> 'the'; 'which which' -> 'which' - derivatives/condition.qmd: 'the the' -> 'the' - derivatives/derivatives.qmd: 'At at' -> 'At' - derivatives/lhospitals_rule.qmd: 'the the' -> 'the' - derivatives/mean_value_theorem.qmd: 'the the' -> 'the' - derivatives/more_zeros.qmd: 'the the' -> 'the' - differentiable_vector_calculus/matrix_calculus_notes.qmd: 3 instances - differentiable_vector_calculus/plots_plotting.qmd: 'The the' -> 'The' - differentiable_vector_calculus/vector_fields.qmd: 'the the'; 'a a' -> 'a' - differentiable_vector_calculus/vector_valued_functions.qmd: 'the the' - differentiable_vector_calculus/vectors.qmd: 'the the' - integral_vector_calculus/div_grad_curl.qmd: 'the the' - integral_vector_calculus/double_triple_integrals.qmd: 'The the'; 'over over' - integral_vector_calculus/line_integrals.qmd: 'the the' - integral_vector_calculus/review.qmd: 2x 'the the' - integral_vector_calculus/stokes_theorem.qmd: 2x 'the the' - integrals/improper_integrals.qmd: 'the the' - integrals/substitution.qmd: 'that that' -> 'that' - integrals/surface_area.qmd: 'the the' - precalc/functions.qmd: 'that that width' -> 'that the width' Article (a/an) corrections: - basics/calculator.qmd: 'A overview' -> 'An overview' - basics/vectors.qmd: 'A example' -> 'An example'; 'are a implemented' -> 'are implemented' - derivatives/optimization.qmd: 'an trigonometry-free' -> 'a trigonometry-free' - derivatives/taylor_series_polynomials.qmd: 'a error' -> 'an error' - differentiable_vector_calculus/scalar_functions_applications.qmd: 'a optimization' -> 'an optimization' - differentiable_vector_calculus/vectors.qmd: 'a another' -> 'another'; 'a an angle' -> 'an angle' - integral_vector_calculus/double_triple_integrals.qmd: 'a azimuthal' -> 'an azimuthal' - integral_vector_calculus/line_integrals.qmd: 'an current' -> 'a current'; 'an simply' -> 'a simply'; 'an rotational' -> 'a rotational' - limits/intermediate_value_theorem.qmd: 'an local' -> 'a local'; 'an minimum' -> 'a minimum' Other typos: - basics/calculator.qmd: 'is is not' -> 'is not'; 'chicken is an unfamiliar' -> 'chicken in an unfamiliar'; 'but you the oven' -> 'but the oven' - differentiable_vector_calculus/matrix_calculus_notes.qmd: 'symmteric' -> 'symmetric'
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@@ -750,7 +750,7 @@ Related but different is the concept of a relative of *local extrema*:
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::: {.callout-note icon=false}
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## Local maximum, local minimum
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A local maxima for $f$ is a value $f(c)$ where $c$ is in **some** *open* interval $I=(a,b)$, $I$ in the domain of $f$, and $f(c)$ is an absolute maxima for $f$ over $I$. Similarly, an local minima for $f$ is a value $f(c)$ where $c$ is in **some** *open* interval $I=(a,b)$, $I$ in the domain of $f$, and $f(x)$ is an absolute minima for $f$ over $I$.
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A local maxima for $f$ is a value $f(c)$ where $c$ is in **some** *open* interval $I=(a,b)$, $I$ in the domain of $f$, and $f(c)$ is an absolute maxima for $f$ over $I$. Similarly, a local minima for $f$ is a value $f(c)$ where $c$ is in **some** *open* interval $I=(a,b)$, $I$ in the domain of $f$, and $f(x)$ is an absolute minima for $f$ over $I$.
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The term *local extrema* is used to describe either a local maximum or local minimum.
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@@ -883,7 +883,7 @@ plot(x -> x * exp(-x), 0, 5)
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##### Example
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The tangent function does not have a *guarantee* of an absolute maximum or an minimum over $(-\pi/2, \pi/2),$ as it is not *continuous* at the endpoints. In fact, it doesn't have either extrema - it has vertical asymptotes at each endpoint of this interval.
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The tangent function does not have a *guarantee* of an absolute maximum or a minimum over $(-\pi/2, \pi/2),$ as it is not *continuous* at the endpoints. In fact, it doesn't have either extrema - it has vertical asymptotes at each endpoint of this interval.
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##### Example
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