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"Expanding the taylor series for ##f(x)##.." (See picture) is this a typo? Aren't we expanding ##f(x + \Delta x)##?
Also, when we evaluate ##f(x)## (coefficients in the expansion), are we assuming ##\Delta x = 0## by setting ##x + \Delta x## (argument of the function) equal to ##x##? Or are we just "replacing" ##x + \Delta x## by ##x##? I'm confused because usually when a function is evaluated at a particular number we just set the variable inside the argument of the function equal to that number, but when we're evaluating at "some other variable", for instance, finding ##f(x^2 + 1)## given ##f(x) = x^5##, we're not implying that ##x^2 + 1 = x^5##.
Also, when we evaluate ##f(x)## (coefficients in the expansion), are we assuming ##\Delta x = 0## by setting ##x + \Delta x## (argument of the function) equal to ##x##? Or are we just "replacing" ##x + \Delta x## by ##x##? I'm confused because usually when a function is evaluated at a particular number we just set the variable inside the argument of the function equal to that number, but when we're evaluating at "some other variable", for instance, finding ##f(x^2 + 1)## given ##f(x) = x^5##, we're not implying that ##x^2 + 1 = x^5##.
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