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Exercise 4: Derivation of difference quotient formulas using Taylor's Theorem.

Taylor's Theorem with remainder states that

provided u has two continuous derivatives with respect to x and

provided u has four continuous derivatives with respect to x. Use these Taylor formulas to derive the following difference quotient formulas.

In each formula, the term with argument is the truncation error; it represents the error incurred in approximating a derivative with a difference quotient, assuming that all arithmetic is infinite precision. Assuming u has three continuous derivatives with respect to x, use Taylor's Theorem with remainder to show