Exercises

Exercises 1.4 Exercises

1.

Let \(f(x)=|x|=g(x)\) and \(h(x)=x^2\text{.}\) Find the derivatives of \((fg)(x)\) and \((h\circ f)(x)\) at \(x=0\text{.}\)

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2.

Let \(f(x) = \begin{cases} -2x & x \ge 0, \\ -3x & x \lt 0, \end{cases}\) and \(g(x) = \begin{cases} 3x & x \ge 0, \\ 2x & x \lt 0. \end{cases}\)

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For each for the following functions, determine whether it is differentiable at \(x=0\) and if so, compute the derivative at \(x=0\text{.}\)

\(\displaystyle (f+g)(x)\)

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\(\displaystyle (fg)(x)\)

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\(\displaystyle (f\circ g)(x)\)

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\(\displaystyle (g\circ f)(x)\)

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3.

A derivation is an operation \(D\) on functions (more general on anything that we can talk about sum and product) satisfying:

\(D(f+g) = Df + Dg\text{;}\) and

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\(\displaystyle D(fg) = (Df)g + f(Dg)\)

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For example, \(\dfrac{d}{dx}\) is a derivation on differentiable functions. Let \(D\) be a derivation. Show that \(D^2(fg) = (D^2f)g + 2(Df)(Dg) + f(D^2g)\text{.}\) Find a general expression for \(D^n(fg)\) for \(n \ge 2\text{.}\)

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4.

Show that \(\ds D=x\frac{d}{dx}\) is a derivation on differentiable functions.

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Show that \(Dc = 0\) for any constant \(c\) and that \(Dx =x\text{.}\)

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Compute \(\ds D \frac{1}{1-x}\text{.}\) Hint: Let \(\ds h=\frac{1}{1-x}\text{.}\) Find \(Dh\) by applying \(D\) to the relation \(1=h(1-x)\text{.}\)

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Compute \(\ds D^2 \frac{1}{1-x}\text{.}\)

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5.

Differentiation alters the parity of a function. That is the derivative of an even (odd) function is odd (even).

Solution.

Let \(f(x)\) be a differentiable even function. So \(f(-x) = f(x)\) for all \(x\) in the domain of \(f\text{.}\) Taking derivative on both sides, yields, according to the Chain Rule,

\begin{equation*} f'(x) =(f(-x))' = -f'(-x). \end{equation*}

This shows that \(f'(x)\) is an odd function.

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Similarly, if \(f(x)\) is an odd function, then

\begin{equation*} -f'(x) = (-f(x))' = (f(-x))' = -f'(-x). \end{equation*}

Thus, \(f'(x) = f'(-x)\) and so \(f'(x)\) is even.

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