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8. DIFFERENTIATION OF REAL VALUED FUNCTIONS

8.2.12. Problem. Let ${\mathit{T}}_{\mathit{a}}\mathrm{:}\mathit{x}\mathrm{↦}\mathit{x}\mathrm{+}\mathit{a}$. The mapping ${\mathit{T}}_{\mathit{a}}$ is called translation by $\mathit{a}$. Note that it is not linear (unless, of course, $\mathit{a}\mathrm{=}\mathrm{0}$). We say that functions $\mathit{f}$ and $\mathit{g}$ in ${{F}}_{\mathit{a}}$ are tangent at $\mathit{a}$ if the functions $\mathit{f}\mathrm{o}{\mathit{T}}_{\mathit{a}}$ and $\mathit{g}\mathrm{o}{\mathit{T}}_{\mathit{a}}$ are tangent at zero.

(a) Let $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\mathrm{3}{\mathit{x}}^{\mathrm{2}}\mathrm{+}\mathrm{1}\mathrm{0}\mathit{x}\mathrm{+}\mathrm{1}\mathrm{3}$ and $\mathit{g}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\sqrt{\mathrm{-}\mathrm{2}\mathrm{0}\mathit{x}\mathrm{-}\mathrm{1}\mathrm{5}}$. Show that $\mathit{f}$ and $\mathit{g}$ are tangent at $\mathrm{-}\mathrm{2}\mathrm{.}$

(b) Develop a theory for the relationship ('tangency at $\mathit{a}$“ which generalizes our work on ""tangency at $\mathrm{0}$“ .

8.2.13. Problem. Each of the following is an abbreviated version of a proposition. Formulate precisely and prove.

(a) ${\mathit{C}}_{\mathrm{0}}\mathrm{+}\mathfrak{O}\mathrm{\subseteq }{\mathit{C}}_{\mathrm{0}}\mathrm{.}$

(b) ${\mathit{C}}_{\mathrm{0}}\mathrm{+}\mathrm{0}\mathrm{\subseteq }{\mathit{C}}_{\mathrm{0}}\mathrm{.}$

(c) $\mathfrak{O}\mathrm{+}\mathrm{0}\mathrm{\subseteq }\mathfrak{O}\mathrm{.}$

8.2.14. Problem. Suppose that $\mathit{f}\mathrm{\simeq }\mathit{g}$. Then the following hold.

(a) If $\mathit{g}$ is continuous at $\mathrm{0}$, so is $\mathit{f}\mathrm{.}$

(b) If $\mathit{g}$ belongs to $\mathfrak{O}$, so does $\mathit{f}\mathrm{.}$

(c) If $\mathit{g}$ belongs to $\mathrm{0}$, so does $\mathit{f}\mathrm{.}$

8.3. LINEAR APPROXIMATION

One often hears that differentiation of a smooth function $\mathit{f}$ at a point $\mathit{a}$ in its domain is the process of finding the best ""linear approximation“ to $\mathit{f}$ at $\mathit{a}$. This informal assertion is not quite correct. For example, as we know from beginning calculus, the tangent line at $\mathit{x}\mathrm{=}\mathrm{1}$ to the curve $\mathit{y}\mathrm{=}\mathrm{4}\mathrm{+}{\mathit{x}}^{\mathrm{2}}$ is the line $\mathit{y}\mathrm{=}\mathrm{2}\mathit{x}\mathrm{+}\mathrm{3}$, which is not a linear function since it does not pass through the origin. To rectify this rather minor shortcoming we first translate the graph of the function $\mathit{f}$ so that the point $\mathrm{\left(}\mathit{a}\mathrm{,}\mathrm{ }\mathit{f}\mathrm{\left(}\mathit{a}\mathrm{\right)}\mathrm{\right)}$ goes to the origin, and then find the best linear approximation at the origin. The operation of translation is carried out by a somewhat notorious acquaintance from beginning calculus $\mathrm{▵}\mathit{y}$. The source of its notoriety is two‐fold: first, in many texts it is inadequately defined; and second, the notation $\mathrm{▵}\mathit{y}$ fails to alert the reader to the fact that under consideration is a function of two variables. We will be careful on both counts.

8.3.1. Definition. Let $\mathit{f}\mathrm{\in }{{F}}_{\mathit{a}}$. Define the function $\mathrm{▵}{\mathit{f}}_{\mathit{a}}$ by

$\mathrm{▵}{\mathit{f}}_{\mathit{a}}\mathrm{\left(}\mathit{h}\mathrm{\right)}\mathrm{:}\mathrm{=}\mathit{f}\mathrm{\left(}\mathit{a}\mathrm{+}\mathit{h}\mathrm{\right)}\mathrm{-}\mathit{f}\mathrm{\left(}\mathit{a}\mathrm{\right)}$

for all $\mathit{h}$ such that $\mathit{a}\mathrm{+}\mathit{h}$ is in the domain of $\mathit{f}$. Notice that since $\mathit{f}$ is defined in a neighborhood of $\mathit{a}$, the function $\mathrm{▵}{\mathit{f}}_{\mathit{a}}$ is defined in a neighborhood of $\mathrm{0}$; that is, $\mathrm{▵}{\mathit{f}}_{\mathit{a}}$ belongs to ${{F}}_{\mathrm{0}}$. Notice also that $\mathrm{▵}{\mathit{f}}_{\mathit{a}}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathrm{.}$

8.3.2. Problem. Let $\mathit{f}\mathrm{\left(}\mathit{x}\mathrm{\right)}\mathrm{=}\mathrm{cos}\mathit{x}$ for $\mathrm{0}\mathrm{\le }\mathit{x}\mathrm{\le }\mathrm{2}\mathit{\pi }\mathrm{.}$

(a) Sketch the graph of the function $\mathit{f}\mathrm{.}$

(b) Sketch the graph of the function $\mathrm{▵}{\mathit{f}}_{\mathit{\pi }}\mathrm{.}$

8.3.3. Proposition. If $\mathit{f}\mathrm{\in }{{F}}_{\mathit{a}}$ and $\mathit{\alpha }\mathrm{\in }\mathbb{R}$, then

$\mathrm{▵}\mathrm{\left(}\mathit{\alpha }\mathit{f}{\mathrm{\right)}}_{\mathit{a}}\mathrm{=}\mathit{\alpha }\mathrm{▵}{\mathit{f}}_{\mathit{a}}\mathrm{.}$

Proof. To show that two functions are equal show that they agree at each point in their domain. Here

$\mathrm{▵}\mathrm{\left(}\mathit{\alpha }\mathit{f}{\mathrm{\right)}}_{\mathit{a}}\mathrm{\left(}\mathit{h}\mathrm{\right)}\mathrm{=}\mathrm{\left(}\mathit{\alpha }\mathit{f}\mathrm{\right)}\mathrm{\left(}\mathit{a}\mathrm{+}\mathit{h}\mathrm{\right)}\mathrm{-}\mathrm{\left(}\mathit{\alpha }\mathit{f}\mathrm{\right)}\mathrm{\left(}\mathit{a}\mathrm{\right)}$

$\mathrm{=}\mathit{\alpha }\mathit{f}\mathrm{\left(}\mathit{a}\mathrm{+}\mathit{h}\mathrm{\right)}\mathrm{-}\mathit{\alpha }\mathit{f}\mathrm{\left(}\mathit{a}\mathrm{\right)}$

$\mathrm{=}\mathit{\alpha }\mathrm{\left(}\mathit{f}\mathrm{\left(}\mathit{a}\mathrm{+}\mathit{h}\mathrm{\right)}\mathrm{-}\mathit{f}\mathrm{\left(}\mathit{a}\mathrm{\right)}\mathrm{\right)}$

$\mathrm{=}\mathit{\alpha }\mathrm{▵}{\mathit{f}}_{\mathit{a}}\mathrm{\left(}\mathit{h}\mathrm{\right)}$

for every $\mathit{h}$ in the domain of $\mathrm{▵}{\mathit{f}}_{\mathit{a}}\mathrm{.}$

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