Problem M-D-1.6
Differential Calculus: Difference Quotients and Commmon Tangents of 2 Functions
Problem Statement
Given are the functions
$$ f:x \mapsto f(x) = x^2+1,~D_f = \mathbb{R} $$
and
$$ g:x \mapsto g(x) = -x^2-1,~D_g = \mathbb{R} $$
- Determine the derivatives \(f^\prime\) and \(g^\prime\) as the limit of the difference quotient.
- Provide the common tangents of \(f(x)\) and \(g(x)\).
Short Solution
a. Determine the derivatives \(f^\prime\) and \(g^\prime\) as the limit of the difference quotient.
$$ \begin{aligned}
f^{\prime}(x_0) &= 2x_0 \\[7pt]
g^{\prime}(x_0) &= -2x_0
\end{aligned} $$
b. Provide the common tangents of \(f(x)\) and \(g(x)\).
$$ \begin{aligned}
t_1(x) &= 2x \\[7pt]
t_2(x) &= -2x
\end{aligned} $$
Comprehensive Solution
a. Determine the derivatives \(f^\prime\) and \(g^\prime\) as the limit of the difference quotient.
If we need to determine the derivative of a function as the limit of the difference quotient, we require the formula for the derivative of \(f\) at the point \(x_0\) (also known as the differential quotient of \(f\) at the point \(x_0\)):
$$ \begin{align}
\tag{1} f^{\prime}(x_0) &= \lim \limits_{h \to 0} \dfrac{f(x_0+h)-f(x_0)}{h}
\end{align} $$
For all \(x_0 \in \mathbb{R}\) and all \(h \neq 0\), the following holds:
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