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•

The definition of a differential is based on Figure 3.4.1, the fundamental diagram of differential calculus. Points A and C are on the line tangent to the red curve at point A. In the righttriangle ΔABC, the angle the hypotenuse (the tangent line) makes with the horizontal is $\mathrm{\θ}$, and by the definition of the derivative at A, $f\prime \=\mathrm{tan}\left(\mathrm{\θ}\right)$. Consequently,

$\mathrm{tan}\left(\mathrm{\θ}\right)$

$\=f\prime$

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$\=\frac{\mathrm{opposite}}{\mathrm{adjacent}}$

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$\=\frac{\mathrm{opposite}}{\mathrm{dx}}$



so that $\mathrm{opposite}\=f\prime \mathrm{dx}\equiv \mathrm{df}$.


Figure 3.4.1 Defining the differential $\mathrm{df}$




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Thus $\mathrm{df}$, the differential of $f\left(x\right)$, is defined as the derivative $f\prime \left(x\right)$ times $\mathrm{dx}$, an increment (large or small) in $x$, the independent variable.

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Figure 3.4.1 then suggests that $\mathrm{df}$ is an approximation to $\mathrm{\Δ}$$f$, the exact change in $f$ as the independent variable changes from $x$ to $x\+\mathrm{dx}$.

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This idea is captured in the notation $\mathrm{\Delta}fequals;f\left(xplus;\mathrm{dx}\right)f\left(x\right)\doteq \mathrm{df}$

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Isolating $f\left(x\+\mathrm{dx}\right)$ leads to the linear approximation $f\left(x\+\mathrm{dx}\right)\doteq f\left(x\right)\+\mathrm{df}$




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The linear approximation is nothing more than the tangentline approximation, that is, the use of the tangent line to approximate values of a nonlinear function.
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The MeanValue theorem (Theorem 3.4.1), whose proof is independent of the relationships in Figure 3.4.1, then states that there is a point $c$ for which the linear approximation is actually an exact equality.
Theorem 3.4.1: MeanValue Theorem

1.

$f\left(x\right)$ is continuous in $\left[a\,b\right]$

2.

$f\left(x\right)$ is differentiable in $\left(a\,b\right)$

⇒
1.

At least one $c$ exists in $\left(a\,b\right)$ for which $f\prime \left(c\right)\=\frac{f\left(b\right)f\left(a\right)}{ba}$




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This form of the MeanValue theorem has a geometric interpretation, namely, that over the interval $\left[a\,b\right]$ there is a point $c$ at which the tangent line is parallel to the secant line connecting $\left(a\,f\left(a\right)\right)$ and $\left(b\,f\left(b\right)\right)$.
If the conclusion of Theorem 3.4.1 is rewritten as
$f\left(b\right)\=f\left(a\right)\+f\prime \left(c\right)\left(ba\right)$
and if $a$ is identified with $x$, and $b$ with $x\+\mathrm{dx}$, then the conclusion of Theorem 3.4.1 becomes
$f\left(x\+\mathrm{dx}\right)\=f\left(x\right)\+f\prime \left(c\right)\mathrm{dx}$
In other words, the analytic content of the MeanValue theorem is that the linear approximation is exact if the differential is evaluated at the special point $c$. However, the point $c$ depends on $x$, so no recipe can be given for finding the value of $c$ that makes the linear approximation exact.
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