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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 right-triangle ΔABC, the angle the hypotenuse (the tangent line) makes with the horizontal is , and by the definition of the derivative at A, . Consequently,
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so that .
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Figure 3.4.1 Defining the differential
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Thus , the differential of , is defined as the derivative times , an increment (large or small) in , the independent variable.
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Figure 3.4.1 then suggests that is an approximation to , the exact change in as the independent variable changes from to .
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This idea is captured in the notation
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Isolating leads to the linear approximation
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The linear approximation is nothing more than the tangent-line approximation, that is, the use of the tangent line to approximate values of a nonlinear function.
The Mean-Value theorem (Theorem 3.4.1), whose proof is independent of the relationships in Figure 3.4.1, then states that there is a point for which the linear approximation is actually an exact equality.
Theorem 3.4.1: Mean-Value Theorem
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is continuous in
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is differentiable in
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At least one exists in for which
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This form of the Mean-Value theorem has a geometric interpretation, namely, that over the interval there is a point at which the tangent line is parallel to the secant line connecting and .
If the conclusion of Theorem 3.4.1 is rewritten as
and if is identified with , and with , then the conclusion of Theorem 3.4.1 becomes
In other words, the analytic content of the Mean-Value theorem is that the linear approximation is exact if the differential is evaluated at the special point . However, the point depends on , so no recipe can be given for finding the value of that makes the linear approximation exact.