Chapter 3: Applications of Differentiation
Section 3.4: Differentials and the Linear Approximation
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 θ, and by the definition of the derivative at A, f′=tanθ. Consequently,
so that opposite=f′dx≡df.
Figure 3.4.1 Defining the differential df
Thus df, the differential of fx, is defined as the derivative f′x times dx, an increment (large or small) in x, the independent variable.
Figure 3.4.1 then suggests that df is an approximation to Δf, the exact change in f as the independent variable changes from x to x+dx.
This idea is captured in the notation Δ f=fx+dx−fx≐df
Isolating fx+dx leads to the linear approximation fx+dx≐fx+df
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 c for which the linear approximation is actually an exact equality.
Theorem 3.4.1: Mean-Value Theorem
fx is continuous in a,b
fx is differentiable in a,b
At least one c exists in a,b for which f′c=fb−fab−a
This form of the Mean-Value theorem has a geometric interpretation, namely, that over the interval a,b there is a point c at which the tangent line is parallel to the secant line connecting a,fa and b,fb.
If the conclusion of Theorem 3.4.1 is rewritten as
and if a is identified with x, and b with x+dx, 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 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.
Approximate 17 by using the differential of the function fx=x.
How accurate is this approximation?
Apply the Mean Value theorem to the function fx=x e−x,0≤x≤3.
Determine c from first principles.
Over what interval would the tangent line at x=3 approximate fx=x e−x with an error no greater than 0.1?
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