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Home Limits, Analytic Geometry, and Approximations Derivatives and Increments Approximating Increments
 
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Approximating Increments

We shall now turn the problem around. Instead of using the increment Δy to approximate the derivative dy/dx, we shall use the derivative dy/dx to approximate the increment Δy. When Δx is small, f(c + Δx) will be close to f(c) + f'(c) Δx even compared to Δx. Part (i) of Theorem 1 gives the error estimate ½M Δx2 for this approximation. This method is especially useful for approximating f(x) when there is a number c close to x such that both f(c) and f'(c) are known.

Example 2

Example 3

INCREMENT THEOREM (Repeated)

Hypotheses f'(c) exists and Δx is infinitesimal.

Conclusion

f(c + Δx) = f(c) + f'(c) Δx + ε Δx

for some infinitesimal ε. which depends on c and Δx.

THEOREM 1 OF THIS SECTION (in an equivalent form)

Hypotheses f"(u) exists and

|f"(u)| ≤ M

for all u between the real numbers c and c + Δx.

Conclusion

f(c + Δx) = f(c) + f'(c) Δx + ε Δx

for some real ε within ½M |Δx| of 0.

Thus Theorem 1 has more hypotheses but also gives more specific information about ε in its conclusion.
Home Limits, Analytic Geometry, and Approximations Derivatives and Increments Approximating Increments

Last Update: 2010-11-25