Suppose f is differentiable. If we use the approximation f(x+h) ≈f(x)+f^'(x) h the error is ε(h)

Name: Problem 49, Chapter 2: Calculus
Uploaded: 2021-06-17T15:13:36-08:00
Duration: 5 min 9 s
Channel: Donald Albin
Description: Suppose f is differentiable. If we use the approximation f(x+h) ≈f(x)+f^'(x) h the error is ε(h)=f(x+h)- f(x)-f^'(x) h . Show that (a) limh →0 ε(h)=0 and (b) limh →0 (ε(h))/(h)=0

step 1 So we have E, the error as a function of H. I guess that's, is that epsilon? I don't know. As a

Suppose f is differentiable. If we use the approximation...

Suppose $f$ is differentiable. If we use the approximation $f(x+h) \approx f(x)+f^{\prime}(x) h$ the error is $\varepsilon(h)=f(x+h)-$ $f(x)-f^{\prime}(x) h .$ Show that $$\text { (a) } \lim _{h \rightarrow 0} \varepsilon(h)=0 \text { and }(\mathrm{b}) \lim _{h \rightarrow 0} \frac{\varepsilon(h)}{h}=0$$

Suppose $f$ is differentiable. If we use the approximation $f(x+h) \approx f(x)+f^{\prime}(x) h$ the error is $\varepsilon(h)=f(x+h)-$ $f(x)-f^{\prime}(x) h .$ Show that
$$\text { (a) } \lim _{h \rightarrow 0} \varepsilon(h)=0 \text { and }(\mathrm{b}) \lim _{h \rightarrow 0} \frac{\varepsilon(h)}{h}=0$$

Key Concepts

Differentiability

Differentiability of a function at a point means that the function can be well approximated by a linear function (its tangent line) near that point. This property implies that the limit of the difference quotient exists, which in turn ensures the smoothness of the function at that point and elaborates on how small changes in the input reflect proportionally small changes in the function's output.

Linear Approximation

Linear approximation, or the tangent line approximation, is the method of approximating the value of a differentiable function near a given point using the function’s value and its derivative at that point. This approximation is expressed as f(x+h) ? f(x) + f'(x)h, illustrating that for small h, the change in the function’s value can be predicted linearly by its derivative.

Error Analysis in Taylor Series

The error term, often referred to as the remainder in a Taylor series approximation, measures the difference between the true value of the function and its linear approximation. In this context, the error ?(h) = f(x+h) - f(x) - f'(x)h quantifies how the linear approximation deviates from the actual function, and analyzing its behavior as h approaches zero confirms the accuracy of the approximation in the limit.

Limit Concepts

Limit concepts are central to understanding both differentiability and the behavior of error terms. By exploring the limit of the error term as h approaches zero, we validate that the error vanishes, and by examining the limit of ?(h)/h as h tends to zero, we further confirm that the linear approximation becomes increasingly accurate, reflecting the formal definition of the derivative.

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