4. Why is f'(x) approximated well by the difference quotient when Ax is very small?

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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**Question 4:**

Why is \( f'(x) \) approximated well by the difference quotient when \(\Delta x\) is very small?

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This question addresses the concept of the derivative in calculus. The difference quotient, given by \(\frac{f(x+\Delta x) - f(x)}{\Delta x}\), is used to approximate the derivative \( f'(x) \). As \(\Delta x\) approaches zero, this quotient becomes a more accurate approximation of the derivative, representing the slope of the tangent line to the curve at point \( x \).
Transcribed Image Text:**Question 4:** Why is \( f'(x) \) approximated well by the difference quotient when \(\Delta x\) is very small? --- This question addresses the concept of the derivative in calculus. The difference quotient, given by \(\frac{f(x+\Delta x) - f(x)}{\Delta x}\), is used to approximate the derivative \( f'(x) \). As \(\Delta x\) approaches zero, this quotient becomes a more accurate approximation of the derivative, representing the slope of the tangent line to the curve at point \( x \).
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