Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 3: Derivatives - Section 3.2 - The Derivative as a Function - Exercises 3.2 - Page 117: 60

Answer

Horizonal axis is X-axis and vertical is Y-axis. You can see that as $h$ approaches to $0$ our graph becomes good approximation. It explains the the formula of derivative for the function $f(x)=x^{3}$ in the graphical form.

Work Step by Step

As the mode of h getting smaller and smaller(i.e., approaches to $0$ from either side) from $2$ to $0.2$ our graph becomes good approximation to $f(x)=3x^{2}$. If you know the formula of derivative, it's the same. If $k(x)=x^{3}$, $\frac{dk}{dx}=3{x}^{2}$ ...(1) And in the form of limit it becomes, $\frac{dk}{dx}=\lim\limits_{h \to 0}\frac{\left(x+h\right)^{3}- x^{3}}{h}$ ...(2) Compare both the equations, $3{x}^{2}=\lim\limits_{h \to 0}\frac{\left(x+h\right)^{3}- x^{3}}{h}$ You can see that ultimately both are the same. So, It's just visualization of our differentiation formula.
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