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Q. 0

Expert-verified
Found in: Page 183

Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

Q. Problem Zero: Read the section and make your own sum-mary of the material.

The Derivative of a Function f at $x=c$is defined as $f\text{'}\left(c\right)=\underset{h\to 0}{\mathrm{lim}}\frac{f\left(c+h\right)-f\left(c\right)}{h}\mathrm{or}f\text{'}\left(c\right)=\underset{z\to c}{\mathrm{lim}}\frac{f\left(z\right)-f\left(c\right)}{z-c}.$

If a function f is differentiable at $x=c$then $f\text{'}\left(c\right)=\underset{h\to 0}{\mathrm{lim}}\frac{f\left(c+h\right)-f\left(c\right)}{h}$must exist.

The left-hand derivative of a function f is defined as $f{\text{'}}_{-}\left(c\right)=\underset{h\to {0}^{-}}{\mathrm{lim}}\frac{f\left(c+h\right)-f\left(c\right)}{h}.$

The right-hand derivative of a function f is defined as $f{\text{'}}_{+}\left(c\right)=\underset{h\to {0}^{+}}{\mathrm{lim}}\frac{f\left(c+h\right)-f\left(c\right)}{h}.$

If a function is differentiable at any point then the function will also be continuous at that point.

The tangent line to the graph of a function f at $x=c$is defined as $y=f\left(c\right)+f\text{'}\left(c\right)\left(x-c\right)$where $f\text{'}\left(c\right)$is the slope.

See the step by step solution

Step 1. Given information.

The topic of the given section is the Formal Definition of the Derivative.

Step 2. Summary of section.

The Derivative of a Function f at $x=c$is defined as role="math" localid="1649815739115" $f\text{'}\left(c\right)=\underset{h\to 0}{\mathrm{lim}}\frac{f\left(c+h\right)-f\left(c\right)}{h}\mathrm{or}f\text{'}\left(c\right)=\underset{z\to c}{\mathrm{lim}}\frac{f\left(z\right)-f\left(c\right)}{z-c}$

If a function f is differentiable at $x=c$then role="math" localid="1649815763221" $f\text{'}\left(c\right)=\underset{h\to 0}{\mathrm{lim}}\frac{f\left(c+h\right)-f\left(c\right)}{h}$must exist.

The left-hand derivative of a function f is defined as role="math" localid="1649815781651" $f{\text{'}}_{-}\left(c\right)=\underset{h\to {0}^{-}}{\mathrm{lim}}\frac{f\left(c+h\right)-f\left(c\right)}{h}.$

The right-hand derivative of a function f is defined as $f{\text{'}}_{+}\left(c\right)=\underset{h\to {0}^{+}}{\mathrm{lim}}\frac{f\left(c+h\right)-f\left(c\right)}{h}.$

If a function is differentiable at any point then the function will also be continuous at that point.

The tangent line to the graph of a function f at $x=c$is defined as role="math" localid="1649815817628" $y=f\left(c\right)+f\text{'}\left(c\right)\left(x-c\right)$where $f\text{'}\left(c\right)$is the slope.