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Q. 0

Expert-verifiedFound in: Page 183

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 $f\text{'}\left(c\right)=\underset{h\to 0}{\mathrm{lim}}\frac{f(c+h)-f\left(c\right)}{h}\mathrm{or}f\text{'}\left(c\right)=\underset{z\to c}{\mathrm{lim}}\frac{f\left(z\right)-f\left(c\right)}{z-c}.$

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

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

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

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 $y=f\left(c\right)+f\text{'}\left(c\right)(x-c)$where $f\text{'}\left(c\right)$is the slope.

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

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

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

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

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

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" $y=f\left(c\right)+f\text{'}\left(c\right)(x-c)$where $f\text{'}\left(c\right)$is the slope.

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