Question

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

mary of the material.

Short answer

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

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

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

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

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.

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

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

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

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

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.