3.1 HW Problems

Table of Contents:

Derivatives: definition and basic rules | Calculus 1 | Math | Khan Academy

Finding a Tangent Line to the Graph of a Function

• The slope of the curve $y=f(x)$ at the point $P(x_0,f(x_0))$ is the number:
  • The tangent line to the curve P is the line through P with this slope.

$$ \lim_{x \rightarrow \, 0} \frac{f(x_0+h)-f(x_0)}{h} $$

Example:

Class Notes:

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Rates of Change: Derivative at a Point

• The expression below is called the difference quotient of f at $x_0$. If the difference quotient has a limit as h approaches zero, that limit is given a special name and notation.

  • Definition: the derivative of a function f at a point $x_0$, denoted by $f’(x_0)$, is:

  $$ f'(x_0)=\lim_{x \rightarrow 0} \frac{f(x_0+h)-f(x_0)}{h} $$

Where a Derivative DOES NOT Exist

1. at a discontinuity