Limits and Derivatives

Understanding Limits and Derivatives

• Concept of Limits:

  • The first example discussed involves a quotient of two differences that represents the slope of a line through two points on the graph of a function

  • As the two points approach the value 3, we approximate the slope of the tangent line to the graph.

  • The slope of a line measures the rate of change of the function with respect to its variable; hence, the slope of the tangent line measures the rate of change of that function.

  • This computation is applicable to any function; by taking two points on the graph, and dividing the vertical difference by the horizontal difference, we establish its behavior around a point.

  • The horizontal difference is expressed as
    $ext{h}$
    representing a small number (but not zero).

  • The numerator consists of the vertical difference.

Definition of Derivative

• If this limit exists, it is termed the derivative of the function at the point, denoted by $f'(a)$ (where $a$ is the point at which we compute the derivative).

• The formal definition is:
  $f'(a) = ext{lim}_{h o 0} rac{f(a+h) - f(a)}{h}$

• If this limit does not exist, then the derivative does not exist as well.

• It is noted that the input variable does not necessarily need to be called $x$. For example, if the position of an object over time is given by a function $s(t)$, then its rate of change (the velocity of the object) can be expressed as $s'(t)$.

Example in Demand Function

• If the price-demand equation is given by $D(p)$, then the derivative $D'(p)$ represents the rate of change of the demand concerning the price.

• This rate of change will later be important for measuring the elasticity of demand.

Derivative Calculation Examples

Step-by-Step Derivative Calculation:

1. Identify the function f(x) to find the derivative:

   • The textbook describes a four-step process:

   1. Find $f(a)$

   2. Find $f(a+h)$

   3. Compute $f(a+h) - f(a)$

   4. Formulate the limit to get $f'(a)$

• Example Result:

  • The derivative tells us that the rate of change of the function is always equal to 5, indicating it grows at a constant rate of 5.

Tangent Lines and Derivatives

• In further examples, one will need to determine the tangent line at a specific point $a$.

  • Key Concept: The slope of the tangent line at point $a$ is equivalent to the value of the derivative $f'(a)$ at that point.

Existence of the Derivative

• The conditions under which the derivative may not exist are critical to understanding its limitations. They are as follows:

  1. The function is not continuous (function may not be defined).

  2. The function has a corner at some point on its graph.

  3. The function has a vertical tangent line at some point.

• It is vital to recognize instances across graphs where the derivative does not exist.

Practical Session: Finding Derivatives

• If given a scenario where the tangent line to $f$ at $x = a$ passes through the point $(p, q)$. Then:

  • We need to find the derivative $f'(a)$ using the steps indicated earlier:

  1. Find the value of the function at that point, $f(a)$.

  2. Find $f(a+h)$, and repeat the steps as required.

• The final task will be obtaining a simplified expression once $h$ approaches zero.