2.7_Derivatives_and_Rates_of_Change_Notes__28Filled_In_29

Derivatives and Rates of Change

Tangent Line Problem

• Objective: Find the equation of the tangent line to the graph of a function at a specific point.

• Given: A function f(x) and a point P(a, f(a)) on its graph.

• Secant Line: The line through points P and Q on the graph.

  • As point Q approaches point P, the secant line resembles the tangent line more closely.

• Slope of Secant Line:

  • Given by:[ m_{PQ} = \frac{f(x) - f(a)}{x - a} ]

• Tangent Line:

  • The line that approaches the secant line in the limit as Q approaches P.

  • The slope of the tangent line:[ m = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} ]

• Alternative Expression: If ( h = x - a ), then the slope of the tangent line can be computed using:[ m_{PQ} = \frac{f(a + h) - f(a)}{h} ]

  • As ( h ) approaches 0:[ m = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} ]

Example 1: Finding Tangent Line Equation

Problem Statement

• Function: ( y = x^3 - 3x + 1 )

• Point: (2,3)

• Objective: Find the equation of the tangent line at this point.

Solution Steps

1. Find the Slope at P(2, 3):[ m_{tan} = \lim_{x \to 2} \frac{x^3 - 3x + 1 - 3}{x - 2} ]

2. Perform Algebraic Manipulations:

   • Factor the numerator and apply limits.

3. Tangent Line Formula:

   • Use ( y = mx + b) to determine b using point P(2, 3).

Velocities

• Position Function: For an object moving in a straight line, displacement from the origin at time t is given by ( s = f(t) ).

Average Velocity

• Calculated over the interval ([t_1, t_2]):[ \text{Average Velocity} = \frac{f(t_2) - f(t_1)}{t_2 - t_1} ]

Instantaneous Velocity

• Defined as:[ v(a) = \lim_{t \to a} \frac{f(t) - f(a)}{t - a} ]

• Interpretation: The derivative of the position function gives the instantaneous velocity.

• Acceleration:

  • Defined as the rate of change of velocity. Can be positive (increasing) or negative (decreasing) based on the context.

Example 2: Height Function

Problem Statement

• Function: ( h(t) = 40t - 16t^2 )

• Velocity Calculation: Find ( v(2) ).

Solution Steps

1. Use Limits:[ v(2) = \lim_{t \to 2} \frac{h(t) - h(2)}{t - 2} ]

2. Simplify and Compute:**

Derivatives

• Definition of Derivative:[ f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} ]

• Alternative form with h:[ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} ]

Example 3: Finding Derivative

Function and Calculation

• Given ( f(t) = 2t^3 + t )

• Find the derivative using the limit definition.

Interpretation of the Derivative

• Slope of Tangent Line: Represents the instantaneous rate of change.

• Example: For a product sold, ( f'(8) ):

  • Represents the rate of change of the quantity sold with respect to price.

  • If negative, indicates a decrease in quantity sold as price increases.