Geometric Interpretation of the Derivative (Review) Geometric - PowerPoint PPT Presentation

Geometric Interpretation of the Derivative (Review)

Geometric Interpretation of the Derivative (Review) The derivative of a function f ( x ) at point x 0 is the slope of the tangent line at that point.

Geometric Interpretation of the Derivative (Review) The derivative of a function f ( x ) at point x 0 is the slope of the tangent line at that point.

Geometric Interpretation of the Derivative (Review) The derivative of a function f ( x ) at point x 0 is the slope of the tangent line at that point.

Geometric Interpretation of the Derivative (Review) The derivative of a function f ( x ) at point x 0 is the slope of the tangent line at that point.

Physical Interpretation of the Derivative Movement at constant veloctity

Physical Interpretation of the Derivative Movement at constant veloctity Consider a train travelling at constant velocity, say, 80km per hour.

Physical Interpretation of the Derivative Movement at constant veloctity Consider a train travelling at constant velocity, say, 80km per hour.

Physical Interpretation of the Derivative Movement at constant veloctity Consider a train travelling at constant velocity, say, 80km per hour.

Physical Interpretation of the Derivative Movement at constant veloctity Consider a train travelling at constant velocity, say, 80km per hour.

Physical Interpretation of the Derivative Movement at constant veloctity Consider a train travelling at constant velocity, say, 80km per hour.

Physical Interpretation of the Derivative Movement at constant veloctity Consider a train travelling at constant velocity, say, 80km per hour.

Physical Interpretation of the Derivative Movement at constant veloctity Consider a train travelling at constant velocity, say, 80km per hour.

Physical Interpretation of the Derivative Movement at constant veloctity

Physical Interpretation of the Derivative Movement at constant veloctity We can plot these points:

Physical Interpretation of the Derivative Movement at constant veloctity We can plot these points:

Physical Interpretation of the Derivative Movement at constant veloctity We can plot these points:

Physical Interpretation of the Derivative Movement at constant veloctity We can plot these points: The slope of this line would be:

Physical Interpretation of the Derivative Movement at constant veloctity We can plot these points: The slope of this line would be: m = ∆ y ∆ x

Physical Interpretation of the Derivative Movement at constant veloctity We can plot these points: The slope of this line would be: m = ∆ y ∆ x = ∆ s ∆ t

Physical Interpretation of the Derivative Movement at constant veloctity We can plot these points: The slope of this line would be: m = ∆ y ∆ x = ∆ s ∆ t That’s average velocity!

Let’s say that the distance traveled by a car is represented in this graph:

Let’s say that the distance traveled by a car is represented in this graph:

Let’s say that the distance traveled by a car is represented in this graph:

Let’s say that the distance traveled by a car is represented in this graph:

Let’s say that the distance traveled by a car is represented in this graph:

Let’s say that the distance traveled by a car is represented in this graph:

Let’s say that the distance traveled by a car is represented in this graph: ∆ y speed = lim ∆ x ∆ x → 0

Let’s say that the distance traveled by a car is represented in this graph: ∆ y speed = lim ∆ x : instantaneous rate of change! ∆ x → 0