BackRates of Change, Tangent Lines, and Differentiability in Calculus
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Rates of Change and Tangent Lines
Average Rate of Change
The average rate of change of a function f on the interval [a, x] is the slope of the corresponding secant line. It measures how much the function changes per unit change in the input over the interval.
Formula:
Secant Line: A line passing through two points on the graph of f, specifically (a, f(a)) and (x, f(x)).
Example: For f(x) = x^2 between a = 2 and x = 3:
Instantaneous Rate of Change and Slope of the Tangent Line
The instantaneous rate of change of f at a is the slope of the tangent line at that point. It is found by taking the limit of the average rate of change as x approaches a (or as h approaches 0).
Formula:
or equivalently,
Tangent Line: A line that touches the graph of f at exactly one point (a, f(a)) and has the same slope as the graph at that point.
Example: For f(x) = x^2 at a = 2:
Equation of the Tangent Line
The equation of the tangent line to the graph of f at the point (a, f(a)) is given by:
Example: For f(x) = x^2 at a = 2:
Derivative of a Function
Definition of the Derivative
The derivative of f at x is the function given by:
If the limit exists, f is said to be differentiable at x.
If f is differentiable at every point of an open interval I, then f is differentiable on I.
Example: Finding the Derivative
Given f(x) = \sqrt{x} - 5x, find f'(x):
Graphical Interpretation of Derivatives
Relationship Between Function and Derivative Graphs
Observing the graph of a function and its tangent lines helps to understand the behavior of its derivative:
If the tangent line is horizontal at a point, at that point.
If the function is increasing, ; if decreasing, .
Points where the function changes from increasing to decreasing (or vice versa) often correspond to zeros of the derivative.
Example Table: (Described in text)
Function with a maximum: at the peak.
Function with a minimum: at the trough.
Function with no horizontal tangent: everywhere.
Differentiability and Continuity
Theorems Relating Differentiability and Continuity
Differentiable Implies Continuous: If f is differentiable at a, then f is continuous at a.
Not Continuous Implies Not Differentiable: If f is not continuous at a, then f is not differentiable at a.
When Is a Function Not Differentiable at a Point?
A function f is not differentiable at a if at least one of the following holds:
Discontinuity: f is not continuous at a.
Corner or Cusp: f has a sharp turn at a.
Vertical Tangent: f has a vertical tangent at a.
Example: Differentiability of Piecewise Functions
Given , determine differentiability at :
Left-side limit: Right-side limit: Since the left and right limits are not equal, is not differentiable at .
Summary Table: Differentiability and Continuity
Condition | Implication |
|---|---|
f differentiable at a | f continuous at a |
f not continuous at a | f not differentiable at a |
f has a corner/cusp at a | f not differentiable at a |
f has a vertical tangent at a | f not differentiable at a |
Practice Problems and Applications
Find the slope of the tangent line to at the point (0,4):
Find the slope of the tangent line to at the point (3,4):
Find for :
Additional info: These notes cover foundational concepts in differential calculus, including the geometric and algebraic interpretation of the derivative, criteria for differentiability, and practical computation of tangent lines and derivatives for various functions.
