Overview
This lecture explains when a function's derivative exists, how to interpret the graph of a derivative, and introduces concepts of concavity and the second derivative.
Conditions for Existence of a Derivative
- The derivative at a point ( a ) exists only if the function exists at ( a ).
- If the function ( f ) has a cusp (a sharp point) at ( x = a ), ( f'(a) ) does not exist.
- If the function is not continuous at ( x = a ), the derivative ( f'(a) ) cannot exist.
- A function is differentiable at ( x = a ) if it is continuous and smooth (no cusps) at that point.
Derivative Graph Interpretation
- The derivative of a linear segment is a constant equal to the slope of that segment.
- Where the graph of ( f ) is flat, the derivative (( f' )) is zero.
- At points with cusps, the derivative does not exist (open circles on the graph).
- The value of ( f' ) corresponds to the slope of ( f ) at each ( x ).
Examples with Specific Functions
- For ( f(x) = \sqrt{x} ), ( f'(x) = 1/(2\sqrt{x}) ) is always positive and gets smaller as ( x ) increases.
- The sign of ( f'(x) ) indicates whether ( f(x) ) is increasing (( f' > 0 )) or decreasing (( f' < 0 )).
Concavity and Second Derivative
- A function is concave down if its slope (( f' )) is decreasing as ( x ) increases.
- Concave up means the slope (( f' )) is increasing as ( x ) increases.
- The second derivative ( f''(x) ) measures the rate of change of the slope of ( f ).
- ( f''(x) ) provides information about the concavity of ( f ).
Key Terms & Definitions
- Derivative (( f'(x) )) — measures the instantaneous rate of change or slope of ( f(x) ).
- Cusp — a sharp point on the curve where the slopes from the left and right are different.
- Continuous — no breaks, holes, or jumps in the function at a point.
- Differentiable — a function is differentiable at a point if its derivative exists there.
- Concave Down — function shape where the slope decreases as you move right.
- Concave Up — function shape where the slope increases as you move right.
- Second Derivative (( f''(x) )) — derivative of the derivative; tells how the slope of ( f(x) ) changes.
Action Items / Next Steps
- Practice sketching the derivative from function graphs, noting points of non-differentiability (cusps, discontinuities).
- Review and memorize the definitions of concave up, concave down, and their relation to first and second derivatives.