Derivatives reveal a function's behavior, showing where it increases, decreases, or levels off. By analyzing the first and second derivatives, we can understand a graph's shape, including its peaks, valleys, and curves.

Applying derivative tests helps classify critical points as maxima or minima. These tools, along with concepts of continuity and differentiability, give us a powerful way to analyze functions and their graphs.

Derivatives and the Shape of a Graph

Relationship of first derivative to graph

• The first derivative $f'(x)$ represents the rate of change or slope of the tangent line at any point on the function $f(x)$
  • Positive first derivative $f'(x) > 0$ indicates the function is increasing at that point (uphill)
  • Negative first derivative $f'(x) < 0$ indicates the function is decreasing at that point (downhill)
  • Zero first derivative $f'(x) = 0$ indicates a horizontal tangent line at that point, known as a critical point (flat)
• The sign of the first derivative determines the monotonicity of the function over intervals
  • Positive first derivative $f'(x) > 0$ for all $x$ in an interval means the function is strictly increasing on that interval (always going up)
  • Negative first derivative $f'(x) < 0$ for all $x$ in an interval means the function is strictly decreasing on that interval (always going down)

First derivative test for extrema

• Critical points are points where the first derivative is either zero $f'(x) = 0$ or undefined
  • Find critical points by solving $f'(x) = 0$ and identifying points where $f'(x)$ is undefined
• The first derivative test classifies critical points as local maxima, local minima, or neither
  • First derivative changes from positive to negative at a critical point indicates a local maximum (peak)
  • First derivative changes from negative to positive at a critical point indicates a local minimum (valley)
  • No sign change in the first derivative at a critical point means neither a local maximum nor minimum (saddle point)

Second derivative and concavity

• The second derivative $f''(x)$ represents the rate of change of the first derivative
• The sign of the second derivative determines the concavity of the function at a point
  • Positive second derivative $f''(x) > 0$ means the function is concave up at that point (opens upward)
  • Negative second derivative $f''(x) < 0$ means the function is concave down at that point (opens downward)
• Inflection points are points where the concavity of the function changes
  • At an inflection point, the second derivative is either zero $f''(x) = 0$ or undefined

Concavity test over intervals

• The concavity test determines the concavity of a function over an open interval
  • Positive second derivative $f''(x) > 0$ for all $x$ in an open interval means the function is concave up on that interval (smiling curve)
  • Negative second derivative $f''(x) < 0$ for all $x$ in an open interval means the function is concave down on that interval (frowning curve)
• Find intervals of concavity by solving inequalities $f''(x) > 0$ and $f''(x) < 0$, and identifying points where $f''(x) = 0$ or is undefined

Function behavior vs derivatives

• The first derivative provides information about the function's increasing/decreasing behavior and critical points
• The second derivative provides information about the function's concavity and inflection points
• Combining information from the first and second derivatives gives a comprehensive understanding of the function's shape and behavior
  • Increasing and concave up (speeding up)
  • Increasing and concave down (slowing down)
  • Decreasing and concave up (slowing down)
  • Decreasing and concave down (speeding up)

Applying Derivative Tests

Second derivative test for extrema

• The second derivative test is an alternative method to classify critical points as local maxima or minima
  • $f'(x) = 0$ and $f''(x) < 0$ at a critical point indicates a local maximum (peak)
  • $f'(x) = 0$ and $f''(x) > 0$ at a critical point indicates a local minimum (valley)
  • $f'(x) = 0$ and $f''(x) = 0$ at a critical point means the test is inconclusive, use the first derivative test instead
• The second derivative test is often easier to apply than the first derivative test, as it only requires evaluating the second derivative at the critical points

Continuity, Differentiability, and Limits

• Continuity is a prerequisite for differentiability, ensuring the function has no breaks or jumps
• Differentiability implies that the function has a well-defined derivative at a point
• The limit of the difference quotient as h approaches zero defines the derivative, connecting the concepts of limits and derivatives