Calculus 1: How Derivatives Affect the Shape of a Graph

Key Concepts

Increasing and Decreasing Functions

• If ( f'(x) > 0 ) on an interval, then ( f(x) ) is increasing.
  • Graphically, tangent lines have positive slopes.
• If ( f'(x) < 0 ) on an interval, then ( f(x) ) is decreasing.
  • Graphically, tangent lines have negative slopes.

Identifying Local Extrema

• First Derivative Test
  • If ( f'(x) ) changes from positive to negative at ( x = c ), ( f(x) ) has a local maximum at ( c ).
  • If ( f'(x) ) changes from negative to positive at ( x = c ), ( f(x) ) has a local minimum at ( c ).
  • Critical numbers occur where ( f'(x) = 0 ) or does not exist.

Concavity and Inflection Points

• Concave Up
  • ( f(x) ) is concave up if the graph lies above its tangent lines.
  • ( f''(x) > 0 ) on the interval.
• Concave Down
  • ( f(x) ) is concave down if the graph lies below its tangent lines.
  • ( f''(x) < 0 ) on the interval.
• Inflection Point
  • A point where the graph changes concavity if ( f(x) ) is continuous at that point.

Second Derivative Test

• Local Minimum
  • If ( f'(c) = 0 ) and ( f''(c) > 0 ), ( f(x) ) has a local minimum at ( c ).
• Local Maximum
  • If ( f'(c) = 0 ) and ( f''(c) < 0 ), ( f(x) ) has a local maximum at ( c ).
• If ( f''(c) = 0 ), the test is inconclusive.

Example Analysis

Example 1

Increasing/Decreasing Intervals

• Given function: Rational function with domain all real numbers.
• Find ( f'(x) ) using quotient rule.
• Critical Values: Where ( f'(x) = 0 ) or does not exist.
• Analyze intervals using a number line and ( f'(x) ).

Concavity and Inflection Points

• Find ( f''(x) ) using quotient rule and simplify.
• Inflection Points: Evaluate where ( f''(x) = 0 ).
• Analyze concavity using ( f''(x) ).

Example 2

Function: ( f(x) = x \sqrt{x+1} )

• Domain: ( x \geq -1 ).
• Find ( f'(x) ) using product rule.
• Critical Values: Only inside the domain.
• First Derivative Test: Number line analysis.

Concavity

• Find ( f''(x) ) using quotient rule.
• Analyze concavity with domain restrictions.
• No inflection points due to domain limits.

Tips

• Always determine the domain first.
• Use number lines to test intervals for ( f'(x) ) and ( f''(x) ).
• Be mindful of critical points at endpoints which cannot be local extrema.

Conclusion

• Understanding derivatives provides insights into the behavior of functions regarding growth, decline, and curvature.
• Use first and second derivative tests to explore function characteristics effectively.

Remember to practice these concepts with more examples and graph observations.