Calculus: Applications of Derivatives – Formula & Concept Review

Objective

This document summarizes key concepts and formulas related to the applications of derivatives in calculus, including average/instantaneous rates of change, mean value theorem, Rolle’s theorem, extrema determination, concavity, L'Hôpital's rule, and Newton's method.

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1. Average Rate of Change

• Formula:
  [ \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a} ]
• Interpretation:
  • Represents the slope of the secant line between two points ((a, f(a))) and ((b, f(b))).
  • Derived from the standard slope formula ((y_2-y_1)/(x_2-x_1)).

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2. Tangent Lines and Instantaneous Rate of Change

• Tangent Line: Touches curve at only one point.
• Slope of Tangent Line:
  • Instantaneous rate of change at (x = c) is (f'(c)) (the first derivative at that point).
• Normal Line:
  • Perpendicular to the tangent line.
  • Slope = negative reciprocal of tangent line’s slope.

Summary Table:

| Line Type | Touches Curve At | Slope Formula | Concept | |----------------|------------------|-----------------------------|-------------------------------------------| | Secant Line | 2 points | Avg rate: (\frac{\Delta y}{\Delta x}) | Average rate of change | | Tangent Line | 1 point | Instant rate: (f'(c)) | Instantaneous rate of change | | Normal Line | 1 point | (-1/f'(c)) | Perpendicular to tangent |

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3. Mean Value Theorem (MVT)

• Conditions:

  1. (f(x)) is continuous on ([a, b])
  2. (f(x)) is differentiable on ((a, b))

• Statement:
  There is (c) in ((a, b)) such that
  [ f'(c) = \frac{f(b) - f(a)}{b - a} ]

• Interpretation:
  At some point, the slope of the tangent equals the slope of the secant line.

• Application:
  Solve (f'(x) = \frac{f(b) - f(a)}{b - a}) for (x) in ((a, b)) to find (c).

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4. Rolle’s Theorem

• Conditions:

  1. (f(x)) is continuous on ([a, b])
  2. (f(x)) is differentiable on ((a, b))
  3. (f(a) = f(b))

• Statement:
  There is (c \in (a, b)) such that
  [ f'(c) = 0 ]

• Interpretation:
  Somewhere in the interval, there is a horizontal tangent (critical point).

• Application:
  Find where (f'(x) = 0) after confirming conditions.

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5. Absolute and Relative Extrema

• Absolute Min/Max: Highest/lowest values on the interval (including endpoints).

• Local (Relative) Min/Max: Highest/lowest point "locally" (not necessarily global).

• Critical Points: Points where (f'(x) = 0) or (f'(x)) is undefined.

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6. Tests for Local Extrema

First Derivative Test

• Local Maximum:
  If (f'(x)) changes from positive to negative at (c), then (f) has a local max at (c).

• Local Minimum:
  If (f'(x)) changes from negative to positive at (c), then (f) has a local min at (c).

Second Derivative Test

• If (f'(c) = 0):
  • If (f''(c) > 0): Relative minimum at (c) (concave up).
  • If (f''(c) < 0): Relative maximum at (c) (concave down).

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7. Concavity & Inflection Points

• Concave Up: (f''(x) > 0), (f'(x)) is increasing
• Concave Down: (f''(x) < 0), (f'(x)) is decreasing
• Inflection Point: Where concavity changes ((f''(x)=0) and (f''(x)) changes sign).

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8. L'Hôpital's Rule

• Purpose: Evaluate limits of indeterminate forms (like (\frac{0}{0}) or (\frac{\infty}{\infty})).
• Statement: [ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} ] (if limit exists and (g'(x) \ne 0) near (x = a)).

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9. Newton’s Method (Approximating Zeros)

• Purpose: Numerically approximate roots of (f(x) = 0).
• Iteration Formula: [ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} ]
• Process:
  1. Start with initial guess (x_0).
  2. Plug into formula to get improved (x_1, x_2, \ldots).
  3. Repeat until desired accuracy.

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References

• Formula sheet (see provided resources for additional formulas and worked examples).
• Example problems and further practice materials (see video description links).

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End of summary.