Applications of Derivatives

Tangent Line

y - f(a) = f'(a)(x - a)

The equation of the tangent line to f(x) at the point (a, f(a)). The slope is the derivative evaluated at x = a.

Tangent & Normal Lines

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Normal Line

y - f(a) = -\frac{1}{f'(a)}(x - a)

The normal line is perpendicular to the tangent line. Its slope is the negative reciprocal of the derivative.

Conditions: f'(a) ≠ 0.

Tangent & Normal Lines

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Critical Points

f'(c) = 0 \text{ or } f'(c) \text{ DNE} \Rightarrow c \text{ is a critical point}

Critical points occur where the derivative is zero or undefined. These are candidates for local extrema.

Curve Analysis

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First Derivative Test

f' \text{ changes } + \to - \Rightarrow \text{local max}; \quad f' \text{ changes } - \to + \Rightarrow \text{local min}

At a critical point c: if f' changes from positive to negative, c is a local max. If f' changes from negative to positive, c is a local min.

Curve Analysis

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Second Derivative Test

f'(c) = 0: \quad f''(c) > 0 \Rightarrow \text{local min}; \quad f''(c) < 0 \Rightarrow \text{local max}

At a critical point where f'(c) = 0: if f''(c) > 0, c is a local minimum (concave up). If f''(c) < 0, c is a local maximum (concave down). If f''(c) = 0, the test is inconclusive.

Curve Analysis

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Concavity

f''(x) > 0 \Rightarrow \text{concave up}; \quad f''(x) < 0 \Rightarrow \text{concave down}

The second derivative determines concavity. Concave up means the curve opens upward (bowl shape). Concave down means it opens downward.

Curve Analysis

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Inflection Points

f''(c) = 0 \text{ (or DNE) and concavity changes at } c \Rightarrow (c, f(c)) \text{ is an inflection point}

An inflection point is where the concavity changes. Find candidates where f''(x) = 0 or is undefined, then verify concavity changes.

Curve Analysis

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Mean Value Theorem

f'(c) = \frac{f(b) - f(a)}{b - a} \text{ for some } c \in (a, b)

If f is continuous on [a,b] and differentiable on (a,b), then there exists at least one c in (a,b) where the instantaneous rate of change equals the average rate of change.

Conditions: f must be continuous on [a, b] and differentiable on (a, b).

Theorems

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Rolle's Theorem

f(a) = f(b) \Rightarrow \exists\, c \in (a,b) : f'(c) = 0

If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then there exists at least one c in (a,b) where f'(c) = 0. This is a special case of MVT.

Conditions: f continuous on [a,b], differentiable on (a,b), and f(a) = f(b).

Theorems

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Related Rates

\frac{d}{dt}[\text{equation}] \text{ -differentiate both sides with respect to time}

Related rates problems involve finding how fast one quantity changes given how fast another changes. Differentiate an equation relating the quantities with respect to time.

Theorems

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Optimization

\text{Find critical points of } f'(x) = 0 \text{, then verify max/min via endpoint or derivative test}

Optimization finds the maximum or minimum value of a function. Set up the objective function, find critical points, and test them.

Applied Problems

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Linearization

L(x) = f(a) + f'(a)(x - a)

The linearization of f at a is the tangent line, used as a linear approximation of f near a. This is the same as the first-degree Taylor polynomial.

Applied Problems

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Differentials

dy = f'(x)\, dx

The differential dy approximates the change in y for a small change dx. Used for error estimation and approximation.

Applied Problems

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Newton's Method

x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

Newton's method iteratively approximates roots of f(x) = 0. Each step uses the tangent line to get a better approximation.

Conditions: f'(xₙ) ≠ 0 at each step. Convergence depends on the initial guess.

Applied Problems

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L'Hopital's Rule (Applications)

\lim_{x \to a} \frac{f(x)}{g(x)} \overset{\frac{0}{0}, \frac{\infty}{\infty}}{=} \lim_{x \to a} \frac{f'(x)}{g'(x)}

L'Hopital's Rule applied to evaluate limits involving indeterminate forms such as 0·∞, ∞-∞, 0⁰, ∞⁰, 1^∞ by algebraic rearrangement to 0/0 or ∞/∞.

Applied Problems

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