Key Concepts in Differential Calculus: Theorems, Tests, and Applications

The Mean Value Theorem

Definition and Application

The Mean Value Theorem (MVT) is a fundamental result in calculus that connects the average rate of change of a function to its instantaneous rate of change. If a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) such that:

• Statement: $f'(c) = \frac{f(b) - f(a)}{b - a}$

• Interpretation: There is at least one point where the tangent to the curve is parallel to the secant line joining (a, f(a)) and (b, f(b)).

• Example: For f(x) = x^2 on [1, 3], there exists c such that $f'(c) = \frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = 4$, so $2c = 4 \implies c = 2$.

Increasing/Decreasing Test

Using the First Derivative

The Increasing/Decreasing Test uses the sign of the first derivative to determine where a function is increasing or decreasing.

• If $f'(x) > 0$ on an interval, then f is increasing there.

• If $f'(x) < 0$ on an interval, then f is decreasing there.

• Critical points occur where $f'(x) = 0$ or $f'(x)$ is undefined.

• Example: For f(x) = x^3 - 3x, $f'(x) = 3x^2 - 3$. Set $f'(x) = 0$ to find critical points at $x = \pm 1$.

Inverse Functions

Definition and Properties

An inverse function reverses the effect of the original function. If f is one-to-one, its inverse f-1 satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

• Derivative of the Inverse: If f is differentiable and has an inverse, then:

$\left( f^{-1} \right)'(y) = \frac{1}{f'(x)}$ where $y = f(x)$

• Example: If $f(x) = e^x$, then $f^{-1}(x) = \ln x$ and $\frac{d}{dx} \ln x = \frac{1}{x}$.

Exponential Functions and Their Derivatives

Key Properties

Exponential functions have the form $f(x) = a^x$ where $a > 0$ and $a \neq 1$.

• Derivative: $\frac{d}{dx} a^x = a^x \ln a$

• Special case: $\frac{d}{dx} e^x = e^x$

• Example: $\frac{d}{dx} 2^x = 2^x \ln 2$

Logarithmic Functions

Definition and Properties

Logarithmic functions are the inverses of exponential functions. The natural logarithm is $\ln x$.

• Domain: $x > 0$

• Range: $(-\infty, \infty)$

• Key property: $\ln(ab) = \ln a + \ln b$

Derivatives of Logarithmic Functions

Formulas and Examples

• Derivative of Natural Logarithm: $\frac{d}{dx} \ln x = \frac{1}{x}$ for $x > 0$

• General Logarithm: $\frac{d}{dx} \log_a x = \frac{1}{x \ln a}$

• Example: $\frac{d}{dx} \ln(x^2 + 1) = \frac{2x}{x^2 + 1}$

Inverse Trigonometric Functions

Definitions and Derivatives

• arcsin(x): $\frac{d}{dx} \arcsin x = \frac{1}{\sqrt{1 - x^2}}$

• arccos(x): $\frac{d}{dx} \arccos x = -\frac{1}{\sqrt{1 - x^2}}$

• arctan(x): $\frac{d}{dx} \arctan x = \frac{1}{1 + x^2}$

• Example: $\frac{d}{dx} \arctan(2x) = \frac{2}{1 + 4x^2}$

Hyperbolic Functions

Definitions and Derivatives

• Definitions:

  • $\sinh x = \frac{e^x - e^{-x}}{2}$

  • $\cosh x = \frac{e^x + e^{-x}}{2}$

• Derivatives:

  • $\frac{d}{dx} \sinh x = \cosh x$

  • $\frac{d}{dx} \cosh x = \sinh x$

• Example: $\frac{d}{dx} \tanh x = \text{sech}^2 x$

Indeterminate Forms and L'Hospital's Rule

Resolving Limits

When evaluating limits, some expressions yield indeterminate forms such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$. L'Hospital's Rule provides a method to resolve these:

• Rule: If $\lim_{x \to a} \frac{f(x)}{g(x)}$ yields $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then:

$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$ (if the limit on the right exists)

• Example: $\lim_{x \to 0} \frac{\sin x}{x} = 1$ (using L'Hospital's Rule)

Maximum and Minimum Values

Finding Extrema

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

• First Derivative Test: Determines if a critical point is a local maximum, minimum, or neither.

• Second Derivative Test: If $f''(x) > 0$ at a critical point, it's a local minimum; if $f''(x) < 0$, it's a local maximum.

• Example: For $f(x) = x^2$, $f'(x) = 0$ at $x = 0$, $f''(0) = 2 > 0$ so $x = 0$ is a local minimum.

How Derivatives Affect the Shape of a Graph

Concavity and Inflection Points

• First Derivative: Indicates increasing or decreasing behavior.

• Second Derivative: Indicates concavity:

  • $f''(x) > 0$: Concave up

  • $f''(x) < 0$: Concave down

• Inflection Point: Where concavity changes (i.e., $f''(x)$ changes sign).

• Example: For $f(x) = x^3$, $f''(x) = 6x$, so inflection at $x = 0$.

Summary of Curve Sketching

Steps for Analyzing and Sketching Functions

• 1. Find domain and intercepts.

• 2. Find critical points using $f'(x)$.

• 3. Determine intervals of increase/decrease.

• 4. Find inflection points and intervals of concavity using $f''(x)$.

• 5. Identify asymptotes and end behavior.

• 6. Sketch the graph using all gathered information.