Calculus I: Differentiation, Applications, and Optimization Study Guide

Techniques of Differentiation

Basic Differentiation Rules

Differentiation is a fundamental operation in calculus, used to find the rate at which a function changes. The derivative of a function at a point gives the slope of the tangent line to the graph at that point.

• Power Rule: For $f(x) = x^n$, $f'(x) = n x^{n-1}$

• Product Rule: For $f(x) = u(x)v(x)$, $f'(x) = u'(x)v(x) + u(x)v'(x)$

• Quotient Rule: For $f(x) = \frac{u(x)}{v(x)}$, $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$

• Chain Rule: For $f(x) = g(h(x))$, $f'(x) = g'(h(x)) \cdot h'(x)$

Example: Find $\frac{d}{dx} \ln(\ln(x)) - x$.

• Apply the chain rule to $\ln(\ln(x))$:

• $\frac{d}{dx} \ln(\ln(x)) = \frac{1}{\ln(x)} \cdot \frac{1}{x} = \frac{1}{x \ln(x)}$

• Derivative of $-x$ is $-1$.

Differentiation of Logarithmic and Exponential Functions

• Natural Logarithm: $\frac{d}{dx} \ln(x) = \frac{1}{x}$

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

• Exponential Function: $\frac{d}{dx} e^{x} = e^{x}$

Example: If $f(x) = \log_3(x^2)$, then $f'(x) = \frac{2}{x \ln(3)}$.

Differentiation of Inverse Trigonometric Functions

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

• Example: For $y = \tan^{-1}\left(\frac{a}{x}\right)$, $\frac{dy}{dx} = \frac{-a}{x^2 + a^2}$

Differentiation Applications

Critical Points and Extrema

Critical points occur where the derivative of a function is zero or undefined. These points are candidates for local maxima or minima.

• Local Maximum: A point where $f'(x) = 0$ and $f''(x) < 0$.

• Local Minimum: A point where $f'(x) = 0$ and $f''(x) > 0$.

• Global Maximum/Minimum: The largest/smallest value of $f(x)$ on a given interval.

Example: Find the maximum value of $f(x) = x^2 + 2x + 4$ on $[-3, 3]$.

• Find critical points by setting $f'(x) = 2x + 2 = 0 \implies x = -1$.

• Evaluate $f(x)$ at $x = -3, -1, 3$:

• $f(-3) = 9 - 6 + 4 = 7$

• $f(-1) = 1 - 2 + 4 = 3$

• $f(3) = 9 + 6 + 4 = 19$

• Maximum is $19$at$x = 3$.

Concavity and Inflection Points

Concavity describes the direction a curve bends. Inflection points are where the concavity changes.

• Concave Up: $f''(x) > 0$

• Concave Down: $f''(x) < 0$

• Inflection Point: Where $f''(x)$ changes sign.

Example: If $f''(x) = x^2 + 1$, then $f''(x) > 0$ everywhere, so $f$ is concave up everywhere.

Optimization Problems

Optimization involves finding the maximum or minimum values of a function, often subject to constraints.

• Set up a function representing the quantity to be optimized.

• Find critical points by setting the derivative to zero.

• Check endpoints if the domain is restricted.

Example: To minimize the sum of squares of two numbers $a$ and $b$ with $a + b = 36$, set $S(a) = a^2 + (36 - a)^2$.

Limits and Continuity

Evaluating Limits

Limits describe the behavior of a function as the input approaches a particular value.

• Direct Substitution: If $f(x)$ is continuous at $a$, $\lim_{x \to a} f(x) = f(a)$.

• Indeterminate Forms: Use algebraic manipulation or L'Hospital's Rule for forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

Example: $\lim_{x \to \infty} x \sin\left(\frac{2\pi}{x}\right)$

• As $x \to \infty$, $\frac{2\pi}{x} \to 0$, so $\sin\left(\frac{2\pi}{x}\right) \approx \frac{2\pi}{x}$

• Thus, $x \sin\left(\frac{2\pi}{x}\right) \approx x \cdot \frac{2\pi}{x} = 2\pi$

Continuity and Differentiability

• Continuous Function: No breaks, jumps, or holes in the graph.

• Differentiable Function: The derivative exists at every point in the domain.

• Intermediate Value Theorem: If $f$ is continuous on $[a, b]$ and $f(a) \neq f(b)$, then $f$ takes every value between $f(a)$ and $f(b)$.

Applications to Motion: Position, Velocity, and Acceleration

Sign Charts and Particle Motion

Sign charts help analyze the behavior of functions and their derivatives, especially in motion problems.

• Position Function: $s(t)$ gives the location of a particle at time $t$.

• Velocity: $v(t) = s'(t)$

• Acceleration: $a(t) = s''(t)$

• Particle is at rest when $v(t) = 0$.

• Particle is slowing down when velocity and acceleration have opposite signs.

Example: If $s'(t)$ changes sign at $t = 2, 5, 8$, analyze intervals for rest and slowing down.

Graphical Analysis of Derivatives

Interpreting the Graph of $f'(x)$

The graph of the derivative $f'(x)$ provides information about the increasing/decreasing behavior and local extrema of $f(x)$.

• $f$ is increasing where $f'(x) > 0$.

• $f$ is decreasing where $f'(x) < 0$.

• Local minima of $f$ occur where $f'(x)$ changes from negative to positive.

• Concavity: $f$ is concave up where $f''(x) > 0$, concave down where $f''(x) < 0$.

Example: Given a graph of $f'(x)$, identify intervals where $f$ is decreasing, local minima, and concave down intervals.

Linear Approximation and Estimation

Linearization

Linearization uses the tangent line at a point to approximate the value of a function near that point.

• Formula: $L(x) = f(a) + f'(a)(x - a)$

• If $f$ is concave up at $a$, the linearization underestimates $f(x)$ for $x > a$.

• If $f$ is concave down at $a$, the linearization overestimates $f(x)$ for $x > a$.

Example: Use the linearization of $f(x) = e^x$ at $x = 0$ to estimate $e^{0.1}$.

Optimization with Constraints

Maximizing Area with Fixed Perimeter

Problems involving maximizing area with a fixed perimeter are classic optimization problems in calculus.

• Express area in terms of one variable using the constraint.

• Differentiate and set to zero to find critical points.

• Check endpoints and interpret results in context.

Example: Given a rectangle with fixed perimeter, maximize area by setting up $A = x y$ and using the constraint $2x + 2y = P$.

Summary Table: Key Differentiation Formulas

Function	Derivative
$x^n$	$n x^{n-1}$
$e^x$	$e^x$
$\ln(x)$	$\frac{1}{x}$
$\tan^{-1}(x)$	$\frac{1}{1 + x^2}$
$\log_a(x)$	$\frac{1}{x \ln(a)}$

Additional info:

• Some questions involve graphical analysis and sign charts, which are essential for understanding the behavior of functions and their derivatives.

• Optimization and linearization are covered, which are key applications of differentiation.

• Motion problems connect calculus concepts to physical applications.