📏Honors Pre-Calculus Unit 12 – Introduction to Calculus

Key Concepts and Definitions

• Calculus branch of mathematics that studies rates of change and accumulation of quantities
• Limit describes the value a function approaches as the input approaches a certain value or infinity
• Continuity property of a function where there are no breaks or gaps in its graph
• Derivative measures the rate of change of a function at a given point (instantaneous rate of change)
• Differentiation process of finding the derivative of a function
  • Involves applying differentiation rules and techniques to compute derivatives
• Integral represents the area under a curve or the accumulation of quantities over an interval
• Integration process of finding the integral of a function (opposite of differentiation)
  • Involves applying integration rules and techniques to compute integrals
• Fundamental Theorem of Calculus establishes the relationship between differentiation and integration

Limits and Continuity

• Limit of a function $f(x)$ as $x$ approaches $a$ is denoted as $\lim_{x \to a} f(x) = L$
  • Means the function values get arbitrarily close to $L$ as $x$ gets closer to $a$
• One-sided limits consider the limit from either the left or right side of a point
  • Left-hand limit: $\lim_{x \to a^-} f(x)$, approaching $a$ from values less than $a$
  • Right-hand limit: $\lim_{x \to a^+} f(x)$, approaching $a$ from values greater than $a$
• Limit laws allow for the evaluation of limits of combined functions (sum, difference, product, quotient)
• Continuity at a point $a$ requires three conditions:
  • $f(a)$ is defined
  • $\lim_{x \to a} f(x)$ exists
  • $\lim_{x \to a} f(x) = f(a)$
• Types of discontinuities include removable, jump, and infinite discontinuities
• Intermediate Value Theorem states that if a function is continuous on a closed interval $[a, b]$ and $f(a) \neq f(b)$, then the function takes on every value between $f(a)$ and $f(b)$ at least once

Derivatives and Differentiation

• Derivative of a function $f(x)$ at a point $a$ is denoted as $f'(a)$ or $\frac{d}{dx}f(x)|_{x=a}$
  • Represents the slope of the tangent line to the graph of $f(x)$ at the point $(a, f(a))$
• Derivative can be found using the limit definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$
• Power rule for differentiating functions of the form $f(x) = x^n$: $\frac{d}{dx}x^n = nx^{n-1}$
• Product rule for differentiating the product of two functions $u(x)$ and $v(x)$: $(uv)' = u'v + uv'$
• Quotient rule for differentiating the quotient of two functions $u(x)$ and $v(x)$: $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$
• Chain rule for differentiating composite functions: $(f(g(x)))' = f'(g(x)) \cdot g'(x)$
• Implicit differentiation technique for finding derivatives of implicitly defined functions

Applications of Derivatives

• Derivatives can be used to find the slope of a tangent line to a curve at a given point
• Marginal analysis in economics uses derivatives to measure the rate of change of cost, revenue, or profit
• Optimization problems involve finding the maximum or minimum values of a function
  • Solved by setting the derivative equal to zero and analyzing critical points
• Related rates problems involve finding the rate of change of one quantity in terms of another
  • Solved by expressing the relationship between quantities and differentiating with respect to time
• Derivatives can be used to analyze the behavior of a function (increasing, decreasing, concavity)
• L'Hôpital's rule for evaluating limits of indeterminate forms (0/0 or $\infty/\infty$) using derivatives

Integrals and Integration

• Indefinite integral of a function $f(x)$ is denoted as $\int f(x) dx$ and represents a family of functions (antiderivatives)
  • Antiderivatives differ by a constant $C$, called the constant of integration
• Definite integral of a function $f(x)$ over the interval $[a, b]$ is denoted as $\int_a^b f(x) dx$
  • Represents the area under the curve $f(x)$ between $x = a$ and $x = b$
• Riemann sums approximate the area under a curve by dividing the interval into subintervals and summing the areas of rectangles
• Integration rules include the power rule for integrals, $\int x^n dx = \frac{1}{n+1}x^{n+1} + C$ for $n \neq -1$
• Substitution rule (u-substitution) is a technique for integrating composite functions
  • Involves substituting a new variable $u$ to simplify the integral
• Integration by parts is a technique for integrating products of functions
  • Based on the product rule for derivatives: $\int u dv = uv - \int v du$

Fundamental Theorem of Calculus

• Part 1: If $f(x)$ is continuous on $[a, b]$, then $\frac{d}{dx} \int_a^x f(t) dt = f(x)$
  • The derivative of the integral of a function is the original function
• Part 2: If $F(x)$ is an antiderivative of $f(x)$ on $[a, b]$, then $\int_a^b f(x) dx = F(b) - F(a)$
  • The definite integral of a function can be evaluated using an antiderivative
• Establishes the relationship between differentiation and integration as inverse operations
• Allows for the evaluation of definite integrals without using Riemann sums or limit definitions
• Enables the calculation of areas, volumes, and other quantities using definite integrals

Problem-Solving Strategies

• Read and understand the problem, identifying the given information and the desired outcome
• Sketch a diagram or graph to visualize the problem, if applicable
• Identify the appropriate concepts, formulas, or techniques needed to solve the problem
• Break down the problem into smaller, manageable steps
  • Solve each step sequentially, using the results from previous steps
• Simplify expressions and equations whenever possible
• Check the reasonableness of the answer by estimating or using common sense
• Verify the solution by substituting it back into the original problem or equation
• Analyze the solution and interpret the results in the context of the problem

Real-World Applications

• Velocity and acceleration can be modeled using derivatives
  • Velocity is the rate of change of position (first derivative)
  • Acceleration is the rate of change of velocity (second derivative)
• Optimization problems in business and economics (maximizing profit, minimizing cost)
• Population growth models use differential equations to describe the rate of change of a population over time
• Marginal analysis in economics (marginal cost, marginal revenue, marginal profit)
• Area between curves can be calculated using definite integrals
• Volume of solids of revolution can be found by integrating cross-sectional areas
• Work done by a variable force can be calculated using definite integrals
• Probability distributions and expected values in statistics involve integration