Basic Calculus Concepts to Know for Math for Non-Math Majors

Calculus helps us understand how things change and accumulate. By exploring functions, limits, derivatives, and integrals, we can analyze real-world situations, optimize outcomes, and model growth or decay, making it relevant for everyday life and various fields.

1. Functions and their graphs

   • A function is a relationship that assigns exactly one output for each input.
   • Graphs visually represent functions, showing how outputs change with inputs.
   • Key features of graphs include intercepts, slopes, and asymptotes.

2. Limits and continuity

   • A limit describes the value a function approaches as the input approaches a certain point.
   • Continuity means a function has no breaks, jumps, or holes in its graph.
   • Understanding limits is essential for defining derivatives and integrals.

3. Derivatives and differentiation rules

   • A derivative represents the rate of change of a function with respect to its variable.
   • Common differentiation rules include the power rule, product rule, and quotient rule.
   • Derivatives can be used to find slopes of tangent lines to curves at specific points.

4. Applications of derivatives (e.g., optimization, rates of change)

   • Derivatives help identify maximum and minimum values of functions (optimization).
   • They are used to calculate instantaneous rates of change in various contexts.
   • Understanding critical points (where the derivative is zero or undefined) is crucial for optimization.

5. Integrals and basic integration techniques

   • An integral represents the accumulation of quantities, such as area under a curve.
   • Basic techniques include the power rule for integration and substitution.
   • Definite integrals provide numerical values, while indefinite integrals yield general functions.

6. Fundamental Theorem of Calculus

   • This theorem connects differentiation and integration, showing they are inverse processes.
   • It states that the integral of a function can be evaluated using its antiderivative.
   • It provides a method for calculating definite integrals efficiently.

7. Applications of integrals (e.g., area under curves, volume)

   • Integrals can be used to calculate the area between curves and the x-axis.
   • They are also used to find volumes of solids of revolution through methods like disk and washer.
   • Applications extend to physics, economics, and biology for modeling real-world scenarios.

8. Exponential and logarithmic functions

   • Exponential functions grow rapidly and are characterized by a constant base raised to a variable exponent.
   • Logarithmic functions are the inverses of exponential functions and help solve equations involving exponentials.
   • Both types of functions are essential in modeling growth and decay processes.

9. Basic differential equations

   • A differential equation relates a function to its derivatives, often describing dynamic systems.
   • Solutions to differential equations can provide insights into behavior over time.
   • Common types include separable equations and first-order linear equations.

10. Sequences and series

    • A sequence is an ordered list of numbers, while a series is the sum of the terms of a sequence.
    • Understanding convergence and divergence is key to analyzing series.
    • Important concepts include arithmetic and geometric series, as well as the use of limits in series.