MATH221 First Semester Calculus Lecture Notes

Introduction to Calculus

Calculus is the mathematical study of change and motion. It is divided into two main branches: differential calculus (concerned with rates of change and slopes of curves) and integral calculus (concerned with accumulation of quantities and areas under curves). In MATH221, we explore fundamental concepts that form the foundation of calculus.

1. Functions and Their Properties

A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Key properties of functions include:

• Domain and Range: The domain is the set of all possible input values, while the range is the set of all possible output values.
• Even and Odd Functions: A function f is even if f(-x) = f(x) and odd if f(-x) = -f(x).
• Periodic Functions: Functions that repeat their values at regular intervals.

2. Limits and Continuity

The concept of a limit is fundamental to calculus. The limit of a function f(x) as x approaches a value a is denoted as:

This means that as x gets arbitrarily close to a, f(x) gets arbitrarily close to L.

Limit Laws

• Limit of sum: lim (f + g) = lim f + lim g
• Limit of product: lim (f g) = lim f lim g
• Limit of quotient: lim (f/g) = lim f/lim g (provided lim g 0)

Continuity

A function f is continuous at a point a if three conditions are met:

1. f(a) is defined
2. lim_xa f(x) exists
3. lim_xa f(x) = f(a)

3. Differentiation

The derivative of a function represents its instantaneous rate of change. Geometrically, it corresponds to the slope of the tangent line to the graph of the function at a given point.

Definition of the Derivative

Differentiation Rules

• Power Rule: If f(x) = xⁿ, then f'(x) = nx^n-1
• Product Rule: (fg)' = f'g + fg'
• Quotient Rule: (f/g)' = (f'g - fg')/g²
• Chain Rule: If y = f(u) and u = g(x), then dy/dx = f'(g(x)) g'(x)

Applications of Derivatives

• Finding maximum and minimum values of functions
• Analyzing the concavity of functions
• Solving optimization problems
• Sketching curves

4. Integration

Integration is the reverse process of differentiation. There are two main types: indefinite integrals (antiderivatives) and definite integrals.

Antiderivatives

An antiderivative of a function f is a function F such that F' = f. We denote the most general antiderivative of f as:

where C is the constant of integration.

Integration Rules

• Constant Rule: k dx = kx + C
• Power Rule: xⁿ dx = xⁿ⁺¹/(n+1) + C, (n -1)
• Exponential Rule: e^x dx = e^x + C
• Logarithm Rule: 1/x dx = ln|x| + C

The Fundamental Theorem of Calculus

This theorem connects differentiation and integration and is perhaps the most important result in calculus. It has two parts:

Definite Integrals

The definite integral _a^b f(x) dx represents the signed area between the graph of f and the x-axis from x = a to x = b. Areas above the x-axis are counted as positive, while areas below are negative.

5. Applications of Integration

Integration has numerous applications in mathematics, science, and engineering:

• Area between curves: The area between two curves y = f(x) and y = g(x) from x = a to x = b is _a^b |f(x) - g(x)| dx
• Volumes of solids: Using methods like disks, washers, and cylindrical shells
• Arc length: The length of a curve y = f(x) from x = a to x = b is _a^b (1 + [f'(x)]²) dx
• Work and physics applications: Calculating work done by variable forces

6. Techniques of Integration

Beyond basic integration rules, several advanced techniques are often needed:

Substitution Method

Used when an integral contains a function and its derivative. If we let u = g(x), then du = g'(x) dx, and f(g(x))g'(x) dx becomes f(u) du.

Integration by Parts

Partial Fractions

A method for integrating rational functions by expressing them as sums of simpler fractions.

7. Transcendental Functions

In calculus, we frequently work with transcendental functions (exponential, logarithmic, trigonometric, and inverse trigonometric functions).

Exponential and Logarithmic Functions

• d/dx(e^x) = e^x
• d/dx(ln x) = 1/x
• e^x dx = e^x + C
• 1/x dx = ln|x| + C

Trigonometric Functions

• d/dx(sin x) = cos x
• d/dx(cos x) = -sin x
• d/dx(tan x) = sec² x
• sin x dx = -cos x + C
• cos x dx = sin x + C
• sec² x dx = tan x + C

8. Approximation Methods

When exact solutions are not possible or practical, approximation methods become essential:

Linear Approximation

The linear approximation of f(x) near x = a is: L(x) = f(a) + f'(a)(x - a)

Taylor Series

This represents a function as an infinite sum of terms calculated from the values of its derivatives at a single point.

Numerical Integration

Methods like the Trapezoidal Rule and Simpson's Rule offer ways to approximate definite integrals when an antiderivative cannot be found analytically.

Conclusion

MATH221 provides a solid foundation in calculus, equipping students with powerful tools for analyzing change and accumulation. The concepts of limits, derivatives, and integrals form a framework for understanding and solving problems across many scientific and engineering disciplines. These fundamental principles are not only essential for higher mathematics but also for modeling real-world phenomena in physics, economics, biology, and beyond.