Introduction to Differential Calculus

Calculus opens doors to understanding how things change in our world. When you want to analyze the speed of a car, the growth of a population, or the trajectory of a rocket, differential calculus gives you the tools to do it.

This course, taught at Occidental Mindoro State College, focuses on the fundamental concepts that will help you understand how quantities change over time or space.

Pro Tip: Don't be intimidated by calculus! It breaks down complex problems into small, manageable pieces—that's actually where the name comes from, as "calculus" originally meant "small stone" used for counting.

Throughout this course, you'll build skills that are essential for engineering, physics, economics, and many other fields.

What is Calculus?

Calculus is all about change—how things grow, shrink, or transform over time. Unlike algebra that deals with static equations, calculus helps us understand motion and dynamic systems.

Calculus has two major branches that work together. Differential calculus finds how quickly functions change by calculating derivatives. For example, when you want to know how fast an object is accelerating at a specific moment, you use derivatives.

Integral calculus does the opposite—it finds the original function when you know its rate of change. This is like figuring out how far a car traveled when you only know its speed at different times.

The two branches are connected by the Fundamental Theorem of Calculus, which shows that differentiation and integration are inverse operations—similar to how multiplication and division relate to each other.

Understanding Functions

Functions are the building blocks of calculus. A function is a relationship that assigns exactly one output value to each input value in its domain. Think of a function as a machine that processes inputs and produces predictable outputs.

For example, in the function y = x² + 1, when you input x = 2, you'll always get y = 5. The domain consists of all possible input values where the function can operate.

Function notation gives us a convenient shorthand to work with functions. When we write f(x) = x² + 1, "f" is the function name, "x" is the input (independent variable), and "f(x)" represents the output (dependent variable) that depends on what value of x you use.

Remember: When a relationship doesn't produce exactly one output for each input, it's not a function. For example, y² = x + 1 is not a function because some x-values give two different y-values.

Function Notation and Evaluation

Function notation gives you a powerful way to express mathematical relationships clearly. When you see f(x) = 3x² + 6x - 4, you can read it as "f of x equals 3x-squared plus 6x minus 4."

The beauty of function notation is its flexibility. The same function can be written as g(x), P(x), or w(x)—the name doesn't matter as long as the relationship stays the same. This lets mathematicians work with multiple functions at once without confusion.

Evaluating functions means finding outputs for specific inputs. For instance, to find f(2) when f(x) = 3x² + 6x - 4, you substitute 2 for every x in the equation: f(2) = 3(2)² + 6(2) - 4 = 3(4) + 12 - 4 = 12 + 12 - 4 = 20

Quick Tip: When evaluating complex inputs like f$x-4$ or f$5x-2$, first substitute the entire expression for every x in the function, then simplify. This technique works for any input, whether it's a number, variable, or expression.

Roots of Functions

Finding the roots of a function means discovering where the function equals zero—the x-values that make f(x) = 0. These are crucial points where graphs cross the x-axis and often represent solutions to real-world problems.

When solving for roots, you're essentially finding the values that make the function disappear. For a function like f(x) = x² - 4, you'd set it equal to zero and solve: x² - 4 = 0, which gives x = ±2. These roots tell you the function crosses the x-axis at x = 2 and x = -2.

Different functions require different solving techniques. Quadratic functions like g(y) = 12y² + 11y - 5 can be solved using the quadratic formula, while polynomial functions might need factoring or numerical methods. Rational functions like f(z) = z/$z-5$ - 4/$z-8$ require algebraic manipulation and careful attention to restrictions.

Challenge Yourself: When you encounter a complex function like R(x) = x⁴ + 6x² - 27, try factoring it as $x² + 9$$x² - 3$ to find its roots more easily. Practice makes perfect!

Domain and Range of Functions

The domain of a function is the set of all valid input values $x-values$ that the function can accept. Understanding domain helps you avoid mathematical errors like division by zero or taking square roots of negative numbers.

For simple functions like f(x) = 6x - 4, the domain includes all real numbers since you can substitute any value for x. But functions with denominators or square roots have restrictions. For example, in f(x) = $x-4$/$x²-2x-15$, you must exclude values that make the denominator zero.

The range is the complete set of output values $y-values$ that a function produces. For a function like b(x) = -2x² + 12x + 5, the range is limited because the parabola opens downward, creating a maximum value.

Practical Tip: When finding domain restrictions, look for denominators (exclude values that make them zero) and even roots (exclude values that make expressions under the radical negative). These are the most common domain limitations you'll encounter.

Function Composition

Function composition combines two functions to create a new function where the output of one becomes the input of another. We write this as (f∘g)(x) = f(g(x)), which means "apply g first, then apply f to the result."

Function composition is like a mathematical assembly line. For example, if f(x) = 2x + 1 and g(x) = -2x² + 12x + 5, then (f∘g)(x) = f(g(x)) = 2$-2x² + 12x + 5$ + 1 = -4x² + 24x + 11. This creates an entirely new function with its own behavior.

When evaluating compositions at specific values, like (f∘g)(5), you first find g(5), then plug that result into f. This two-step process ensures you're following the correct order of operations.

Remember: Function composition is not commutative, meaning (f∘g)(x) is usually different from (g∘f)(x). Always pay attention to the order in which you apply the functions!

Inverse Functions

An inverse function undoes what the original function did—it's like pressing "undo" on your calculator. If f takes x and gives y, then f⁻¹ takes y and gives back x. We write the inverse as f⁻¹(x).

You can verify if two functions are inverses by checking if (f∘g)(x) = x and (g∘f)(x) = x. For example, if f(x) = 3x - 2 and g(x) = $x+2$/3, they're inverses because applying one after the other always gives you back the original input.

To find an inverse function, follow these steps: replace f(x) with y, swap x and y to get x = f(y), solve for y, and then replace y with f⁻¹(x). For example, to find the inverse of f(x) = 3x - 2, write y = 3x - 2, swap to get x = 3y - 2, solve for y to get y = $x+2$/3, and finally write f⁻¹(x) = $x+2$/3.

Visual Insight: Inverse functions are reflections of each other across the line y = x. This symmetry means that if the point (a,b) lies on the graph of f, then the point (b,a) lies on the graph of f⁻¹.

Trigonometric Functions

Trigonometric functions relate angles in a right triangle to the ratios of its sides. These functions are fundamental to describing periodic phenomena in physics, engineering, and many natural processes.

The six main trigonometric functions are:

• sine (sin): opposite side ÷ hypotenuse
• cosine (cos): adjacent side ÷ hypotenuse
• tangent (tan): opposite side ÷ adjacent side (or sin ÷ cos)
• cosecant (csc): hypotenuse ÷ opposite side (or 1 ÷ sin)
• secant (sec): hypotenuse ÷ adjacent side (or 1 ÷ cos)
• cotangent (cot): adjacent side ÷ opposite side (or 1 ÷ tan)

The mnemonic SOH CAH TOA helps you remember the three primary functions: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Application Tip: When solving real-world problems, identifying the right angle and labeling the sides correctly in relation to the angle is crucial for selecting the appropriate trigonometric function.

Trigonometric Identities

Trigonometric identities are equations that are true for all values where the functions are defined. These powerful relationships help simplify complex expressions and solve challenging problems.

The reciprocal identities show the relationships between pairs of functions: sin(θ) = 1/csc(θ), cos(θ) = 1/sec(θ), and tan(θ) = 1/cot(θ). These are useful when converting between different trigonometric expressions.

The Pythagorean identities connect the squared values of functions: sin²(θ) + cos²(θ) = 1 is the most fundamental one. From this, we can derive tan²(θ) + 1 = sec²(θ) and 1 + cot²(θ) = csc²(θ).

Sum and difference formulas like sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β) are essential for breaking down complex angles. These formulas, along with double angle formulas like sin(2θ) = 2sin(θ)cos(θ), help solve problems involving multiple angles.

Strategy Insight: When tackling trigonometric problems, look for ways to use identities to transform expressions into more manageable forms. Often, several approaches are possible—the key is finding the most efficient path to the solution.