Basic Calculus Mini-Course — Learn Calculus Online

Welcome to the Course

Six self-contained lessons. Study at your own pace.

This mini-course is designed for absolute beginners. You'll learn the fundamental concepts of calculus in a clear, friendly way. Each lesson builds on the previous one, but you can also jump around.

📚 What You'll Learn

Lesson 1: What calculus is and why it matters.
Lesson 2: Limits — the foundation of calculus.
Lesson 3: Derivatives — rates of change.
Lesson 4: Applications of derivatives — optimization, motion, and more.
Lesson 5: Integrals — accumulation and area.
Lesson 6: Applications of integrals — volume, averages, and real-world problems.

🛠️ Tools & Resources

Explore these interactive tools to deepen your understanding and practice calculus concepts.

🧮

Scientific Calculator

Perform advanced calculations with trig, log, exponential, factorial, and more.

Launch Calculator →

📈

Graphing Tools

Visualize functions, tangent lines, and areas under curves with interactive graphs.

View Graphs →

🎯 Ready to Begin?

Use the menu above to navigate, or click the button below to start with Lesson 1.

Lesson 1

Introduction to Calculus

What calculus is, why it matters, and the big ideas behind limits, derivatives, and integrals.

What is Calculus?

Calculus is the branch of mathematics that studies change and motion. While algebra and geometry give us tools to describe static situations, calculus lets us understand how quantities evolve over time, how they accumulate, and how they relate to one another.

Calculus is traditionally divided into two main branches:

Differential calculus — studies rates of change (derivatives).
Integral calculus — studies accumulation (integrals).

These two ideas are connected by the Fundamental Theorem of Calculus, which we'll explore in later lessons.

Why Learn Calculus?

Calculus is the language of the physical world. It's used in physics, engineering, economics, biology, computer science, and many other fields. Here are a few examples:

📈 Predicting population growth
🚀 Calculating rocket trajectories
📊 Optimizing business profits
🧬 Modeling the spread of diseases

The Three Big Ideas

Throughout this mini-course, we'll focus on three core concepts:

1. Limits

A limit describes what happens to a function as the input approaches a certain value. It's the foundation of both derivatives and integrals.

$$ \lim_{x \to 2} (x^2 + 1) = 5 $$

2. Derivatives

A derivative measures the instantaneous rate of change of a function. It tells us how steep a curve is at any point.

$$ \frac{d}{dx} x^2 = 2x $$

3. Integrals

An integral measures the accumulation of a quantity — the area under a curve, the total distance traveled, or the total growth over time.

$$ \int_0^1 2x \, dx = 1 $$

What's Next?

In the next lesson, we'll dive deeper into limits — the building block of all calculus.

Lesson 2

Limits

The foundation of calculus — understanding what happens as variables approach a value.

What is a Limit?

A limit describes the value that a function approaches as the input approaches some value. Limits are the bedrock of calculus — they make derivatives and integrals possible.

We write limits like this:

\lim_{x \to a} f(x) = L

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

Understanding Limits Intuitively

Consider the function $ f(x) = \frac{x^2 - 1}{x - 1} $. This function is undefined at $ x = 1 $, but what happens as $ x $ gets closer to 1?

Let's test some values:

$ x = 0.9 \Rightarrow f(0.9) = 1.9 $
$ x = 0.99 \Rightarrow f(0.99) = 1.99 $
$ x = 0.999 \Rightarrow f(0.999) = 1.999 $
$ x = 1.001 \Rightarrow f(1.001) = 2.001 $
$ x = 1.01 \Rightarrow f(1.01) = 2.01 $

As $ x $ gets closer to 1, $ f(x) $ gets closer to 2. So we say:

\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2

One-Sided Limits

Sometimes we care about approaching from the left or the right:

$ \lim_{x \to a^-} f(x) $ — left-hand limit (from below)
$ \lim_{x \to a^+} f(x) $ — right-hand limit (from above)

If both one-sided limits exist and are equal, the two-sided limit exists.

Key Limit Rules

Sum rule: $ \lim (f+g) = \lim f + \lim g $
Difference rule: $ \lim (f-g) = \lim f - \lim g $
Product rule: $ \lim (f \cdot g) = \lim f \cdot \lim g $
Quotient rule: $ \lim \frac{f}{g} = \frac{\lim f}{\lim g} $ (if $ \lim g \neq 0 $)
Constant multiple rule: $ \lim (c \cdot f) = c \cdot \lim f $

Practice

Try evaluating these limits:

$ \lim_{x \to 3} (2x + 1) $
$ \lim_{x \to 0} \frac{\sin x}{x} $
$ \lim_{x \to \infty} \frac{1}{x} $

Lesson 3

Derivatives

Rates of change, slopes of curves, and the rules that make differentiation powerful.

What is a Derivative?

A derivative measures the instantaneous rate of change of a function. Geometrically, it's the slope of the tangent line to the curve at a given point.

The derivative of $ f(x) $ with respect to $ x $ is denoted:

f'(x) = \frac{d}{dx} f(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}

Basic Differentiation Rules

Power Rule

For any real number $ n $: $ \frac{d}{dx} x^n = n \cdot x^{n-1} $

Example: $ \frac{d}{dx} x^3 = 3x^2 $

Constant Rule

The derivative of a constant is zero: $ \frac{d}{dx} c = 0 $

Sum & Difference Rules

$ \frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x) $

Product Rule

$ \frac{d}{dx} [f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x) $

Quotient Rule

$ \frac{d}{dx} \left[ \frac{f(x)}{g(x)} \right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2} $

Common Derivatives

$ \frac{d}{dx} \sin x = \cos x $
$ \frac{d}{dx} \cos x = -\sin x $
$ \frac{d}{dx} e^x = e^x $
$ \frac{d}{dx} \ln x = \frac{1}{x} $

Practice Problems

Find the derivative of each function:

$ f(x) = 3x^2 + 2x - 5 $
$ g(x) = \sin(x) \cdot x^2 $
$ h(x) = \frac{x^2 + 1}{x} $

Lesson 4

Applications of Derivatives

Optimization, motion, tangent lines — putting derivatives to work in the real world.

Tangent Lines

The derivative $ f'(a) $ gives the slope of the tangent line to $ f(x) $ at $ x = a $. The equation of the tangent line is:

$$ y - f(a) = f'(a)(x - a) $$

Rates of Change

Velocity: $ v(t) = s'(t) $ where $ s(t) $ is position.
Acceleration: $ a(t) = v'(t) = s''(t) $.
Marginal cost/revenue: the derivative of cost/revenue functions.

Optimization

Derivatives help us find maximum and minimum values of functions. To find extrema:

Find $ f'(x) $.
Set $ f'(x) = 0 $ to find critical points.
Use the second derivative test or sign analysis.

Second Derivative Test

If $ f''(x) > 0 $, the function is concave up → local minimum.
If $ f''(x) < 0 $, the function is concave down → local maximum.
If $ f''(x) = 0 $, the test is inconclusive.

Real-World Example

Problem: A farmer wants to fence a rectangular field with 200 m of fencing. What dimensions maximize the area?

Solution: Let width = $ x $, length = $ y $. The perimeter is $ 2x + 2y = 200 \Rightarrow y = 100 - x $.

Area: $ A(x) = x(100 - x) = 100x - x^2 $.

$ A'(x) = 100 - 2x $. Set $ A'(x) = 0 \Rightarrow x = 50 $.

Then $ y = 50 $. The maximum area is $ 50 \times 50 = 2500 $ m².

Visualizing Calculus with Graphs

Graphs help us see the behavior of functions, their slopes, and the areas under curves. Below are two common examples:

📈 Parabola $ f(x) = x^2 $ with tangent at $ x = 1 $

The tangent line at $ x = 1 $ has slope $ f'(1) = 2 $.

🌊 Sine wave $ f(x) = \sin(x) $ and area under one arch

The definite integral $ \int_0^\pi \sin(x) \, dx = 2 $ gives the shaded area.

Lesson 5

Integrals

Accumulation, area under curves, and the Fundamental Theorem of Calculus.

What is an Integral?

An integral measures the accumulation of a quantity. The definite integral of a function over an interval gives the net area between the curve and the x-axis.

We write the definite integral as:

\int_a^b f(x) \, dx

This represents the area under $ f(x) $ from $ x = a $ to $ x = b $.

The Fundamental Theorem of Calculus

This theorem connects differentiation and integration — it's the most important idea in calculus!

Part 1

If $ F(x) $ is an antiderivative of $ f(x) $, then:

$$ \int_a^b f(x) \, dx = F(b) - F(a) $$

Part 2

If $ F(x) = \int_a^x f(t) \, dt $, then $ F'(x) = f(x) $.

Basic Integration Rules

Power rule: $ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C $ (for $ n \neq -1 $)
Constant multiple: $ \int c \cdot f(x) \, dx = c \int f(x) \, dx $
Sum rule: $ \int [f(x) + g(x)] \, dx = \int f(x) \, dx + \int g(x) \, dx $

Common Integrals

$ \int \sin x \, dx = -\cos x + C $
$ \int \cos x \, dx = \sin x + C $
$ \int e^x \, dx = e^x + C $
$ \int \frac{1}{x} \, dx = \ln |x| + C $

Example

Evaluate $ \int_0^2 (3x^2 + 2x) \, dx $.

Solution:

\int_0^2 (3x^2 + 2x) \, dx = \left[ x^3 + x^2 \right]_0^2 = (8 + 4) - (0 + 0) = 12

Lesson 6

Applications of Integrals

Finding areas, volumes, averages, and solving real problems with integration.

Area Between Curves

To find the area between two curves $ f(x) $ (top) and $ g(x) $ (bottom) over $ [a, b] $:

A = \int_a^b [f(x) - g(x)] \, dx

Volume of Revolution

Rotate a region around an axis to create a 3D solid. Two common methods:

Disk Method

Rotate around the x-axis:

$$ V = \int_a^b \pi [f(x)]^2 \, dx $$

Washer Method

Rotate around the x-axis with a hole:

$$ V = \int_a^b \pi ([f(x)]^2 - [g(x)]^2) \, dx $$

Average Value of a Function

The average value of $ f(x) $ on $ [a, b] $ is:

f_{avg} = \frac{1}{b-a} \int_a^b f(x) \, dx

Real-World Examples

Distance traveled: $ s = \int_a^b v(t) \, dt $
Work done: $ W = \int_a^b F(x) \, dx $
Fluid pressure: $ P = \int_A \rho g h \, dA $
Probability: $ P(a \le X \le b) = \int_a^b f(x) \, dx $

Example: Area Between Curves

Find the area between $ f(x) = x^2 $ and $ g(x) = x $ from $ x = 0 $ to $ x = 1 $.

A = \int_0^1 (x - x^2) \, dx = \left[ \frac{x^2}{2} - \frac{x^3}{3} \right]_0^1 = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}

🎉 Course Complete!

🧮 Tool

Scientific Calculator

Perform advanced calculations with ease — trigonometric, logarithmic, exponential, and more.

📖 About This Calculator

This scientific calculator supports basic arithmetic, trigonometric functions (sin, cos, tan), logarithms (log base 10, natural log ln), square root, exponentiation (x^y), factorial, and constants π and e. It's designed to help you verify your calculus homework and explore mathematical functions interactively.

🧰 How to Use

Basic operations: Use + - * / for addition, subtraction, multiplication, and division.
Trig functions: Click sin, cos, or tan then enter the angle in radians (e.g., sin(π/2)).
Logarithms: log is base 10, ln is natural log (base e).
Powers & roots: Use ^ for exponentiation (e.g., 2^3) and √ for square root.
Factorial: Type a number then click ! (e.g., 5!).
Constants: π (pi) and e (Euler's number).
Parentheses: Use ( and ) to group expressions.
Clear: Press AC to reset the display.
Evaluate: Press = to compute the result.

⚡ Example Calculations

sin(π/2) → 1
cos(0) → 1
ln(e) → 1
log(100) → 2
√(16) → 4
2^5 → 32
5! → 120
(3+4)*2 → 14

Errors (like division by zero or invalid input) will show "Error".

Basic Calculus Made Simple

Welcome to the Course

📚 What You'll Learn

🛠️ Tools & Resources

Scientific Calculator

Graphing Tools

🎯 Ready to Begin?

Introduction to Calculus

What is Calculus?

Why Learn Calculus?

The Three Big Ideas

1. Limits

2. Derivatives

3. Integrals

What's Next?

Limits

What is a Limit?

Understanding Limits Intuitively

One-Sided Limits

Key Limit Rules

Practice

Derivatives

What is a Derivative?

Basic Differentiation Rules

Power Rule

Constant Rule

Sum & Difference Rules

Product Rule

Quotient Rule

Common Derivatives

Practice Problems

Applications of Derivatives

Tangent Lines

Rates of Change

Optimization

Second Derivative Test

Real-World Example

Visualizing Calculus with Graphs

Integrals

What is an Integral?

The Fundamental Theorem of Calculus

Part 1

Part 2

Basic Integration Rules

Common Integrals

Example

Applications of Integrals

Area Between Curves

Volume of Revolution

Disk Method

Washer Method

Average Value of a Function

Real-World Examples

Example: Area Between Curves

Scientific Calculator

📖 About This Calculator

🧰 How to Use

⚡ Example Calculations

Basic Calculus
Made Simple