Calculus Companion
Welcome to Your Calculus Companion!
This is your interactive guide to help you review, practice, and master the core concepts of Calculus.
We've translated a standard calculus roadmap into an interactive experience. This widget is fully responsive and designed to work on both desktop and mobile. You'll find:
- Interactive Tables: Click buttons to see the rules for differentiation (derivatives).
- Dynamic Charts: Visualize how integrals (Riemann Sums) approximate the area under a curve.
- Working Practice Terminals: Test your knowledge with "fun" questions and get instant feedback.
- Key Theorems: A special section for the "big ideas" that tie everything together, like the Fundamental Theorem of Calculus.
Use the menu to navigate between topics. Let's get started!
1. What is Calculus? (Limits)
Calculus is the study of change. It all begins with the Limit, which is the idea of getting "infinitely close" to a value without ever *quite* reaching it.
The Intuitive Idea
Imagine you are 10 feet from a wall. You walk half the distance (5 ft), then half the remaining distance (2.5 ft), then half again (1.25 ft), and so on.
- Question: Will you ever *reach* the wall?
- Answer: No, you'll only ever get halfway there each time.
- But: The *limit* of your distance from the wall is zero. You can get infinitely close to it.
- Calculus is built on this idea of "infinitely close."
Limit Notation
We use special notation to describe this. Don't let it scare you! It's just a sentence written in math.
lim f(x) = L
x→a
This reads: "The limit of the function f(x) as x approaches the number a is equal to L."
Example: For the function f(x) = x + 2, the limit as x approaches 10 is 12. As x gets infinitely close to 10 (e.g., 9.9, 9.999), the function's value gets infinitely close to 12.
When Limits Fail to Exist
A limit *does not exist* (DNE) if the function doesn't approach a single, finite number. This happens in 3 main ways:
# 1. A "Jump" # The function approaches one value # from the left, but a different # value from the right. # (Like a broken staircase) # 2. A "Vertical Asymptote" # The function shoots off to # positive or negative infinity. # (Like f(x) = 1/x at x=0) # 3. "Oscillation" # The function bounces wildly # between values and never # settles down.
Practice Time
Q1: A superhero, 'The Approacher,' never *reaches* the city, but gets infinitely close. What calculus concept is this?
> Terminal ready. Awaiting answer...
2. Derivatives (The Slope)
The Derivative is the first "big idea" in calculus. It's a function that tells you the instantaneous rate of change (the slope) of *another* function at any point.
| Rule | If f(x) is... | The Derivative f'(x) is... |
|---|
Practice Time
Q1: What is the derivative (slope) of the simple line y = 3x? (Hint: what's the slope?)
> Terminal ready. Awaiting answer...
3. Applications of Derivatives
Why do we care about slope? Because it tells us where a function is increasing, decreasing, or at a peak/valley. This is called optimization.
Finding Min/Max (Optimization)
The peak of a hill (a "local maximum") or the bottom of a valley (a "local minimum") has a flat slope.
# The slope is flat. # This means the derivative is zero! f'(x) = 0 # How to find the max profit: # 1. Get the Profit(x) function. # 2. Find the derivative, Profit'(x). # 3. Set Profit'(x) = 0 and solve # for x. This is your # "critical point"!
Concavity (The 2nd Derivative)
The Second Derivative (f''(x), the derivative *of the derivative*) tells us about concavity.
# f''(x) is POSITIVE # # The graph is "smiling" :) # (Like a cup holding water) # # The slope is increasing.
# f''(x) is NEGATIVE # # The graph is "frowning" :( # (Like a cup spilling water) # # The slope is decreasing.
Practice Time
Q1: To find the *peak* of a hill (a maximum) on a graph, you'd find where the derivative (slope) is equal to what number?
> Terminal ready. Awaiting answer...
4. Integrals (The Area)
The Integral is the second "big idea" in calculus. It is the *opposite* of the derivative, and it helps us find the total accumulated change (the area under a curve).
The Indefinite Integral
This is just the "anti-derivative." You're working the Power Rule backwards.
# Derivative (Power Rule): # f(x) = x³ # f'(x) = 3x² # Integral (Anti-derivative): # f'(x) = 3x² # f(x) = ∫3x² dx = x³ + C # We add "+ C" (a constant) # because the derivative of any # constant (like 5, -10, 2.3) # is zero. We lost that info!
The Definite Integral
This finds the *exact* area under a function f(x) between two points, a and b.
∫ f(x) dx
(from a to b)
Fundamental Theorem of Calculus:
If F(x) is the antiderivative of f(x), then the area is just:
F(b) - F(a)
This is the magic link! To find the area, you just find the antiderivative and plug in the endpoints.
Practice Time
Q1: The derivative of f(x) is 2x. What is the *antiderivative* (integral) of 2x? (Hint: Power rule backwards, ignore the + C)
> Terminal ready. Awaiting answer...
5. Visualizing Integrals (Riemann Sums)
How do we find the area of a curvy shape? We can't use Area = h * w. But we can *approximate* it by filling the curve with tiny rectangles. This is called a Riemann Sum.
Visualizing the area under y = x + 2 from x=0 to x=4.
Practice Time
Q1: The chart shows 'Riemann Sums.' As you use *more* rectangles (n increases), does the area approximation get better or worse?
> Terminal ready. Awaiting answer...
6. Key Theorems (The "Big Ideas")
These are the "power moves" of calculus. They are the famous theorems that connect all the ideas we've learned.
The Fundamental Theorem of Calculus (FTC)
This is the big one!
It proves that Derivatives and Integrals are opposites. It's the "magic" that links "finding the slope" to "finding the area."
# Part 1: The derivative of an # integral is just the original # function. # d/dx [∫ f(t) dt] = f(x) # (They cancel each other out) # Part 2: To find the area under # f(x), you just need to use its # antiderivative, F(x). # # Area = F(b) - F(a)
L'Hôpital's Rule
This is a "power move" for solving tricky limits that look like 0/0 or ∞/∞ (which are "indeterminate forms").
The Process
# You want to find the limit of f(x)/g(x) # but it's 0/0. What to do? # Take the derivative of the top # AND the derivative of the bottom # *separately*. # lim [f(x) / g(x)] = lim [f'(x) / g'(x)] # This *new* limit is often # much easier to solve! # (Note: This is NOT the quotient rule)
Mean Value Theorem (MVT)
This is the "Average Speed" theorem. It's more intuitive than it sounds.
The Idea (Road Trip)
# You drive 120 miles in 2 hours. # Your *average speed* is 60 mph. # The MVT guarantees that for at # least *one instant* (one "mean # value") during that trip, your # speedometer *must* have read # exactly 60 mph. # Math terms: # The *average slope* between two # points [f(b)-f(a)]/[b-a]... # ...must equal the *instantaneous # slope* f'(c) at some point c.
Practice Time
Q1: L'Hôpital's Rule helps find tricky limits that look like 0/0 or ∞/∞. (True/False)
> Terminal ready. Awaiting answer...