From Functions to Probability — Interactive learning made simple! Master essential math concepts with hands-on exercises and visual demonstrations.
Section 1 of 5
Think of a function like a magic machine!
• You put a number IN one side
• The machine does something to it
• A new number comes OUT the other side
Input number:
⬇️ The machine squares it! ⬇️
Output: 9
Pizza Area Function: f(radius) = π × radius²
If your pizza radius is 5 inches, the area is π × 5² = about 78 square inches!
Imagine you're riding a bike on a hill...
• Going up a steep hill = positive steep number
• Going down = negative number
• Flat road = zero!
The derivative tells you exactly how steep the hill is at any point!
Click anywhere on the hill to see how steep it is there!
Power Rule: If f(x) = x³, then f'(x) = 3x²
Simple Pattern: Bring down the power, subtract 1 from the power
Try it: x⁴ becomes 4x³, x² becomes 2x
Sometimes we want to find the BEST answer:
• Highest point on a hill = maximum
• Lowest point in a valley = minimum
• We use derivatives to find these special spots!
Let's find the lowest point of: f(x) = x² - 6x + 5
Step 1: Find the derivative: f'(x) = 2x - 6
Step 2: Set it equal to zero: 2x - 6 = 0
Step 3: Solve: x = 3
Step 4: Check if it's a minimum or maximum
Probability tells us how likely something is to happen!
• 0 = Impossible (like rolling a 7 on a normal die)
• 0.5 = 50/50 chance (like flipping heads on a coin)
• 1 = Definitely will happen (like the sun rising tomorrow)
One Die: Chance of rolling a 4 = 1/6 ≈ 0.167
Two Dice: Chance of rolling a total of 7 = 6/36 = 1/6
Coin Flip: Chance of heads = 1/2 = 0.5
Let's test what you've learned!
Question 1: If f(x) = x², what is f(4)?
Question 2: What's the derivative of x³?
Question 3: What's the probability of rolling a 6 on a fair die?