Glossary
Chain Rule
A fundamental rule in calculus used to differentiate composite functions, crucial when differentiating terms with respect to time in related rates problems.
Example:
When differentiating with respect to time, the chain rule gives .
Constants (in differentiation)
Values in an equation that do not change with respect to the variable of differentiation (e.g., time in related rates problems), and thus their derivative is zero.
Example:
In the sliding ladder problem, the length of the ladder (13 meters) is a constant, so its derivative with respect to time is 0.
Implicit Differentiation
A technique used to differentiate equations where variables are not explicitly defined as functions of each other, typically applied when differentiating with respect to time in related rates problems.
Example:
To find for , you'd use implicit differentiation to get .
Pythagorean Theorem
A fundamental geometric theorem stating that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides ($a^2 + b^2 = c^2$).
Example:
If a ladder is 13 feet long and its base is 5 feet from a wall, the Pythagorean Theorem helps you find the height it reaches on the wall: .
Pythagorean Triple
A set of three positive integers $a, b, c$ such that $a^2 + b^2 = c^2$, representing the side lengths of a right-angled triangle.
Example:
The numbers (3, 4, 5) form a Pythagorean Triple because .
Related Rates Problems
Problems that involve finding the rate at which a variable changes concerning the rate of change of another related variable, often with respect to time.
Example:
Imagine a balloon being inflated; a related rates problem might ask how fast its radius is increasing when its volume is increasing at a certain rate.