Rearranging and Domains

Worked Example: The Fencing Constraint

In calculus, we often need to link variables. Imagine you have 40 feet of fencing to build three sides of a rectangular yard against a wall. Your constraint is 2x plus y equals 40. We isolate y by subtracting 2x from both sides. Now we know y equals 40 minus 2x. Our goal is to find the area, which is x times y. We substitute our new expression for y to get area in terms of one variable: 40x minus 2x squared.

The Constraint Logic

Optimization problems follow a pattern. You have a fixed limit, or constraint, and an objective you want to maximize. But watch out for the sign trap. If you subtract a group, like 10 minus the group x plus 2, you must distribute that negative to every term.

Practice: Rearranging for Volume

Let's practice rearranging. Solve this volume formula for the height, h. Click the step you would take first. That's legal, but it leaves the one-third in the way. Clearing the fraction first is cleaner. Try multiplying by 3. Excellent. Multiplying by 3 gives us 3V equals pi-r-squared times h. Now, just divide by the pi-r-squared group to isolate h.

Worked Example: Finding the Domain

To find the domain of the square root of x minus 5, remember that the inside cannot be negative. We set the expression inside the root to be greater than or equal to zero. Adding 5 to both sides tells us that x must be 5 or greater.

The Two Deal-Breakers

In calculus, most domain restrictions you'll meet come from these two rules. First, denominators can never be zero. Second, even roots cannot have a negative inside. If you see both at once, the rules combine.

Practice: Denominator Domains

What is the domain of this function? It's easiest if you factor the denominator first. Click the values that must be excluded from the domain. Remember, the entire denominator cannot be zero. Factor x-squared minus 9 to find both 'deal-breaker' values. Correct. Since x-squared minus 9 is zero when x is 3 or negative 3, the domain is all real numbers except those two.

Mixed Review: Roots and Exponents

Let's combine skills. First, rewrite this function using a negative exponent. Then, think about the domain restriction. Great. The square root is the 1/2 power, and being in the denominator makes it negative. Now, which is the domain: x greater than or equal to negative 2, or x strictly greater than negative 2? Exactly. Because the root is in the denominator, the inside cannot be zero. We lose the 'equal to' part of the rule.

The Calculus Problem: The Sliding Ladder

A 10-foot ladder leans against a wall. The relationship is x-squared plus y-squared equals 100. Solve for y in terms of x. We get y equals the square root of 100 minus x-squared. Now, what is the physical domain for x? Remember, x is a distance and the ladder is only 10 feet long. Algebraically, x could be between -10 and 10. But since x is a physical distance, it can't be negative. The domain is zero to 10.

Takeaways & Definitions

Let's recap. We use constraints to simplify our functions before we ever start the calculus. We also check the domain to ensure our variables stay within the rules of denominators and roots.