For what number does the principal square root exceed eight times the number by the largest amou

For what number does the principal square root exceed eight...

Key Concepts

Difference Function Analysis

When a problem involves finding where one function exceeds another by the largest amount, a difference function can be defined. Analyzing this function often involves setting up an expression as the difference between the two original functions and then optimizing that expression using calculus techniques.

Principal Square Root

The principal square root is the nonnegative value that, when squared, equals the original number. This function is essential in many mathematical problems, especially where only the nonnegative solution is meaningful, and it forms a fundamental part of various equations and optimization tasks.

Optimization in Calculus

Optimization involves determining the maximum or minimum values of a function. In calculus, this typically includes defining a function that models the problem, computing its derivative to find critical points, and analyzing those points to identify where the function achieves its optimal value.

Derivative and Critical Points

The derivative represents the rate of change of a function and is used to identify points where the function's slope is zero or undefined. These critical points are potential candidates for local maxima or minima, and examining them helps in determining the optimal solution in a given problem.

Transcript

00:01 We wish to find a number whose principal square root exceeds eight times the number by the largest amount.

00:08 So principal square root just means the positive square root of a number.

00:12 Okay, so what we're really looking for is we're looking to maximize the difference between square root of a number.

00:20 So let's call that x and eight times the number.

00:24 So that would be eight x.

00:26 Okay, so let's call this function f of x, which is the difference between the number.

00:31 The square root of x and eight times x.

00:33 So since we're trying to maximize this, that tells us we should go ahead and find its critical values, which means we need to find the derivative of f.

00:43 So let's go ahead and do that.

00:44 Derivative of f is equal to one half times x, the power of negative one half minus eight.

00:52 And we know that critical values happen where f prime of x is equal to zero or f prime of x doesn't exist.

01:00 Okay, so notice.

01:01 That when x is equal to zero, f prime of x doesn't exist.

01:07 But the square root of zero and a times zero is just zero.

01:12 So we'll keep that in the back of our head, but that's probably not going to be the largest difference.

01:20 So let's go ahead and find the other critical values, which is when we set f prime of x equal to zero.

01:26 So that's going to be, so let's rewrite this as one over two root x minus eight.

01:32 Okay, so let's go ahead and isolate 8.

01:35 We get 8 is equal to 1 over 2 square root x, or multiplying both sides by 2 gives us 16 is equal to 1 over square root x.

01:48 Okay, and then we're just going to flip both sides...

Please give Ace some feedback