Recording-2025-03-24T14:57:29.268Z

Understanding Square Roots

• Square root of a number is a value, when multiplied by itself, gives the original number.

• For example, √64 = 8.

• The square root of 65 is slightly more than 8 because 65 is just above 64.

Approximation of Square Roots

• Before calculators, mathematicians had strategies for approximating square roots.

• An initial estimation can be made by comparing the number with squares of nearby integers.

  • For example, since 64 is 8², √65 is approximately 8.

Using Calculus for Better Approximations

• The lesson introduces the idea of using tangent lines to approximate functions, particularly the square root function.

• Key Idea: "Lines are good."

  • Tangent lines are straightforward, characterized by a slope and a y-intercept, making them easy to work with.

  • These lines can be used to approximate more complex functions.

Tangent Line to Approximate Square Roots

• To find the square root of 65, draw the square root curve and the tangent line at the point where x = 64 (where the square root is exactly known as 8).

• Use calculus to derive the slope at this point:

  • The derivative of the square root function gives the slope of the tangent line at any point.

Finding the Tangent Line Equation

• Find the point on the curve corresponding to x = 64: f(64) = 8.

• Compute the derivative at x = 64 to find the slope:

  • Slope (m) = 1/16.

• Using point-slope form:

  • The equation of the tangent line:[ y - 8 = \frac{1}{16}(x - 64) ][ y = \frac{1}{16}x + 4 ]

  • Find the output for x = 65 to get the approximation:

    • Plugging in x = 65 gives approximately 8.0625.

Improving Estimates

• Approximation is less accurate further away from the tangential point.

  • Using Newton's Method can enhance accuracy further.

  • Tangent lines point toward roots, allowing iterative refinement.

Newton's Method Framework

• Reframe the problem: find roots of the polynomial f(x) = x² - 65, which intersects the x-axis at √65.

• Start with an initial guess (e.g., x₀ = 10).

• Compute the derivative and find the tangent line's x-intercept to generate a new guess.

  • Iterate this process to improve approximation:

    1. First Guess: x₀ = 10 → slope = 20 → tangent intercept = 8.25.

    2. Second Guess: x₁ = 8.25 → slope = 16.5 → tangential intercept = ~8.0644.

Iteration Continuation

• Repeating these steps gets you even closer to the exact value with each iteration.

• After a few iterations using a good calculator or graphing tool (like Desmos), you see improvements in precision.

Final Thoughts

• This method allows the calculation of roots much more efficiently than evaluating square roots directly without a calculator.

• Important takeaway: The approximation using tangent lines is not just a one-time calculation but is a dynamic iterative process useful in various mathematical applications.