Precalculus Question

Question

For the following function, y = f(x), simplify the following expression so that h is no longer a factor in the denominator, then calculate the result of setting h = 0 in this simplified expression: (f(x + h) − f(x))/h .

f(x) = √ (25 − x^2).

Answer 1

f(x) = √(25 - x²)

[f(x+h) - f(x)] / h = [√(25 - (x+h)²) - √(25 - x²)] / h

Multiply and divide by √(25 - (x+h)²) + √(25 - x²) to obtain:

[(25 - (x+h)² - (25 - x²)] / [h(√(25 - (x+h)²) + √(25 - x²))]

= [x² - x² - 2xh - h²] /[ h(√(25 - (x+h)²) + √(25 - x²))]

= h(-2x - h) / [h(√(25 - (x+h)²) + √(25 - x²))]

= (-2x - h) / [√(25 - (x+h)²) + √(25 - x²)]

Replace h by 0 to get -2x / [2√(25 - x²)] = -x / √(25 - x²)