Compute Δy and dy for the given values of x and dx = Δx.
y = x² - 6x, x = 5, Δx = 0.5

Solution:

Let y = f(x) = x² - 6x --- equation(1)

∵ Δy = f (x + Δx) - f(x)

Substitute the values of x = 5, Δx = 0.5

= f(5 + 0.5) - f(5)

= f(5.5) - f(5)

By substituting the values in the equation (1) and simplifying it, we get

Δy = 5.5² - 6(5.5) - [5² - 6(5)]

= - 2.75 - (- 5)

Δy = 2.25

Differentiating both sides of equation (1) with respect to ‘x’.

dy = f’(x) = 2x - 6 dx

Substitute the values of x = 5 and dx = Δx = 0.5

dy = f’(5) = [2(5) - 6] (0.5)

= 4(0.5)

dy = 2

Δy = 2.25 and dy = 2

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Compute Δy and dy for the given values of x and dx = Δx.
y = x² - 6x, x = 5, Δx = 0.5

Summary:

The values of Δy and dy for the given values of x and dx = Δx.

y = x² - 6x, x = 5, Δx = 0.5 is 2.25 and 2 respectively.