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Home Limits, Analytic Geometry, and Approximations Derivatives and Increments Examples Example 1
See also: Example 1 (Concluded) 
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Example 1

Consider the curve

f(x) = x1/3.

Then

f'(x) = ⅓x-2/3.

At the point x = 8, we have

x = 8, f(x) = 2, f'(x) = 1/12 = 0.0833 ....

Thus

05_limits_g_approx-471.gif

This is the slope of the line tangent to the curve at the point (8,2). As Δx approaches zero, the slope of the secant line through the two points (8, 2) and (8 + Δx, (8 + Δx)1/3) will approach 1/12. We make a table showing the slope of the secant line for various values of Δx.

Δx

Δy = (8 + Δx)1/3 - 2

05_limits_g_approx-472.gif = slope of secant line

05_limits_g_approx-473.gif

10

0.6207

0.0621

0.0212

1

0.0801

0.0801

0.0032

1/10

0.00829

0.0830

0.0003

-10

-3.2599

0.3260

0.2427

-1

-0.0871

0.0871

0.0038

-1/10

-0.00837

0.0837

0.0004

Home Limits, Analytic Geometry, and Approximations Derivatives and Increments Examples Example 1

Last Update: 2010-11-25