Solution We follow the recipe in [4.5].
2. f(a + h)- f(a) = (a3 + 3crh + 3ah2 + h3) - a3 = 3a~h + 3ah2 + h3
5. As h tends to 0, so 3ah + h2 will also tend to 0; hence, the entire expression 3a2 + 3ah + hr tends to 3a2. Therefore, f'(a) = 3a2.
We have thus shown that the graph of the function fix) = x3 at the point x = a has a tangent with slope 3a2. Note that f'(a) = 3a2 > 0 when a # 0, and /'(0) = 0. The tangent points upwards to the right for all a ^ 0, and is horizontal at the origin. You should draw the graph of fix) = x3 to confirm this behavior.
It is easy to use the recipe in [4.5] on simple functions. However, the recipe becomes difficult or even impossible if we try to use it on slightly more complicated functions such as f(x) = V3x2 + x + I. The next chapter develops rules for computing the derivative of quite complicated functions, without the need to use [4.5]. Before presenting such rules, however, we must examine the concept of a limit a little more carefully. This is done in Section 4.4.
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