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By George Boole
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F(x ϩ ⌬x) Ϫ f(x) 6x ⌬x ϩ 3(⌬x)2 ϩ 4⌬x ϭ ϭ 6x ϩ 3⌬x ϩ 4 ⌬x ⌬x fr(x) ϭ lim (6x ϩ 3 ⌬x ϩ 4) ϭ 6x ϩ 4 ⌬x S 0 If f(x) ϭ 3x2 ϩ 4x Ϫ 5 were a curve, the slope of the tangent line at any point on the curve would be found by multiplying the x value by 6 and adding 4. The Basics EXAMPLE 3 CONTINUED— Find the slope of the tangent line to the curve f(x) ϭ 3x2 ϩ 4x Ϫ 5 at the point (3, 34). The slope m ϭ 6x ϩ 4 ϭ 6(3) ϩ 4 ϭ 22. E X A M P L E 3 C O N T I N U E D, C O N T I N U E D — Find the equation of the line tangent to f(x) ϭ 3x2 ϩ 4x Ϫ 5 at the point (3, 34).
And |y Ϫ L| Ͻ ε means L Ϫ ε Ͻ y Ͻ L ϩ ε. Given ε Ͼ 0, if we take any x value such that 0 Ͻ |x Ϫ a| Ͻ ␦, the interval on the x axis, and find the corresponding y ϭ f(x) value, then this y value must be within ε of L, that is, |f(x) Ϫ L| Ͻ ε. 13 14 BOB MILLER’S HIGH SCHOOL CALCULUS FOR THE CLUELESS N OT E The point (a, L) may or may not be actually there, depending on the function. But remember, we are only interested in values of x very, very close to a but not exactly at a. Take a smaller ε2; we can find ␦2 such that if 0 Ͻ |x Ϫ a| Ͻ ␦2, then |f(x) Ϫ L| Ͻ ε2.
Now take smaller and smaller positive ε’s; we can find smaller and smaller ␦’s. In the limit as the x circle shrinks to a, the f(x) circle shrinks to L. Read this a number of times!!! TRANSLATION 3 Let us see the real picture. y ϭ f(x). |x Ϫ a| Ͻ ␦ means a Ϫ ␦ Ͻ x Ͻ a ϩ ␦. And |y Ϫ L| Ͻ ε means L Ϫ ε Ͻ y Ͻ L ϩ ε. Given ε Ͼ 0, if we take any x value such that 0 Ͻ |x Ϫ a| Ͻ ␦, the interval on the x axis, and find the corresponding y ϭ f(x) value, then this y value must be within ε of L, that is, |f(x) Ϫ L| Ͻ ε.