By By (author) Bob Miller
With Bob Miller at your facet, you by no means must be clueless approximately math back!
Algebra and calculus are difficult on highschool scholars such as you. Professor Bob Miller, with greater than 30 years' educating event, is a grasp at making the complicated basic, and his now-classic sequence of Clueless learn aids has helped tens of hundreds of thousands comprehend the harsh subjects.
Calculus-with its integrals and derivatives-is recognized for tripping up even the fastest minds. Now Bob Miller-with his 30-plus years' adventure educating it-presents highschool calculus in a transparent, funny, and fascinating way.
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Additional resources for Bob Miller's High School Calc for the Clueless - Honors and AP Calculus AB & BC
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.