Introduction to Calculus ("SX" T version)

Eh, calculus, said for the time being not very deep research ... Although there are teaching in high school, but it seems that the order of small watermelon learning is not the same, um ... Shouldn't you learn the limits before you teach calculus? Regardless, this article according to Silly X Teng's understanding to engage. Limit...... University of Things Oh, let's get to know a symbol: LIM, which says that the limit is usually written in the lower part of Lim (for example, n→0 means n tends to 0,n→∞ means n tends to infinity), and then follows a formula that asks you to find out when a variable is approaching a value. The value of the subsequent equation. I believe that a watermelon approach to 0 IQ is impossible to directly understand the above text, so, examples come better: Ask Lim (n→0) 1/n, we all know, N is more than 0 and very small time, the value of 1/n is very large, then ... When n approaches 0, 1/n is approaching infinity, so Lim (n→0) 1/n=∞. But if n tends to be 0, is it approaching from a part greater than 0, or from a part less than 0? The difference between the two is very large, so we will add a + or-number after 0 to indicate the two cases, just now the formula is actually Lim (n→0+) 1/n=∞, and Lim (n→0-) 1/n=-∞. Another case, Lim (n→∞) ∑ (1≤i≤n) I/2^i, just learned that will be tangled this problem for a long time, later heard that high school has a common method: dislocation phase elimination. So I found I was silly, set the original for S, then S=2s-s=lim (n→∞) (∑ (1≤i≤n) i/2^ (i-1)-∑ (1≤i≤n) i/2^i) =1+lim (n→∞) ∑ (1≤i≤n) 1/2^i=1+1=2 (the gods do not taunt the watermelon is water ...... ）。 Extreme learning has a lot of things, such as the limits of trigonometric functions (Lim (n→0) sin (n) =0, etc.), and then, the limit has a very good law of Lofa, it directly gives when the ∞/∞,0/0,0*∞ and other conditions of the formula (derivative-related), and then can be easily calculated, Let's talk about these things later in the diary. Next go to the Chase, calculus. The jokes invented by calculus (including the arguments of Newton and Leibniz) believe that everyone knows, I will not say (actually lazy). Well? Calculus is divided into derivatives and integrals, which are reciprocal things. Differentiation is the knowledge of the original function, requiring its derivative function (that is, the rate of change of the original function), according to the definition of the need to use the limit: F (x) ' =lim (n→0) [F (X+n)-f (x)]/n, the rest is the limit calculation. It is clear that the primary function of a function is the constant (f (x) =kx, then f (x) ' =k). Two times the function I give an example: F (x) =x^2, then f (x) ' =lim (n→0) [(x+n) ^2-x^2]/n=lim (n→0) (x^2+n^2+2xn-x^2)/n=lim (n→0) (N^2+2XN)/n=lim (n→0) n+2x=2x, other analogy. Special, (e^x) ' =e^x (using the definition of e E=lim (n→∞) (1+1/n) ^n can be easily obtained), (ln x) ' =1/x (the process of constructing an e out of the line). If the derivative of ln x is introduced, then the general logarithmic function can also be introduced (Loga x) ' =x^ ( -1)/lna, Gong, the construction of the logarithm is also easy to find out: (a^x) ' = (LNA) * (a^x). In addition, there is the formula (x^n) ' =n*x^ (n-1), as well as the derivative of the composite function = Outer Guide * Inner layer Guide, the two function add-on function is obviously added, multiply the words is two derivative functions are multiplied by the original function (self-seeking limit can be proved). The integral is the inverse of the differential, that is, the original function of knowing the derivative function of a function to seek it. In fact, the function value of a point of the original function is the area of the Guide function image up to this point and the axis, the formula can be obtained by dividing The matrix: ∫ (A, B) f (x). Dx=lim (n→∞) ∑ (1≤i≤n) F (A + (B-A) *i/n) * (b-a)/n, which indicates that the image , b] on the area truncated, its operations on differential reciprocal, there is nothing to say (after all, watermelon so water), this log first stop, and may do further in the future (in fact, the class is immediately). Finish...... by SWM_SXT from Enceladus

Introduction to Calculus ("SX" T edition)