Calculus Volume 1 · 免费预览
牛顿法
牛顿法通过迭代切线与曲线的交点来近似方程的根；当从单根附近出发时，其收敛速度惊人，而当切线给出误导时，它也会明确地失败。
1. The tangent line crosses the axis
The tangent line crosses the axis
2. Convergence and the mean value theorem
Convergence and the mean value theorem
3. When Newton fails
When Newton fails
4. A worked approximation
A worked approximation
5. Newton iteration formula
Newton's method for approximating a root of f(x) = 0 updates the current guess x_n by which formula?
6. First step for square root of two
Applying Newton's method to f(x) = x^2 - 2 from x_0 = 1, what is x_1?
7. Speed of convergence
Near a simple root, Newton's method exhibits quadratic convergence. What does that mean for the error?
8. Failure near an extremum
Why can a Newton guess near a local maximum or minimum derail the iteration?
9. Basis of the root's existence
Before Newton's method hunts a root of a continuous f, which theorem certifies that a root exists between a sign change?
10. Cycling failure
For f(x) = x^3 - 2x + 2, Newton's method started at x_0 = 0 oscillates between 0 and 1 forever. What does this illustrate?
11. Newton update for x cubed minus two
For f(x) = x^3 - 2, one Newton update is x_1 = x_0 - (x_0^3 - 2)/(3 x_0^2). Starting from x_0 = 1, what is x_1? Answer with a fraction like 4/3.
12. Babylonian special case
Newton's method on f(x) = x^2 - a gives the update x_{n+1} = (x_n + a/x_n)/2, the Babylonian square root iteration. For a = 2 and x_0 = 1, what is x_1? Answer with a number.
13. Iterates to five digits
In the sqrt(2) computation, x_2 is about 1.4167 and x_3 about 1.4142, already correct to five digits. How many iterations were needed from x_0 = 1? Answer with a number.
14. Simple root requirement
Quadratic convergence requires f prime of the root r to be nonzero, meaning r is a simple root. As a number, what is f prime of r for a simple root? Answer with a number.