一些常用函数及其泰勒(Taylor)展开式的图像

图 1 y?exp(x)及其 Taylor 展开式

其中，

y?exp(x)?ex;y1?P1(x)?1?x;x2y2?P2(x)?1?x?;2! x2x3y3?P3(x)?1?x??;2!3!x2x3x4y4?P4(x)?1?x???。2!3!4!

Figure 1 y=exp(x) and its Taylor expansion equation25yy1y215y3y42010Y50-5-3-2-10X123

图 2 y?sin(x)及其 Taylor 展开式

其中，

y?sin(x);y1?P1(x)?x;x3y3?P3(x)?x?;3! x3x5y5?P5(x)?x??;3!5!x3x5x7y7?P7(x)?x???。3!5!7!

Figure 2 y=sin(x) and its Taylor expansion equation86420-2-4-6-8-4yy1y3y5y7Y-3-2-10X1234

图 3 y?cos(x)及其 Taylor 展开式

其中，

y?cos(x);x2y2?P2(x)?1?;2!x2x4y4?P4(x)?1??;2!4! x2x4x6y6?P6(x)?1???;2!4!6!x2x4x6x8y8?P8(x)?1????。2!4!6!8!

Figure 3 y=cos(x) and its Taylor expansion equation4yy22y4y6y80-2Y-4-6-8-4-3-2-10X1234

图 4 y?ln(1?x)及其 Taylor 展开式

其中，

y?ln(1?x);y1?P1(x)?x;x2y2?P2(x)?x?;2 x2x3y3?P3(x)?x??;23x2x3x4y4?P4(x)?x???。234

Figure 4 y=ln(x) and its Taylor expansion equation3yy1y2y31y420Y-1-2-3-1-0.500.5X11.52

图 5 y?sh(x)及其 Taylor 展开式

其中，

ex?e?xy?sh(x)?;2y1?P1(x)?x;x3y3?P3(x)?x?;3! x3x5y5?P5(x)?x??;3!5!x3x5x7y7?P7(x)?x???。3!5!7!

Figure 5 y=sh(x) and its Taylor expansion equation806040200-20-40-60-80-5yy1y3y5y7Y-4-3-2-10X12345