Taylor formula is an important theorem in mathematical analysis, which provides an approximate method for calculating functions. When the independent variable changes around a certain point, the function can be approximately expressed by Taylor formula. Taylor formula is also widely used in college entrance examination mathematics. The following are some applications of Taylor formula in college entrance examination mathematics.
1. Approximate calculation: In some college entrance examination questions, some complex function values may need approximate calculation. At this time, the function value can be approximately calculated by Taylor formula. For example, when calculating the derivative of a function, Taylor formula can be used to approximate the derivative value of the function, so as to get the answer.
2. Solving the limit: Taylor formula can provide an effective method for solving some limit problems. For example, when the independent variable x→0, Taylor formula can be used to expand some trigonometric functions, exponential functions and other functions into infinite series, thus turning the limit problem into a problem of seeking series convergence.
3. Find the zero point of the derivative function: in some cases, the zero point of the derivative function is required. At this time, Taylor formula can be used to approximate the derivative function, so as to find the zero point of the derivative function. For example, when solving the extreme point of a function, Taylor formula can be used to approximate the derivative function of the function, and then the zero point of the derivative function can be obtained, so as to get the extreme point of the function.
4. Approximate calculation of definite integral: In some cases, it is necessary to approximately calculate the definite integral of a function. At this time, Taylor formula can be used to expand the function into infinite series, and then the integration interval can be divided into several cells, and each cell can be integrated by series expansion. Finally, the integration results between all units are accumulated to obtain approximate calculation results.
5. Solving differential equations: In some cases, it is necessary to solve a differential equation. At this time, Taylor formula can be used to approximately solve the function. For example, when solving a second-order ordinary differential equation, we can use Taylor formula to expand the solution function into infinite series, and then substitute it into the differential equation to solve it.
In a word, Taylor formula is widely used in college entrance examination mathematics. It can help us to approximately calculate the function value, find the limit, find the zero point of the derivative function, approximately calculate the definite integral and solve the differential equation. In practical application, it is necessary to choose the appropriate expansion mode and usage method according to specific problems in order to achieve the best effect.