1, Newton-Leibniz formula: This is one of the most basic formulas in calculus, which shows the relationship between the cumulative effect of indefinite integral and differential. A BF (x) dx = f (b)-f (a), where F (x) is the original function of F(x). This means that the integral of function f(x) at a and b is equal to the difference between the values of its original function at b and a. This formula is one of the most important formulas in calculus, because it establishes the relationship between definite integral and indefinite integral, thus transforming the calculation of definite integral into the problem of finding the original function.
2. Chain rule: This is a basic rule in differential calculus, which allows us to find the derivative of compound function. If u=g(x) and y=f(u), then dy/dx=dy/du*du/dx. This formula is widely used. It can be used to find the derivative of complex compound function and solve problems involving multiple variables. Through this formula, we can better understand the change rate of function and solve various problems in scientific research, engineering technology and economic analysis.
3. Partial integral formula: This is an important formula in integral calculus, which is used to find the definite integral of complex variable function. ∫uv'dx=uv-∫u'*v dx, where U and V are differentiable functions. This formula is widely used and can be used to solve the integration problem involving multiple variables and deal with complex integrand functions. Through this formula, we can transform the complex integral problem into a simpler integral problem, so as to find out the original function of the integrand more conveniently, and then find out the value of the definite integral.
The application of calculus;
1, Application in Geometry: Calculus is widely used in geometry. Calculus can be used to study the image, area, volume and approximation of functions, which plays an irreplaceable role in engineering drawing and design. For example, finding the area of a plane figure can be realized by definite integral, which is equal to the algebraic sum of the figure area surrounded by a function, a straight line and an axis.
2. Application in economics: Calculus is also widely used in economics. For example, marginal analysis and optimization problems can be solved by differential calculus, while economic indicators such as total cost, total revenue and total profit can be calculated by integral calculus. Calculus also plays an important role in forecasting modeling tools of economics, such as cost-benefit analysis and risk assessment using calculus.
3. Application in biology: Calculus is also widely used in biology. For example, in ecology, calculus can be used to describe the changing law of population number; In biomedicine, calculus can be used to describe the spread process of diseases and the mathematical model in epidemiology; In biophysics, calculus can be used to describe the structure and dynamic behavior of biomolecules.