Examples of differential in the following topics:
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- Differentials are the principal part of the change in a function y=f(x) with respect to changes in the independent variable.
- The differential dy is defined by:
- The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular differential form, or a particular analytical significance if the differential is regarded as a linear approximation to the increment of a function.
- Higher-order differentials of a function y=f(x) of a single variable x can be defined as follows:
- Use implicit differentiation to find the derivatives of functions that are not explicitly functions of x
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- Differential equations can be used to model a variety of physical systems.
- Differential equations are very important in the mathematical modeling of physical systems.
- Many fundamental laws of physics and chemistry can be formulated as differential equations.
- In biology and economics, differential equations are used to model the behavior of complex systems.
- Give examples of systems that can be modeled with differential equations
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- Differential equations are solved by finding the function for which the equation holds true.
- Differential equations play a prominent role in engineering, physics, economics, and other disciplines.
- Solving the differential equation means solving for the function f(x).
- The "order" of a differential equation depends on the derivative of the highest order in the equation.
- You can see that the differential equation still holds true with this constant.
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- In the previous atom, we learned that a second-order linear differential equation has the form:
- When f(t)=0, the equations are called homogeneous second-order linear differential equations.
- In general, the solution of the differential equation can only be obtained numerically.
- Linear differential equations are differential equations that have solutions which can be added together to form other solutions.
- Identify when a second-order linear differential equation can be solved analytically
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- The rules of differentiation can simplify derivatives by eliminating the need for complicated limit calculations.
- When we wish to differentiate complicated expressions, a possible way to differentiate the expression is to expand it and get a polynomial, and then differentiate that polynomial.
- In many cases, complicated limit calculations by direct application of Newton's difference quotient can be avoided by using differentiation rules.
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- Implicit differentiation makes use of the chain rule to differentiate implicitly defined functions.
- Implicit differentiation makes use of the chain rule to differentiate implicitly defined functions.
- However, we can still find the derivative of y with respect to x by using implicit differentiation.
- For example, given the expression y+x+5=0, differentiating yields:
- Use implicit differentiation to find the derivatives of functions that are not explicitly functions of x
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- A second-order linear differential equation has the form dt2d2y+A1(t)dtdy+A2(t)y=f(t), where A1(t), A2(t), and f(t) are continuous functions.
- Linear differential equations are of the form Ly=f, where the differential operator L is a linear operator, y is the unknown function (such as a function of time y(t)), and the right hand side f is a given function of the same nature as y (called the source term).
- where D is the differential operator dtd (i.e.
- When f(t)=0, the equations are called homogeneous second-order linear differential equations.
- A simple pendulum, under the conditions of no damping and small amplitude, is described by a equation of motion which is a second-order linear differential equation.
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- Differentiation, in essence calculating the rate of change, is important in all quantitative sciences.
- Given a function y=f(x), differentiation is a method for computing the rate at which a dependent output y changes with respect to the change in the independent input x.
- Equations involving derivatives are called differential equations and are fundamental in describing natural phenomena.
- In every aspect of life in which something changes, differentiation and rates of change are an important aspect in understanding the world and finding ways to improve it.
- Give examples of differentiation, or rates of change, being used in a variety of academic disciplines
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- Direction fields and Euler's method are ways of visualizing and approximating the solutions to differential equations.
- Direction fields, also known as slope fields, are graphical representations of the solution to a first order differential equation.
- The slope field is traditionally defined for differential equations of the following form:
- Then, from the differential equation, the slope to the curve at A0 can be computed, and thus, the tangent line.
- Describe application of direction fields and Euler's method to approximate the solutions to differential equations
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- Separable differential equations are equations wherein the variables can be separated.
- Non-linear differential equations come in many forms.
- A separable equation is a differential equation of the following form:
- The original equation is separable if this differential equation can be expressed as:
- This is the easiest variety of differential equation to solve.