Examples of spring constant in the following topics:
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- If F is constant, in addition to being directed along the line, then the integral simplifies further to:
- This calculation can be generalized for a constant force that is not directed along the line, followed by the particle.
- Let's consider an object with mass m attached to an ideal spring with spring constant k.
- When the object moves from x=x0 to x=0, work done by the spring would be:
- The spring applies a restoring force (−k⋅x) on the object located at x.
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- In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: F⃗=−kx⃗, where k is a positive constant.
- The system under consideration could be an object attached to a spring, a pendulum, etc.
- If F is the only force acting on the system, the system is called a simple harmonic oscillator, and it undergoes simple harmonic motion: sinusoidal oscillations about the equilibrium point, with a constant amplitude and a constant frequency.
- ω0 is called angular velocity, and the constants A and ϕ are determined from initial conditions of the motion.
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- An indefinite integral is defined as ∫f(x)dx=F(x)+C, where F satisfies F′(x)=f(x) and where C is any constant.
- We can add any constant C to F without changing the derivative.
- With this in mind, we define the indefinite integral as follows: ∫f(x)dx=F(x)+C , where F satisfies F′(x)=f(x) and C is any constant.
- Therefore, all the antiderivatives of x2 can be obtained by changing the value of C in F(x)=(3x3)+C, where C is an arbitrary constant known as the constant of integration.
- Apply the basic properties of indefinite integrals, including the constant, sum, and difference rules
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- A complete solution contains the same number of arbitrary constants as the order of the original equation.
- Since our example above is a first-order equation, it will have just one arbitrary constant in the complete solution.
- Therefore, the general solution is f(x)=Ce−x, where C stands for an arbitrary constant.
- You can see that the differential equation still holds true with this constant.
- For a specific solution, replace the constants in the general solution with actual numeric values.
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- If f(x) is a constant, then f′(x)=0, since the rate of change of a constant is always zero.
- By extension, this means that the derivative of a constant times a function is the constant times the derivative of the function.
- The known derivatives of the elementary functions x2, x4, ln(x), and ex, as well as that of the constant 7, were also used.
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- A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable.
- where m and b designate constants.
- In this particular equation, the constant m determines the slope or gradient of that line, and the constant term b determines the point at which the line crosses the y-axis, otherwise known as the y-intercept.
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- A partial derivative of a function of several variables is its derivative with respect to a single variable, with the others held constant.
- Usually, the lines of most interest are those which are parallel to the xz-plane and those which are parallel to the yz-plane (which result from holding either y or x constant, respectively).
- To find the slope of the line tangent to the function at P(1,1,3) that is parallel to the xz-plane, the y variable is treated as constant.
- By finding the derivative of the equation while assuming that y is a constant, the slope of f at the point (x,y,z) is found to be:
- For the partial derivative at (1,1,3) that leaves y constant, the corresponding tangent line is parallel to the xz-plane.
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- As the derivative of a constant is zero, x2 will have an infinite number of antiderivatives, such as 3x3+0, 3x3+7, 3x3−42, 3x3+293, etc.
- Therefore, all the antiderivatives of x2 can be obtained by adding the value of C in F(x)=3x3+C, where C is an arbitrary constant known as the constant of integration.
- If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G(x)=F(x)+C for all x.
- C is called the arbitrary constant of integration.
- If the domain of F is a disjoint union of two or more intervals, then a different constant of integration may be chosen for each of the intervals.
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- Functions of the form cex for constant c are the only functions with this property.
- If a variable's growth or decay rate is proportional to its size—as is the case in unlimited population growth, continuously compounded interest, or radioactive decay—then the variable can be written as a constant times an exponential function of time.
- Explicitly for any real constant k, a function $f: R→R$ satisfies $f′ = kf $ if and only if f(x)=cekx for some constant c.
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- A volume integral is a triple integral of the constant function 1, which gives the volume of the region D.
- of the constant function 1 calculated on the cuboid itself.
- Triple integral of a constant function 1 over the shaded region gives the volume.
- Calculate the volume of a shape by using the triple integral of the constant function 1