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How are the components of a vector-valued function integrated?

Each component is integrated separately, treating each as its own integral problem.

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How are the components of a vector-valued function integrated?

Each component is integrated separately, treating each as its own integral problem.

What do the constants of integration represent when integrating a vector-valued function representing velocity?

The initial position of the particle.

What information can be obtained by integrating a vector-valued acceleration function?

The velocity function, plus a constant of integration representing initial velocity.

How do you integrate a vector-valued function r(t)=⟨f(t),g(t),h(t)⟩r(t) = \langle f(t), g(t), h(t) \rangle?

∫r(t)dt=⟨∫f(t)dt,∫g(t)dt,∫h(t)dt⟩\int r(t) dt = \langle \int f(t) dt, \int g(t) dt, \int h(t) dt \rangle

What is the distance formula in 2D?

(x2−x1)2+(y2−y1)2\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

What is a vector-valued function?

A function whose domain is a set of real numbers and whose range is a set of vectors.

What does integrating a vector-valued function give?

The antiderivative of the vector-valued function, which can represent position given a velocity function.

What is the form of a 2D vector-valued function?

Can be written as r(t)=f(t)i+g(t)jr(t) = f(t)i + g(t)j or r(t)=⟨f(t),g(t)⟩r(t) = \langle f(t), g(t) \rangle.