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  2. AP Calculus
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What is a vector-valued function?

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

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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)jr(t)=f(t)i+g(t)j or r(t)=⟨f(t),g(t)⟩r(t) = \langle f(t), g(t) \rangler(t)=⟨f(t),g(t)⟩.

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) \rangler(t)=⟨f(t),g(t),h(t)⟩?

∫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∫r(t)dt=⟨∫f(t)dt,∫g(t)dt,∫h(t)dt⟩

What is the distance formula in 2D?

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

How do you find the position of a particle at time t given its velocity vector and initial position?

  1. Integrate the velocity vector to find the position vector. 2. Use the initial position to solve for the constant of integration. 3. Plug in t to find the position at that time.

How do you find the distance of a particle from the origin at time t, given its velocity vector and initial position?

  1. Integrate the velocity vector to find the position vector. 2. Use the initial position to solve for the constant of integration. 3. Plug in t to find the position at that time. 4. Use the distance formula to find the distance from the origin.