Parametric functions, vector-valued functions, planar motion, matrices as transformations, and matrix operations and inverses as in the AP Precalculus Unit 4 topics.
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- What is a parametric equation?
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An equation in which both x and y are expressed as separate functions of a third variable (the parameter, usually t).- In the parametric equations x = cos(t), y = sin(t), what does the parameter t represent?
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The angle measured from the positive x-axis on a circle of radius 1.- What does 'eliminating the parameter' mean?
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Finding a single equation relating x and y by removing the parameter t from a pair of parametric equations.- How does eliminating the parameter from x = 2t, y = 4t relate the variables?
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Solving for t from one equation and substituting into the other gives y = 2x.- What is the formula for dy/dx when a curve is given parametrically?
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dy/dx = (dy/dt) / (dx/dt), provided dx/dt is not zero.- What does the direction of a parametric curve show?
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As t increases, the direction indicates how the point moves along the curve (the orientation).- How do you find d²y/dx² for a parametric curve?
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d²y/dx² = d/dt(dy/dx) / (dx/dt).- How do you write a parametric equation for a line segment from (1, 2) to (5, 8)?
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x = 1 + 4t, y = 2 + 6t, where t goes from 0 to 1.- What is a vector-valued function?
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A function that assigns a vector to each input value, usually written as r(t) = (f(t), g(t)).- How do you express the parametric curve x = cos(t), y = sin(t) as a vector-valued function?
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r(t) = (cos(t), sin(t)).- What does the position vector r(t) represent in planar motion?
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The location of a moving point at time t, given as r(t) = (x(t), y(t)).- How do you find the velocity vector from the position vector r(t)?
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Take the derivative: v(t) = dr/dt = (dx/dt, dy/dt).- How do you find the acceleration vector from the position vector r(t)?
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Take the second derivative: a(t) = d²r/dt² = (d²x/dt², d²y/dt²).- What is the relationship between the velocity vector and the tangent line to a curve?
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The velocity vector is parallel to the tangent line; its direction gives the tangent direction.- What is the speed of a moving object with velocity vector v(t) = (vx, vy)?
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speed = sqrt(vx^2 + vy^2), the magnitude of the velocity vector.
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