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Vectors and Calculus

Vectors and Calculus. Section 11-B. If a smooth curve C is given by the equation Then the slope of C at the point (x, y) is given by Where and the second derivative is given by :. Vectors and Derivatives.

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Vectors and Calculus

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  1. Vectors and Calculus Section 11-B

  2. If a smooth curve C is given by the equation Then the slope of C at the point (x, y) is given by Where and the second derivative is given by : Vectors and Derivatives

  3. The derivative may be interpreted as the slope of the tangent line to the curve C, or as the slope of the path of a particle traveling along the curve C, or as the rate of change of y with respect to x. The second derivative is the rate of change of the slope of the curve C with respect to x. Vectors and Derivatives

  4. Is the rate at which the x-coordinate is changing with respect to t or the velocity of the particle in the horizontal direction. Is the rate at which the y-coordinate is changing with respect to t or the velocity of the particle in the vertical direction. Velocity

  5. Is the position vector at any time t. Is the velocity vector at any time t. Is the acceleration vector at any time t. Position Vector

  6. Is the speed of the particle or the magnitude (length) of the velocity vector Is the length of the arc (or arc length) of the curve from t = a to t = b or the distance traveled by the particle from t = a to t = b. Speed of the particle

  7. A particle moves in the xy-plane so that at any time t, the position of the particle is given by a) Find the velocity vector when t = 1

  8. A particle moves in the xy-plane so that at any time t, the position of the particle is given by b) Find the acceleration vector when t = 2

  9. 2) A particle moves in the xy-plane so that at any time t, t ≥ 0, the position of the particle is given by Find the magnitude of the velocity vector when t = 1 Note: The formula for the magnitude of the velocity vector is the same as the formula for the speed of the vector.

  10. A particle moves in the xy-plane so that The path of the particle intersects the x-axis twice. Write an expression that represents the distance traveled by the particle between the two x-intercepts. Do not evaluate

  11. Home Work Worksheet: 11-B

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