V ECTORS AND C ALCULUS Section 11-B. Vectors and Derivatives If a smooth curve C is given by the equation Then the slope of C at the point (x, y) is given.

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Presentation transcript:

V ECTORS AND C ALCULUS Section 11-B

Vectors and Derivatives 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 The derivativemay 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.

Velocity 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.

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

Speed of the particle 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.

1)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

1)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

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.

3)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

H OME W ORK Worksheet: 11-B