Warm Up Determine the anti-derivative. Then differentiate your answer to check your work. 1. 2. Evaluate the definite integral: 3.

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Presentation transcript:

Warm Up Determine the anti-derivative. Then differentiate your answer to check your work. 1. 2. Evaluate the definite integral: 3.

Homework Questions…

Applications of the Derivative Particle Motion

Particle Motion Lab

You drive to the beach. If your trip is exactly 200 miles and it takes you 4 hours to get there, what is your average speed?

Velocity is the slope of position. Average velocity = Average rate of change of position = slope of the secant line …between the beginning time and ending time…

Instantaneous velocity is… the derivative of the position function or slope of the tangent line.

The graph of a position function is shown. Show the graph of velocity.

The position function is an equation that gives the position of an object relative to something at a specified time. Notation is usually f(x), x(t) or y(t). Displacement is the change in position from time a to time b. In other words, Displacement = x(b) – x(a)

1) Evaluate and explain what is meant by x(0), x(1) and x(3). In driving from New York to Boston, your position function, x(t), given in miles from New York is described by the function where t is the number of hours since the trip began. It takes you 8 hours to get to Boston. 1) Evaluate and explain what is meant by x(0), x(1) and x(3). 2) Determine the displacement from t = 0 to t = 8. What does this value mean in terms of the situation? 3) Find the average velocity for the trip from New York to Boston 4) Find the velocity at time t = ½ hour.

Velocity versus speed Velocity is a directed speed. Positive velocity means you are moving forward (up) and negative velocity means you are meaning backward (down). Speed = | velocity |

What is my speed at t = 5 hours? In driving in a straight line from New York to Boston, your position function given in miles from New York is described by the function where t is the number of hours since the trip began. It takes you 8 hours to get to Boston. What is my speed at t = 5 hours? Do I ever “stop moving” during this trip? 7) Do I ever backtrack (change directions) during this trip?

Your turn… NO Calculator A particle moves vertically(in inches)along the y-axis according to the position equation x(t) = t3 – 6t2 + 9t – 4., where t represents seconds. Find 1) The velocity at t = 2 2) The speed when t = 2 3) The average velocity on the interval [2,4] 4) When the particle changing direction 5) When the particle moving left? Right? 6) The displacement from t = 0 to t = 4.

END OF DAY 1