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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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Homework Questions…

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**Applications of the Derivative**

Particle Motion

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Particle Motion Lab

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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?

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**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…

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**Instantaneous velocity is… **

the derivative of the position function or slope of the tangent line.

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**The graph of a position function is shown.**

Show the graph of velocity.

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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)

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**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.

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**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 |

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**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?

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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.

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END OF DAY 1

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2.1 Derivatives and Rates of Change. The slope of a line is given by: The slope of the tangent to f(x)=x 2 at (1,1) can be approximated by the slope of.

2.1 Derivatives and Rates of Change. The slope of a line is given by: The slope of the tangent to f(x)=x 2 at (1,1) can be approximated by the slope of.

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