Lesson Plan #6
Lesson Plan #14
Class: AP Calculus Date: Thursday October 7th, 2010
Topic: Rates of Change
Aim: How can we find the average velocity of an object?
Objectives:
1) Students will be able to find average and instantaneous velocity of an object
HW# 14:
1) Find the average rate of change of the given function over the indicated interval
A) [pic] [pic]
B) [pic] [pic]
C) [pic] [pic]
2) Use the following position function for free falling objects to answer the subsequent questions.
[pic]
A) A pebble is dropped from a height of 600 feet. Find the pebble’s velocity when it hits the ground.
B) A ball is thrown straight down from the top of a 220 foot building with an initial velocity [pic]feet per second.
i) What is the velocity after 3 seconds?
ii) What is the velocity after falling 108 feet?
Do Now:
If an object is dropped from a height of 100 feet and falls freely, its height [pic]feet at time [pic]is given by the function
[pic].
Find the height of the object after 1 second.
Find the height of the object after 2 seconds.
Write the Aim and Do Now
Get students working!
Take attendance
Give back work
Go over the HW
Collect HW
Go over the Do Now
What was the change in height?
What was the change in time?
What was the average velocity, in feet per second?
Using the position function in the Do Now, for the interval [1, 1.5], find the average rate of change.
Using the position function in the Do Now, for the interval [1, 1.1] find the average rate of change.
Find the average rate of change of the given function over the indicated interval.
1) [pic] [pic]
2) [pic] [pic]
3) The height [pic], in feet, at time [pic], in seconds, of a silver dollar dropped from a height of 1350 feet is given by [pic]. Find the average velocity in the interval [1,2]
Suppose we had to find the velocity at the instant [pic], or the instantaneous velocity at [pic]. To find the instantaneous velocity it’s like finding the average velocity [pic], but there is no change in time. It is during an instant. So we need to evaluate the limit, [pic], which we know from our previous work is the definition of the derivative, or in this case [pic].
Example:
For the height function[pic], representing the height, s of an object at time t, find the instantaneous velocity at [pic] and at [pic]:
How long, to the nearest tenth of a second, will it take the object to hit the ground?
In general, the position of a free falling object under the influence of gravity on earth can be represented by the equation
[pic],where[pic]is the initial height, and [pic]is the initial velocity.
We consider velocity to be positive for upward motion and negative for downward motion
Note: Speed is the absolute value of velocity, so speed is always positive.
Example:
A projectile is shot upward from the surface of the earth with an initial velocity of 384 feet per second. What is the velocity after 5 seconds?
Example:
To estimate the height of a building, a stone is dropped from the top of a building into a pool of water at ground level. How high is the building if the splash is seen 6.8 seconds after the stone is dropped?
Example:
A dynamite blast blows a heavy rock straight up with a launch velocity of 160ft/sec. It reaches a height of [pic]ft. after [pic]seconds.
A) What is the height of the rock after 3 seconds?
B) What is the maximum height reached by the rock?
C) What is the instantaneous velocity at 4 seconds?
D) How long is the rock in the air?
E) How fast is the rock traveling when it first reaches 256 feet above the ground?
Exercise:
A diver jumps from a diving board 32 feet high. The position of the diver is given by [pic], where s is measured in feet and t is measured in seconds.
1) After the diver jumps, how many seconds later will he hit the water?
2) After the diver jumps, how many seconds will it take for the diver to reach his maximum height?
3) What is the maximum height attained by the diver?
4) What is the diver’s velocity at impact with the water?
5) What is the diver’s velocity 1 second after jumping?
6) How many seconds after jumping is the diver’s height 32 feet?
7) What is the diver’s velocity when he reaches 32 feet heading downward?
-----------------------
Definition of (Instantaneous) Velocity: If [pic]is the position function for an object moving along a straight line, then the instantaneous velocity of the object at time [pic]is given by [pic]
Definition of Average Velocity: If [pic]gives the position at time [pic]of an object moving in a straight line, then the average velocity of the object in the interval [pic] is given by
Average Velocity = [pic]
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