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Monday 9/21
Today we went over investigation 3.5a), our introduction to particle motion. If a function is used to describe the position of a particle moving along a horizontal line at any time t, the velocity of the particle is described by the derivative of the position function, and the acceleration of the particle is described by the derivative of the velocity function or the second derivative of the position function. Here is an example of those ideas: Position: x(t) = 4t³ -5t² - 6t +8 Velocity: v(t) = x '(t) = 8t² -10t - 6 Acceleration: a(t) = v ' (t) = x " (t) = 16t - 10

Since velocity indicates direction and speed, we can find out whether the particle changes direction by looking to see whether the velocity changes sign; this is likely to occur when the velocity is zero. The sign of the velocity, **combined with the sign of the acceleration**, will indicate whether the particle is slowing down or speeding up.

Solutions to the exploration can be found under Friday of last week.

Tuesday 9/22
An unexpected day off due to the flooding of parts of Atlanta.

Wednesday 9/23
Today we began class with a problem of the day, shown below.

We then discussed legal calculator operations.

__Legal Calculator Operations:__ You may do the following "without work" but set-ups must be shown. 1. Graph a function 2. Evaluate f '(c) 3. Evaluate a definite integral over a defined integral 4. Solve an equation

For the remainder of class we went over a motion problem sheet.





Homework: Complete Motion Problem Sheet

Thursday 9/24
Problem of the day:

[[image:smallstones09p2:Problems_of_the_day_24.gif width="579" height="818" align="left"]]
Use of the power rule twice.

Rewrite the expression using a negative exponent before using the power rule.

Rewrite the expression using a negative, rational exponent before using the power rule.

Remember to deliver the answer in the same form as the question was asked.

Recognize that the function in this question is not a power function; it is an exponential function for which we have no derivative rule. We will have a rule soon!

Use the nDeriv command to find the derivative of a function at a particular value of its domain nDeriv is found on the MATH menu at #8. The parameters needed are function, x, value of x.

Next task was to finish the problems from yesterday. The last slides is shown here.

REMEMBER: speed is the magnitude or absolute value of velocity

Next we began an exploration aimed at discovering how to find the derivative of the 1) sine function and 2) a composition of two functions. We found, by graphing and conjecture, that if f(x) = sin(x) then f '(x) = cos(x). From there we tried to predict the derivative of f(x)= sin(3x). Most of us thought f ' (x) would be cos(3x) but, when we graphed the derivative and our conjecture, we realized we needed a stretch factor of 3 for our conjecture. Consequently, the derivative of sin(3x) must be 3cos(3x)

Friday:
On Friday, we looked at more sin and cos functions and their derivative functions. We began with our problem of the day:



We then found the derivatives of some more of the sin/cos functions trying to come up with a generalized chain rule.

Then we did some more problems to try to reach our conjecture about the chain rule formula, and we came upon our answer.