new unit: derivative tools in this unit you will develop tools to better understand curves. you will...

New Unit: Derivative Tools• In this unit you will develop tools to

better understand curves. You will learn to distinguish between maximums, minimums, and points of inflection. You will also learn to differentiate more complex functions such as composites and trigonometric reciprocals.

Two Main Learning Targets

• You will use the derivative in a variety of applied situations including velocity, acceleration, and optimization.

• You will learn techniques to find the derivative of more complicated functions involving products, quotients, and composite functions.

What is the starting position?

Today you will find velocity and acceleration from

a position function.

Spitwads

• Help Marisol by writing an equation for this data. Use h and t as your variables.

In her science class, Marisol often got into trouble for shooting spitwads. Because she was sneaky about it, she rarely got caught. One day, someone bumped her arm, causing a spitwad to shoot straight up in the air. Out of curiosity, Marisol decided to find an equation for the height (above her head) of the spitwad as a function of time. She collected the data in the chart below at right.

Spitwads

• Marisol decided to find

and . What does this mean? Explain the notation.

In her science class, Marisol often got into trouble for shooting spitwads. Because she was sneaky about it, she rarely got caught. One day, someone bumped her arm, causing a spitwad to shoot straight up in the air. Out of curiosity, Marisol decided to find an equation for the height (above her head) of the spitwad as a function of time. She collected the data in the chart below at right.

dh

dt2

2

d h

dt

Spitwads

• Find and• What does each tell

you about the spitwad?

• What are the units?

In her science class, Marisol often got into trouble for shooting spitwads. Because she was sneaky about it, she rarely got caught. One day, someone bumped her arm, causing a spitwad to shoot straight up in the air. Out of curiosity, Marisol decided to find an equation for the height (above her head) of the spitwad as a function of time. She collected the data in the chart below at right.

dh

dt

2

2

d h

dt

Spitwads

In her science class, Marisol often got into trouble for shooting spitwads. Because she was sneaky about it, she rarely got caught. One day, someone bumped her arm, causing a spitwad to shoot straight up in the air. Out of curiosity, Marisol decided to find an equation for the height (above her head) of the spitwad as a function of time. She collected the data in the chart below at right.

dh

dtUsing , explain why the acceleration is negative. Notice that acceleration is negative even when the spitwad is moving up (in the positive direction).

Spitwads

In her science class, Marisol often got into trouble for shooting spitwads. Because she was sneaky about it, she rarely got caught. One day, someone bumped her arm, causing a spitwad to shoot straight up in the air. Out of curiosity, Marisol decided to find an equation for the height (above her head) of the spitwad as a function of time. She collected the data in the chart below at right.

Notice that the acceleration is constant. What does this mean? What causes this constant acceleration.

/ sec.ft

Closure

• How does the position function reveal information about initial velocity and initial position?

• On Earth, is there a constant force of gravity?

Assignment

HW N

See you tomorrow!