#### Packet:curve-analysis-packet ; Textbook:textbook :

**Daily Lessons**

**Wed. Dec 13th**

- Start of next unit : place answers in your notebook
- In class we defined(page 2 in packet) terms 1 – 6. These definitions can be found further down on the website.
- Work on page 3 questions 1 – 5 (You will need a graphing calculator)
- HW watch this video.

##### Videos:

**Supplementary materials**

**How to verify answers:**

- Relative min at x=1 f ‘(1)=0 and f'(x) changes from negative to positive at x=1
- Relative max at x=3 f ‘(3)=0 and f'(x) changes from positive to negative at x=3
- Interval f(x) is decreasing f ‘ (x) < 0 in that interval
- Interval f(x) is increasing f ‘ (x) > 0 in that interval
- Inflection points f ” (x) = 0 at those points
- f(x) is concave down f “(x) < 0 in that interval
- f(x) is concave up f “(x) > 0 in that interval

#### summary notes

- Link that draws in the derivative graph
- answers to internet activity
- Test your skill at drawing f(x) from derivative graphs
- Concavity
- 2nd derivative graphs

**Definitions:**

A **C****ritical point****(**number) is where the derivative is zero or undefined.

The **absolute extreme **is either the highest or lowest y-value of a function in an interval. To find the absolute min and max in an interval, calculate the y-values at the endpoints in the interval. Also find any point in the interval in which the derivative=0 because the hill and valleys occur when f ‘(x) = 0. Find the y-values for when f ‘(x) = 0. The absolute min will be the lowest y-value of all the points you calculated and the max will be the greatest y-value.

The **relative extremes** are the highest and lowest points** local** to the surrounding points. They are the points on top of the hills or on the bottom of the valleys. The relative extremes can occur where the derivative=0 or is undefined. In the graph there is a relative min at x=-3 , x=2 and relative max x=-5 , x=0 & x = 4.If you have a function and want to find the relative min or max. Take the derivative of the function and set it = 0. Draw a sign chart of the derivative. The sign chart will indicate if the function is increasing or decreasing. If the sign chart produces a pattern + | – the point is a relative max. If the sign chart has a pattern – | + the point is a relative min. If the sign chart is either – | – or + | + , there is no sign change so the point is nether a relative min or max but is a plateau.You can also use the 2nd derivative to determine if the relative extreme is a min or max.

If f'(x) = 0 and f”(x) > 0 (concave up) it is a min

If f'(x) = 0 and f “(x) < 0 (concave down) it is a max

** Decreasing Intervals:** Where f(x) has negative slope. f ‘ (x) < 0 or negative

**Increasing Intervals:** Where f(x) has positive slope. f ‘ (x) > 0 or positive.

**Plateau:** Where f ‘(x) = 0 , but it is neither a relative min or max.

**Inflection point:** The point when the function changes it’s concavity.f”(x) = 0 at this point.

**Concave up:** ** f “(x) > 0 . ** **Concave down: **** f “(x) < 0**

**Monotonic:** When the function is entirely increasing or decreasing. f ‘(x) is always positive or negative.

**Differentiable:** The function can be differentiable at a point or over the entire domain. This means that the derivative exists either at that point or everywhere.

#### Rational example:

#### Velocity and acceleration revisited

- Where object stops –> v(t) = 0
- Object traveling down, left or backwards —> v(t) <0
- Object traveling up, right forwards —> v(t) > 0
- Displacement –> p(t2) – p(t1) ; change in position
- Total distance –> Need to split up the intervals the object is moving forwards and back. Find the distance for each and add the absolute values.
- average velocity, do not use velocity equation; displacement/time
- average speed, total distance / time
- velocity decreasing —> a(t) < 0
- velocity increasing —-> a(t) > 0
- Speed decreasing, velocity and acceleration have opposite signs
- Speed increasing , both velocity and acceleration have same sign