Unit 6 Curve Analysis

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

Definitions:

A Critical 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.  absolute extremesTo 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.  abs and relative extremesThey 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.plateau

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

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

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:

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