c1 1 2 3 function analysis critical values intervals of increase decrease first derivative test n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
C1.1,2,3 – Function Analysis – Critical Values, Intervals of Increase/Decrease & First Derivative Test PowerPoint Presentation
Download Presentation
C1.1,2,3 – Function Analysis – Critical Values, Intervals of Increase/Decrease & First Derivative Test

Loading in 2 Seconds...

play fullscreen
1 / 15

C1.1,2,3 – Function Analysis – Critical Values, Intervals of Increase/Decrease & First Derivative Test - PowerPoint PPT Presentation


  • 159 Views
  • Uploaded on

C1.1,2,3 – Function Analysis – Critical Values, Intervals of Increase/Decrease & First Derivative Test. IB Math HL/SL - Santowski. (A) Important Terms. Recall the following terms as they were presented in a previous lesson: turning point : points where the direction of the function changes

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'C1.1,2,3 – Function Analysis – Critical Values, Intervals of Increase/Decrease & First Derivative Test' - lindsey


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
c1 1 2 3 function analysis critical values intervals of increase decrease first derivative test

C1.1,2,3 – Function Analysis – Critical Values, Intervals of Increase/Decrease & First Derivative Test

IB Math HL/SL - Santowski

a important terms
(A) Important Terms
  • Recall the following terms as they were presented in a previous lesson:
  • turning point: points where the direction of the function changes
  • maximum: the highest point on a function
  • minimum: the lowest point on a function
  • local vs absolute: a max can be a highest point in the entire domain (absolute) or only over a specified region within the domain (local). Likewise for a minimum.
  • increase: the part of the domain (the interval) where the function values are getting larger as the independent variable gets higher; if f(x1) < f(x2) when x1 < x2; the graph of the function is going up to the right (or down to the left)
  • decrease: the part of the domain (the interval) where the function values are getting smaller as the independent variable gets higher; if f(x1) > f(x2) when x1 < x2; the graph of the function is going up to the left (or down to the right)
  • "end behaviour": describing the function values (or appearance of the graph) as x values getting infinitely large positively or infinitely large negatively or approaching an asymptote
b review graphic analysis of a function
(B) Review – Graphic Analysis of a Function
  • We have seen functions analyzed given the criteria intervals of increase, intervals of decrease, critical points (AKA turning points or maximum or minimum points)
  • We have also seen graphically how the derivative function communicates the same criteria about a function  these points are summarized on the next slide:
b review graphic analysis of a function1
f(x) has a max. at x = -3.1 and f `(x) has an x-intercept at x = -3.1

f(x) has a min. at x = -0.2 and f `(x) has a root at –0.2

f(x) increases on (-, -3.1) & (-0.2, ) and on the same intervals, f `(x) has positive values

f(x) decreases on (-3.1, -0.2) and on the same interval, f `(x) has negative values

(B) Review – Graphic Analysis of a Function
c analysis of functions using derivatives a summary
(C) Analysis of Functions Using Derivatives – A Summary
  • If f(x) increases, then f `(x) > 0
  • If f(x) decreases, then f `(x) < 0
  • At a max/min point, f `(x) = 0
  • We can also state the converse of 2 of these statements:
  • If f `(x) > 0, then f(x) is increasing
  • If f `(x) < 0, then f(x) is decreasing
  • The converse of the third statement is NOT true  if f `(x) = 0, then the function may NOT necessarily have a max/min  so for now, we will call any point that gives f `(x) = 0 (i.e. produces a horizontal tangent line) aCRITICAL POINTS or EXTREME POINTS
d first derivative test
(D) First Derivative Test
  • So if f `(x) = 0, how do we decide if the point at (x, f(x)) is a maximum, minimum, or neither (especially if we have no graph?)
  • Since we have done some graphic analysis with functions and their derivatives, in one sense we already now the answer:  see next slide
e first derivative test graphically
At the max (x = -3.1), the fcn changes from being an increasing fcn to a decreasing fcn  the derivative changes from positive values to negative values

At a the min (x = -0.2), the fcn changes from decreasing to increasing  the derivative changes from negative to positive

(E) First Derivative Test - Graphically
f first derivative test algebraically
(F) First Derivative Test - Algebraically
  • At a maximum, the fcn changes from being an increasing fcn to a decreasing fcn  the derivative changes from positive values to negative values
  • At the minimum, the fcn changes from decreasing to increasing  the derivative changes from negative to positive
  • So to state the converses:
  • If f `(x) = 0 and f the sign of if `(x) changes from positive to negative, then the critical point on f(x) is a maximum point
  • If f `(x) = 0 and f the sign of if `(x) changes from negative to positive, then the critical point on f(x) is a minimum point
  • So therefore, if the sign on f `(x) does not change at the critical point, then the critical point is neither a maximum or minimum  we will call these points STATIONARY POINTS
g first derivative test example 1
(G) First Derivative Test – Example #1
  • Find the local max/min values of y = x3 - 3x + 1 (Show how to use inequalities to analyze for the sign change)
  • f `(x) = 3x2 – 3
  • f `(x) = 0 for the critical values
  • 0 = 3x2 – 3
  • 0 = 3(x2 – 1)
  • 0 = 3(x – 1)(x + 1)
  • x = 1 or x = -1
  • Now, what happens on the function, at x = + 1?  let’s set up a chart to se what happens with the signs on the derivative so that we can determine the sign on the derivative so that we can classify the critical points
g first derivative test example 12
(G) First Derivative Test – Example #1
  • Since the derivative changes signs from +ve to –ve, the critical point at x = -1 is a maximum (the original function changing from being an increasing fcn to now being a decreasing fcn)
  • Since the derivative changes signs from -ve to +ve, the critical point at x = 1 is a minimum (the original function changing from being a decreasing fcn to now being an increasing fcn)
  • Then, going one step further, we can say that f(-1) = 3 gives us a maximum value of 3 and then f(1) = -1 gives us a minimum value of -1
  • And going another step, we can test the end behaviour of f(x):
    • lim x-∞ f(x) = -∞
    • lim x ∞ f(x) = +∞
  • Therefore, the point (-1,3) represents a local maximum (as the fcn rises to infinity “at the end”) and the point (1,-1) represents a local minimum (as the fcn drops to negative infinity “at the negative end”)
h in class examples
(H) In Class Examples
  • Ex 2. Find the local max/min values of g(x) = x4 - 4x3 - 8x2 - 1
  • Ex 3. Find the absolute minimum value of f(x) = x + 1/x for x > 0
  • Ex 4. Find the intervals of increase and decrease and max/min values of f(x) = cos(x) – sin(x) on (-,)
  • Ex 5. Find the critical numbers, intervals of increase & decrease and max/min values of y = csc(x) – cot(x) on (-/2,3/2)
  • Ex 6. Find the intervals of increase/decrease and max/min points of f(x) = x2e-x
  • Ex 7. Find the local and absolute maximum & minimum points for f(x) = x(ln(x))2
i internet links
(I) Internet Links
  • Visual Calculus - Maxima and Minima from UTK
  • Visual Calculus - Mean Value Theorem and the First Derivative Test from UTK
  • First Derivative Test -- From MathWorld
  • Tutorial: Maxima and Minima from Stefan Waner at Hofstra U
j homework
(J) Homework
  • Handout from Stewart, 1997, Chap 4.2, p279-281, Q3-6,7-14,25-46,47-50