Tip:
Highlight text to annotate it
X
CU2L4b Average Rate of Change Good day students in this is really going
over two examples on how to find the average rate of change the slope equation of secant
line related to the function I'm going to be showing you how to use the TI eighty-nine
calculator to simplify the process considerably okay right so let's are going to take a look
at question number one this is the highest at times here and object follows in the air
is given by the height function ten t minus four ninety square whenever closely over the
time interval from equals one of which equals one point five right equation of the secant
line on this interval and sketched determine secant line okay so units when you're talking
about the average velocity to the average rate of change over this interval so you don't
velocity as the average rate of change So the formula for the average velocity which
is average rate of change three change is also equal to the same formula for the average
velocity average velocity the money average velocity on the interval the okay so the only
average velocity on interval (a,b) is the eight oh be the final time minus H of the
initial time ÷ divided by b minus a this is the same formula as we remember the slope
formula algebra 2 write y2 to minus y one one over X minus X one so that exactly what
the what we doing here Right so they haven't okay so what were going to do now on is determine
how this formula applies to this situation is dealing with over here ok so lets go ahead
and write it down of the average the average the on let's see from one two one point five
is going to be H of be which is eight one point five minus H of one divided by one point
five minus one okay you what was once we noted that in this problem a is equal to one first
value in the interval is a and the second value in the interval is be okay so that the
b normally comes first when you are inputting into the function minus H of a divided by
b minus a as indicated in this formula okay so now there are different ways we can do
this is your do this by hand is the something like this H of one point five equals ten times
one point five minus point nine times one point five square and then you do the whole
process well this is the really complicated so you might need a calculator but it was
easier ten time one minus four point nine times one square okay only calculator how
to make this process considerably easy you have a TI83 or less you can computes this
process using this process is write it computes values is process with you TI eighty-nine
let me show you a trick can use right and is the procedure I am showing you also applicable
to the TI eighty-nine titanium okay so though one nice thing about calculator is that you
can create function function and you can store them from what you do is an is the f1 menu
define h of t okay so the first and for option one is defined so to find typing h of t our
H parenthesis t] L I N K (equals you are going to type in what h of t is the problem case
the problem is tends he minus (is that we can see minus one point nine to raised to
the second power okay press enter it will tell you done going so what we just the way
this distance towards this function in this in this assignment is that there is h of ten
t minus one ninety square starting h of t okay so is verify for input is correct start
typing h of t see what the calculator gives us the second so go off H parentheses to enter
from the users the original function of enter into it nine You Want to Approximate Form
with This One Is on Diamond Enter There Is Exactly What We Want ten t Minus Four Ninety
Square Okay so Let's Get Started Now Is the start doing the Computation so We Want to
Calculate Each of One Point Five He so We Simply Do Is move your cursor to the Right
Going to parenthesis delete t and Typing One Point Five enter That the Decimal That the
Fractional Result Want to Do This decimals here diamond Enter Three Point Nine Seven
Five Six Is a Final Is Process over and over Again you can just do it in one step and you
also need h of One h of One of the cursor to the Right go into the parenthesis Doing
One Point Five Negative One Enter My Decimal so Five Point One Okay so You See How We Can
Quickly Computes the Our Volume this function for Multiple Inputs Just by Changing the Argument
of Our h of t Function so I am going to put in 3.97 for h of 9.5 Five for Eighty-One Point
Five and Five Point One for h of One Interval and then well Computer Quotient Our Resulting
Quotient Rights alright so let's write that down so We Have Eighty-One Point Five Is Three
Point Nine seven five h of One Is Five Point One and One Point Five Is One easy to Determine
Simply Zero Point Five Right Now We Think of You This by Hand against What Allusion
a Strategic Do You Calculator, Only Can You Determines That Individual Functions We Can
Also Conduct Operations in Function so I Can Type in This into Esxpression
Calculator to determine what the final Answer Is a Solution to Do That to parenthesis the
Numerator Write This Is Go out to Age of One Point Five Minus Log to Each of One Enter
the Numerator Expression Here Divided by) in the Denominator One Point Why Minus One
Lead Really Careful When Using Three Now Long Expression like This in the Calculator You
Have Grouping Area See How Have To Separate the Numerator and Denominator Using) to Group
Them into the Type the Numerator (and I Think the Denominator (Okay, the Answer Now EARLIER
Expression Is an Calculator Examine Your Pretty Prints to Ensure That the Format Is Consistent
with What You Looking for in the Result Is Too Much for Assigned Is Only One in the Answer
Is Negative Two Point Two Five Idiocy about This Is That She's You Computation Clearly
a Look Likely to Make Arithmetic Errors Second so There You Go Negative Two Point Two Five
the Average Velocity Isolate Spread at the Negative to Points to Five Right in the Unit
Here This Is the We so Is Be Meters per Second Meters per Second I Silicon Valley Have the
RHD the Next Equation the Problem Tells Us to Find the Equation of the Secant) Say Right
Right Equation Is Secant Line Okay so Equation of the Line What We Looking for Here Secant
Line so What Equation of the Line Well If You Want to Find Equation Line It up Points
in the Slope Use of Formula 1 Minus Why One Equal and X Minus
This Is the Points One of Equation of a Solution the Slope Just What the Slope Is the Slope
of Secant Line Is the Valley the Average Speed Which Is Negative Two Point Two Five Okay
and We the Point X One Why One Why One What We Going to Use This Is Going
to Use of Wine and H of One Is One Eighty-One or One Point Five Rageful (Five Is the Second
so One Is One One Is Eighty-One What You Get an Opportunity Is One into Your Function Remember
the That Earlier Each of One Right Here Is Exactly Equals Five Point so Each of One Is
Five Point Can't so This Is X One... Why Is Why One Okay Right so Now We Have All the
Ingredients We Need to Write Equation of the Line Right Here Is X One Why One so Equation
of Secant Line from the the Equation of the Secant Line Is Why Minus Why One Which Is Why Minus Five
Point One Equal and Negative Two Point Two Five Times X Minus
X Minus One Okay so That's the Equation of Your Secant Line in the One Slope Right Now
Lastly We Asked Sketch the Graph of the Situation Negative Incentive Function and Also Indicates
Secant Line There to Okay so to Graph within Make the Meeting Use of Our Graph Calculator
so All the Ninjas Title of What We Right Here the Graph so All What We Grafting the Graph
the Original Function the Graph h of t Which Equals Ten Be Minus Four Point Ninety Square
Emilio All Also Going to Graph the Secant Line Why Minus Five Point One Equals Negative
Two Point Two Five Times X Minus One Is System Isolate Going to Denominator Is the Function
to Calculator so If You're at the Screen Use Press Diamond of One Versus the Graph Is Negative
Let's Enter the Function Ten X Minus Four Point Nine Square Answer by My One or Minus
Using See over the Graph the Manual Your Independent Variable X on a Function Is an Independent
It Was to Matter What You Call the Variable Going over the Internet the Independent Will
Always value X okay so is this is exactly the same as is negative to write to look ahead
and graph) and when the graph in addition look like this okay to that goes be function
that position function of the object altering the air so why does will focus zooming is
area right here from around one two two one zero two two because the graph in on the interval
from 1 to 1 point five okay to exhibit a little bit portion that enables drug the graph summarizing
is. Right here so I will press F2 *** box on have to offer (of left corner to something
there to the left of zero in a variety now it is negative sketch here sites I get the
snapshot of what is the sketch right so that if I get enter right sell it so much like
the graph so that the degree that is what my graph looks likes the let's see how hundred
and so treated three portrays the Valley five point zero five point one okay in a goes from
one so slightly are about to isolate limited sketch using what we got from the left calculator
so a goes your y-axis the goes flying X axis by force here one two three four
one two three four it's still using that because all my that the spread out considerably case
of this is why two three four five I have my X with more marked tick marks for unit
so let us exaggerate my graph in the X direction okay so let's for our long here Wednesday
intersected some here Venecia for the what the left side looks like I'll is that zero
okay so zero question zero two definitely sketch of your function nothing perfect just
right through the sketch in that and then in and then comes down to that this is looks
like I not only have the graph of the function of the function mislabel it is is h of t equals
ten minus one ninety square now we also have to graph or secant line remember secant line
went through one and one point five right so this is one and raise one two three four
one so this is right here is one point five we have to points when here that's one is
when three point nine seven five in the point five notice it is one and then we have one
point five right here that interested when I simplify and then that like this so they haven't so you have to points
and then we just first secant line for secant line that is what the exact need to bring
our you the points with his remember one point five was three point nine seven five so CBC
somewhere is going on here but this fifty perfectly okay so it's just that something like this
is point nine seven five is one point five and five point one is one one okay so the
secant line this is to secant line and then is going to write and what the equation of
the secant line so there so there goes the group the the graph of the secant okay let's
take a caller times I think it does but like that okay so... Secant line all let's write
down the equation to this is your secant line why minus five point one equals negative two
points by times X minus one this is a goes the secant line in the slope right here that's
that we calculated before that the average the average rate of change this indicate that
in the graph also so this is term right here here here is point that right there is is
your Delta X each of he over Delta T right and that you average velocity is negative
two point negative two point two five meters per second right so they haven't right to
take a look at the question to in this example really help you see the power of the formula
that, should the earlier can so let us were to accidentally of the curriculum silver time
interval, one two one point zero why want negative one one two one point is your one
into the left point I negative one one nine nine 9 to 1 point nine nine nine nine to what
you this is negative thinking EQUALS one can't so slowly can we actually determine what the
speed of this of object is X equals one using with these left and right limiting values
of the average velocity thinking that's the question is well with you the benefit of the
people to help organize my data in a we use a calculator to do it with the so i dont spend
all my energy involved in arithmetic Right so the first on the first of first: politically
answer in the interval's so calling one is negative interval's interval of from eight
to be in I and then on those addition, the answer the value of the of the average reduction
and so Colón in the second column and ASAP average average velocity remember the average
velocity is simply that the everything changing answer the average velocity is built the h
of t Delta h of t divided by the open T which is the formula h of d minus H is a divided
by B minus eight Isolate going into the first first interval 1 to 0 point one andwithlongway
zero one so they go from one one point zero one Right so the average velocity is the average
velocity infinity see what the age of be is one point zero why minus each of wine divided
by one point zero one minus one okay and after we can keep that winter the value on the last
call amplitude prehearing right so let's call the calling for about here is the value to
widely get we looking at the Valley here isolate the into this this is the first interval start
see on the scene out remember we are in the final variable-rate these two women same problem
so let's the complex of function is our h of t use the there okay so that the denominator
to get approximate hapless I to does nothing but on the right in the function limit every
for graph: answer in this nation was following so cynical hl why one of the information practices
parentheses out for h of one point zero one.
how is this minus out for hl one can that will the numerator divided by one point zero
one minus one okay i still yes i express center them INTERCEDING APPROXIMATE THIS IS POINT
ONE FIVE OKAY TO THE VALUES POINTS WILL DIVISOR