Exploration 3.1.13.1.2-1
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Sharpstown H S**We aren't endorsed by this school
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GEOM A PREAP 5
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Mathematics
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Dec 16, 2024
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3
Uploaded by ChefHippopotamus6820
Discovery Precalculus: A Creative and Connected ApproachLesson 3.1: A Special NumberExploration 3.1.1: A Number between 2 and 3In this activity we will investigate an accumulation function related to thealgebraicmeasureof the area under the curve of the function. Byalgebraic measure, we arereferring to the measure of the area between the curve and thex-axis on a giveninterval.f(t) =1tWe will restrict our activities to the intervalt∈[1,3].1.On graph paper, create coordinate axes with a scale of0.1unit on each axis. Thevalues on thef(t)axis should range from0to1.1, and the values on thet-axisshould range from 0 to3.1. Plot the graph off(t)on the domain interval[1,3]byplotting at least 8 evenly spaced values on the interval. Carefully draw acontinuous curve through the values.2.The function that represents the accumulated area underf(t)on the interval[1, x]wherex∈[1,3]will be calledL(x). What is the value ofL(1)?3.With a scale of 0.1 on each of your axes, what is the value represented by thearea of each square of the grid on your graph paper? Use this fact toapproximate the value ofL(2).4.Approximate the value ofL(3).5.Use what you have discovered so far to estimate the value forx∈[1,3]such thatL(x) = 1. Do you know the special name given to this value?Research Extension:6.SinceL(x)is acontinuousfunction, there is a theorem from calculus thatguarantees that there is a unique valuex∈[1,3]such thatL(x) = 1. Researchthis theorem and explain how it applies to this problem.5
Discovery Precalculus: A Creative and Connected ApproachExploration 3.1.2: More on that Special Number1.Write a generic exponential growth functionA(t)that has initial valuePand anannual growth rate (expressed as a decimal) ofr.2.The function that you wrote in part 1 describes compounding the growth onceper year. Adjust this function to compound the growthntimes per year. Forexample, ifr= 0.12(12%)once per year, lettingn= 2would mean that the valueofAincreases by6%each of 2 times per year.3.Your function from part 2 is called the compound interest function. Strictlyspeaking, this is not a continuous function. For example, if12%interest iscompounded twice a year, then the value of the investment would only take on 3discrete values during that year — the original value,106%of the original value,and106%of106%of the original value. Sketch graphs of the compound interestfunction over a two-year period forn= 2andn= 4. You can letPtake on somearbitrary value, such as 100.6
Discovery Precalculus: A Creative and Connected Approach4.Describe at least two things that happen to the “bars” on the graph asnincreases.5.Describe how the appearance of the graph would change as you letnapproachinfinity.6.Let’s define a quantityx=nr. For a fixed value ofr, what happens toxasnapproaches infinity?7.Now go back to your function from part 2. Ifx=nr, then we can replacernwith, and we can replace thenin the exponent with.Make these substitutions and write the revised formula here.8.Your formula in part 7 should contain the expression1 +1xx. In a graphingcalculator, enter this expression underY1, go to TBLSET (2nd WINDOW), andchange the independent variable from Automatic to Ask. Now when you go tothe TABLE, you can type in the values ofxthat you want to substitute into thefunction instead of having the calculator pick them for you. Now use the Table toevaluate this expression forx= 10,100,1000,10000,etc. Describe what happensto this expression asxgets very large.9.The number that this expression approaches asxgets very large is callede, anirrational number that is approximately equal to. Substituteeinto your function from part 7. Describe the difference between what thisfunction describes and what your function from part 2 describes.7