integer problems adding multiplying dividing subtract,solving multiple variable polynomial equation,prealgebra solving equations by multiplying and dividing worksheet,solving second order nonlinear differential equations,fraction and decimal multiplying and dividing worksheets,exponents Simplifying Multiplication Expressions Lesson Plan,solving second order non-homogeneous differential equations,combining like terms in algebraic expressions worksheets,solving cubed polynomials equations,help with the algebraic expression four times a number divided by three,factoring polynomials with a cubed term, tutorial,Multiplying rational expression fractions solver,free advanced algebra add subtract rational expressions,What are sqaure root method of Quadratic Equations?,pre algebra distributive property prentice hall,techniques in getting the least common denominator in college algebra,simplify radical expressions calculator root,solving second order nonhomogeneous differential equation,convert decimal to fraction algebra worksheet,simplifying complex rational expressions solver,solving second order linear difference equation,convert decimal to square root fraction,convert mixed number to decimal,multi root quadratic equation solver TI-83,quadratic equations square root property calculator,Uniqueness of forcing terms in linear partial differential equations,simple pre-algebra equations,solve my algebra problem with a calculator and fractions,solving 2nd order differential equation,online calculator that solves complex trinomials,method to solve third order polynomials,Algebra I help turning rational equations into linear equations,adding ,subtracting ,multiplying ,and dividing fractions,fractions formula adding subtracting multiplying,solving nonlinear differential equations matlab,how do you divide and times add subtract integers worksheet,multiplying,dividing,adding,subtracting integers,solving quadratics by factoring powerpoints,Like Terms Simplifying Algebraic Expressions Junior High,adding and subtracting positive and negative integers free worksheets,solving addition and subtraction equations,solving radicals and square roots,solve nonlinear equation system maple symbolic,how use my Casio Calculator for solving linear equations,solving nonhomogeneous 2nd order differential equations,solving nonhomogeneous second order linear differential equation,solving quadratic equations 3 variables,test for adding and subtracting negitive and positive numbers,greatest common factor of two numbers is 871,solving partial differential equations by factoring,algebraic formula percentage number variable,online factoring of complex quadratic equations with variables only,multiplying dividing adding subtracting fractions,how to solve multivariable algebra equation with fractions,find vertex of absolute value equations,math poems with the words, prime numbers, common multiples,common factors,pre algebra adding and subtracting integers worksheet,Combining Like Terms pre-algebra worksheet,linear equation in two variables subject to linear constraint inequalities,adding subtracting dividing multiplying integers games,Java method Convert Decimal Numbers to time,slow steps for balancing chemical equations with least common multiples,find suare root of real numbers,order of fractions from least to greatest Glencoe,McGraw-Hill,slope solving by elimination online calculator,simplifying exponents and square root calculator,quadratic equation extracting square root,dividing fractions and mix numbers cheat problem solver,simplified radical form square roots,solving quadratic equations by completing the square,multiply and divide rational expressions calculator,convert decimal to fraction - formula,how to solve first order linear differential equations using laplace transform,graphing linear equations on the coordinate plane powerpoint,solving third order linear equations,worksheet solving addition and subtraction equations,solving quadratic equations completing the square,combining like terms powerpoint lesson,Simplifying a sum of radical expressions calculator,greatest common factor for three number with variables,Pre-Algebra chapter 2 evaluate expressions, worksheet triangle expressions answers,integers adding, subtracting, multiplying, dividing worksheet,how do you do the square root on the TI-83 plus graphic calculator?,worksheets solving addition and subtraction equations,math help containing equivalent fractions decimals and percentages,error 13 dimension on a ti 86 graphing calculator solving linear equations,adding subtracting multiplying dividing integers,solve my algebra problem with a calculator and fractions for free,examples of quadratic equations with fractional exponential,ti-83 plus solving systems of linear equations in three variables,example of pictures of plotting points on the graphing calculator,adding subtracting multiplying and dividing integers,square roots and cube roots activity,least common denominator calculator,Solving Non Linear Simultaneous Equations,worksheets on add subtract multiply divide fractions,Adding, subtracting, multiplying and dividing Integer worksheets,Like Terms Simplifying Algebraic Expressions Activities Lessons,converting base 2 fraction to decimal fraction,solving simultaneous equations with excel equation solver,equation of the line solver ordered pairs,value for variable expression radicals roots,worksheet on adding,subtracting,multiplying,dividing integers,rules for adding variables in a square root,simplify dividing algebra expressions online calculator free,solving basic algebra equations homework word problem,dividing algebraic terms online free calculator,adding and subtracting fractions with like denominators worksheet,square odd integer multiple of 8 plus one triangle number,7th grade level variables and equations explanations Equations Substitution variables,free help on a 4>3 solve the inequality. graph the solutions on a number line,Maths worksheet for highest common factor & least common factor for year 7,finding the least common denominator for two rational expressions,solving absolute value and radical equations using restrictions,simplify square root of difference of two squares,how to solve second order linear nonhomogeneous differential equations,slow steps balancing chemical equations with least common multiples,solve nonlinear differential equation first order,practice worksheets on adding, subtracting,multiplying,and dividing decimals. for 6th grade,adding, subtracting, multiplying, dividing integers,Quadratic Equation Calculator factor,HOW DO I DIVIDE A FRACTION BY A RADICAL FRACTION
Thank you for visiting our site! You landed on this page because you entered a search term similar to this: how to solve exponential probability in TI-83 Plus calculator, here's the result:
Weight W in newtons Length l in meters
20 2042
40 2086
60 2132
80 2176
100 2219

If we were to plot this data on a graph of weight versus length, we would see that our data points would lie on (or very close to) a straight line. We would therefore conclude that a linear relationship exists between W and l. Many relationships exist in nature that are linear, but many relationships in nature are nonlinear as well. Bacteria, for example, grow in an exponential fashion, as will be covered later in this section.

We will now use the data given above in an actual example involving Hooke’s law. One form of Hooke’s law can be written in the following form

l = m W + l

The problem is to find values for m and l based on the data given above. We will then obtain a model for which we can calculate the extension of the wire produced by a weight of 75 newtons. Since l = m W + l is a linear relationship between l and W, then the graph of this relation will be a straight line with slope m and y-intercept l. If we plot the experimental data as a graph and draw the best fitting line connecting all of the data points, we see that the line crosses the l axis at l = 2000 meters. l is therefore given the value 2000 meters, which is to be expected since the unextended length of the wire was given as 2 meters. Using the points (0, 2000) and (100, 2219), we can compute the line’s slope as follows:

m =

We can therefore rewrite our mathematical model as l = 000219W + 20 00

If we evaluate our model by setting W = 75 newtons, l becomes 2164 meters. This implies that the wire’s length has been extended by 0164 meters (1.64 mm) since 2164 - 2000 = 0164. If we take the ration of the extended length to the unextended length, we see that the wire’s length has increased by much less than 1% of its original length, i.e., 2164/2000 = 1082.

Interpolation and extrapolation are two very critical concepts in data analysis and the modeling of physical phenomena based on experiment. We now examine these concepts in relation to the linear problem posed above. Plot the data given above on a graph of l versus W, and label each point P , …, P respectively from top to bottom in our data table.

Now suppose that we are interested in the value of l for an imposed load somewhere between 40 and 60 newtons. Intuitively, we conclude that this value of l should lie somewhere between 2086 and 2132 meters given the data above. Now assuming Hooke’s law is valid, we can connect the points P and P with a straight line. We can now read directly from the graph the value of l which corresponds to a given value of W. The method used to obtain information between collected data points in this fashion is called linear interpolation. The term linear is used here to imply that we are using the method of interpolation with a linear relationship.

Notice that the experiment was conducted using weight values of up to 100 newtons. Suppose that we are interested in a certain value of l given by a value of W greater than 100 newtons. We would use the method of linear extrapolation in this case. The method is as follows. By Hooke’s law, we can connect P and P with a straight line. We can then extend the line to any value W desired to obtain a corresponding value for l.

It is important to realize that the methods of analysis outlined above give approximations and not precise values. Part of this is due to the fact that the values of l measured in an experiment may contain errors of varying degrees. We may however want to use interpolation or extrapolation to actually calculate more precise values of the dependent variable in question. To do so, we use the interpolation/extrapolation formula.

To use the interpolation/extrapolation formula, we must first have an experimentally based data set. Let’s call our independent variable x and our dependent variable y. Let’s also assume that we have at least two values for x, which we will denote x and x, and two for y, denoted y and . Now suppose that we are interested in the value of y, denoted , that corresponds to some value of x between x and x, which we will call . Note that the distance - x is just a fraction (- x)/( x - x) of the distance x - x. We can therefore conclude that the corresponding distance - y is the same fraction of - y. Setting these two ratio’s equal to each other and rearranging gives

, which gives

This is the interpolation/extrapolation formula, and it works well for both methods. Now let’s use this formula to calculate the extended length of the same steel wire as before, but for an attached weight of 30 newtons. We therefore take x=20 and x=40 newtons. We also take y=2042 and = 2086 meters, with =30 newtons in this case. Thus

= 2042+ = 2064 meters.

Notice that our answer makes perfect sense since the calculated value lies in between y and . Note also that we used interpolation in this case. To solve a similar problem using extrapolation, we would have chosen to be some weight value greater than those used in the experiment, say 200 newtons for instance.

One final note regarding interpolation and extrapolation: not all relationships are entirely linear. It turns out that the relationship between weight and the corresponding values for extended length of a steel wire is linear for a certain range of weights. As the weight load gets heavier and the wire gets closer to breaking, the graph of this relationship becomes nonlinear and more exponential in nature. This therefore implies that linear extrapolation may involve an associated amount of error as a result of working outside the scope of the data. Linear interpolation is useful for approximating values of the dependent variable for intermediate values of the independent variable. In both cases we are dealing with approximations, hence care should be taken when using either method.

Lesson 2: Bacterial Growth and Models Involving Exponentials

Standards/Benchmarks Addressed: 1-5

Suppose a microbiologist is interested in the rate at which Escherichia coli (a type of bacteria) grows under optimal growing conditions. Now lets suppose that the microbiologist measures the number of bacteria cells every ten minutes, and creates the following table based on these observations:

Time Elapsed (min) 0 10 20 30 40 50 60 70 80 90 100
Number of cells 7 10 14 20 27 39 54 76 108 151 213