The data in the table are generated by the function f . Graphically approximate the unknown cons

The data in the table are generated by the function f ....

Key Concepts

Logarithmic Transformation

Logarithmic transformation is a technique used to linearize an exponential model. Taking the natural logarithm of both sides of the equation converts the exponential relationship into a linear one, which simplifies the task of estimating the decay rate 'b' using linear regression methods.

Graphical Analysis for Estimation

Graphical analysis involves plotting the data to visually estimate the parameters of the function. By examining the graph, one can approximate the intercept and the rate of decrease, which correspond to the parameters in the exponential model, and refine these estimates by identifying patterns or trends.

Parameter Estimation

Parameter estimation is the process of determining the constants in a model based on observed data. In the context of exponential functions, it involves finding values for 'a' and 'b' that best describe the behavior of the data, often through methods such as curve fitting or regression analysis.

Exponential Function Modeling

Exponential functions describe processes where changes occur at a rate proportional to the current value. They are used widely in modeling decay or growth, where the function takes the form f(x) = a * e^(-b * x), with 'a' representing the starting value and 'b' the decay rate.

Transcript

00:01 All right, in 8 .1, number 57, they give us a table of data for a function that has this form a times e to the negative bx.

00:14 So it's an exponential function.

00:17 And they're asking us to find the parameters of a and b, so that leading coefficient and the constant that multiplies up in the numerator.

00:27 They're asking us to find those to four decimal places.

00:32 So because they're asking us to find two unknown values, what we'll need to do is set up a system of equations to help us do that.

00:45 And they're also asking us to solve or represent those values graphically.

00:54 So we'll talk about that.

00:57 Now, to set up a system of equations, all i'm going to do is take some ordered pairs.

01:02 Which represent the x and the y value and make a couple of equations.

01:07 So i'm going to choose to use the first ordered pair 1 .0 .80487, and i will substitute that value into my equation for the x and the y value.

01:26 So x is going to go in as 1, which really is just e to the negative b.

01:33 And remember your properties of exponents, a negative exponent bumps that base to the denominator, makes the exponent positive.

01:44 So that side would look like a over e to the b.

01:49 And then for your second equation, you can use any of the other three ordered pairs.

01:54 I'm just choosing to use the second ordered pair from the table, just going in order here.

02:01 And we'll set up a similar equation.

02:06 And we'll plug the two in.

02:10 So again, we have a negative exponent.

02:12 We'll kind of clean this up and get rid of our negative exponent.

02:17 That would bump the e to the 2b to the denominator.

02:22 And now what we would want to do to solve this graphically using a graphing calculator or maybe an online calculator like desmos is we would want to solve both equations for that easy common variable.

02:37 We would want to isolate a.

02:40 And that's going to be like the y value that we would be solving for.

02:48 So that's pretty easy.

02:50 If i want to isolate a, i would just multiply the first equation by e to the b.

02:56 And i would multiply the second equation on both sides by e to the 2b.

03:02 So that we clear that denominator.

03:04 And these are the two equations that we would want to be putting in to our graphing calculator or desmos calculator.

03:17 And i have done that here.

03:19 I'll show you a screenshot of what happens when we substitute each of these into, or plug each of them into our graphing calculator application...

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