Math 178, Spring 2019, Term Test I (Version 1)

Mathematical Modeling

Date: March 6, 2019

Time: 12:00-13:15

Section: 002

Instructor: Giang Le

Last Name:                                                                  

First Name:                                                                  

SJSU Student ID Number:                                                                  

Instructions

1. Fill out this cover page completely.

2. Answer questions in the space pro-vided, using scratch paper for rough work.

3. Show all the work required to obtain your answers.

1. Short Answer:

(a) Suppose we have a data set {(xi , yi), i = 1, . . . , n}. State the formula for the sum of squared errors of the data set relative to model predictions ˆyi , i = 1, . . . , n.

(b) Using the least squared criterion, what is the best fitting line y = kx to dataset {(xi , yi), i = 1, . . . , n}?(i.e what is the formula for k?)

(c) Write down the first five terms x0, ..., x4 of the following sequence:

(d) For the following difference equation, find an equilibrium value (fixed point) if one exists:

(e) For the following system of difference equations, find all fixed points

(f) A sum of $1000 is invested at 8% interest compounded quarterly. When does the investment double?

(g) A bank offers a savings account with an annual percentage yield (APY) of 2%. Assuming interest compounds monthly, determine the rate at which the principal compounds each month.

(h) A linear model y = ax + b is fitted to the dataset {(xi , yi), i = 1, . . . , n}. State the linear system of equations for the parameters a and b which minimizes the sum of squared errors.

2. Difference Equations

Formulate a dynamical system that models change exactly for each described situation (you do not need to solve the equation):

(a) You currently have $400 in a savings account that pays 0.4% interest each month. You add another $200 each month.

(b) You owe $1000 on a credit card that charges 2% interest each month. You pay $100 each month and you make no new charges.

(c) You left a bucket of radium at Grand Canyon park. Radium decays at the rate of 1% every 25 years. (Let rn be the amount of radium left after n of the 25-year periods, where r0 is the initial amount).

3. Student Loan

A recent graduate needs to pay off his student loan balance of $10, 000. The lender asks for interest of 0.6% compounded monthly with a payment of (at least) $150/month.

(a) Let xn denote the amount of principal owed on the student loan in month n. Formulate a difference equation model for the monthly change in principal owed. (Note: Do not forget about x0).

(b) Determine the explicit solution xn for the model.

(c) how long will it take to pay off the loan?

(d) Suppose the person gets a good job and can be able to pay $210 a month. How long does it take him to pay off the loan?

(e) (Extra credit, 2 points) What is the total amount of money he has to pay for the loan if he pays $150 a month?

4. Linear Regression:

Suppose you want to know if money makes people happy, you download the Better Life Index data from the OECD’s website as well as stats about GDP per capita from the IMF’s website. The following table shows some part of the data (for the year 2017) you can get:

(a) Fit a linear model y = ax + b to the data, where y is the life satisfaction index and x is GDP per capita.

(b) Compute the sum of squared errors for this model relative to the data set given.

(c) Using your model, predict the life satisfaction index for Canada, given the 2017 GDP per capita of Canada is 41.92 (x1000 USD)

5. Data Transformation

Consider the following data for the growth of yeast in a culture.

(a) Use a logarithmic transformation and the sum of squared error to fifind the best fitting line model of the form y = be^ax, where y is the observed yeast biomass and x is the time.

(b) Compute the sum of squared errors for the optimal model found in part (a).

(c) Predict the yeast mass at 7th hour.

THIS PAGE IS FOR ROUGH WORK