MATH 240 Midterm: MATH 240 NIU intro

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turquoisegnu128

15 Feb 2019

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An introductory lecture on linear algebra by prof. john beachy. Linear algebra is the study of linear functions and operations. In calculus, functions of the form f (x) = mx + b are called linear. In this course, we focus on similar functions, with more variables, but with a minor change in terminology. The important part of a linear function is the multiplication, and in our new terminology, only functions of the form f (x) = mx will be called linear. We restrict our de nition to functions whose graph is a straight line through the origin. To show that the input is a single variable, and the output is a single variable, we write f : r r. If we allow two input variables, then we have a function from r2 to r. if the function is linear, then its graph should be a plane through the origin, with f (x1, x2) = m1x1 + m2x2.

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must understand functions to be able linearize. First let's review what graphs of certain functions looks like. Sketch the shape of each type of y vs. x function below. k is listed as a generic constant of proportionally. Linear y = kx Inverse y = k/x Inverse Square y = k/x^2 Power y = kx^2 You will notice that only the linear function is a straight line. We can easily find the slope of our line by measuring the rise and dividing it by the run of the graph or calculating it using two points. The value of the slope should equal the constant, k, from the equation. Finding k is a bit more challenging in the last three graphs because the slope isn't constant. This should make sense since your graphs aren't linear. So how do you calculate the constant, k? You need to transform the non-linear graph into a linear graph in order to calculate a constant slope. You can accomplish this by transforming one or both of the axes for the graph. The hardest part is figuring out which axes to change and how to change them. The easiest way to accomplish this task is to solve your equation for the constant. Note in the examples from the last page there is only one constant, but this process could be done for other equations with multiple constants. Instead of solving for a single constant, put all of the constants on one side of the equation. When you solve for the constant, the other side of the equation should be in fraction form. This fraction gives the rise and run of the linear graph. Whatever is in the numerator is the vertical axis and the denominator is the horizontal axis. If the equation is not in fraction form, you will need to inverse one or more of the variables to make a fraction. First let's solve each equation to figure out what we should graph. Then look below at the example and complete the last one, a sample AP question, on your own. State what should be graphed in order to produce a linear graph to solve for k. Inverse Graph Vertical Axis: _____ Horizontal Axis: _____ Inverse Square Graph Vertical Axis: _____ Horizontal Axis: _____ Power (Square) Graph Vertical Axis: _____ Horizontal Axis: _____

Student Homework on Composition of Functions Each horse you own eats 4 bales of hay per month. Each bale of hay costs $15.

Create a function called f whose input is the number of horses you own and whose output is the number of bales of hay eaten per month. Use x as your input variable. Create a function called g whose input is the number of bales of hay eaten per month and whose output is the total cost. Use x as your input variable.

Create a function called h whose input is the number of horses you own and whose output is the total cost. Use x as your input variable.

If you own 5 horses, use your functions f, g and h to find out how many bales of hay are eaten per month and how much money this will cost. Draw a diagram of the three functions f, g, and h. Show the input and output of each function, and their relationship.

All shirts are on sale at 25% off (to 75% of their original price) and are then subject to sales tax of 8.5%.

Write the function f, which pertains to the 25% off. The input is a price and the output is a sales price.

Write the function g, which pertains to the sales tax of 8.5%. The input is a price and the output is the price along with the sales tax.

Write the function h, which is the composition of these two functions. Draw a diagram, which shows the functions f, g and h.

The number of students enrolled at SFCC is the input to the function h, and the total tuition generated is the output. The president of SFCC has been working with this function to keep the budged balanced. But now he wants this function broken down. He knows that the average number of credits taken by an SFCC student each semester is 8 (use this to create function f whose input is the number of students and whose output is the total number of credits taken by these students). Tuition is $44.50 per credit hour (use this to create function g whose input is the number of credit hours at SFCC and whose output is the tuition created by these credit hours). Write out the functions f, g and h. Show their relationship with a diagram.

Thanks in advance.

MATH 240 Midterm: MATH 240 NIU intro

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