Prof. Bodrero s Guide to Derivatives of Trig Functions (Sec. 3.5) Name: Objectives: Understand how the derivatives of the six basic trig functions are found. Be able to find the derivative for each of the six basic trig functions (in conjunction with other derivative rules). Below is the graph of the sine function. Select at least 8 different points on the curve. At each of these points, estimate the slope of the curve (put estimates in table below). x slope Next, graph the points created by the table ( x, slope at x ) on the blank graph below. What does this suggest about the derivative of y = sin(x)? Recall the trig identity: sin( A + B ) = Use this trig identity and limit definition of the derivative to prove your conjecture about d (sin(x) ). dx Hint: lim h 0 cos(h) 1 h = 0.
Below is the graph of the cosine function. Select at least 8 different points on the curve. At each of these points, estimate the slope of the curve (put estimates in table below). x slope Next, graph the points created by the table ( x, slope at x ) on the blank graph below. What does this suggest about the derivative of y = cos(x)? Using only the derivatives of sine and cosine (and time-saving rules), how would you find the derivative of the four other basic trig functions? Do it! Compare your results with your groupmates. d d d (tan(x)) (cot(x)) (sec(x)) d (csc(x)) dx dx dx dx
Find the following derivatives: d dx (x) (sin ) d e x dx ((cos2 (x))) d 2 dx 2 (sec(x)) (2nd deriv.) Note: I bring in a spring-mass system and do some whole-class discussion with this next part. Many real-life phenomena exhibit simple harmonic motion (follow a wave pattern). Waves, an oscillating mass hanging form a spring, etc. are common examples. A mass on a spring is pulled 4 cm below its equilibrium (resting) point and released. The position as a function of time is s(t) =. Find the velocity function for the mass on a spring and graph it to the right. v(t) = How is the velocity of the object related to the position of the object at a given time? Why does this make sense physically? Find the acceleration function for the mass on a spring and graph it to the right. a(t) = How is the acceleration of the object related to the position of the object at a given time? How is the acceleration of the object related to the velocity of the object at a given time? acceleration (cm/s 2 ) velocity (cm/s) position (cm)
Note: this next problem is to tie in to the chain rule, to be learned next class. Below is the graph of y = sin( x 2 ). Estimate slopes of the function to graph the derivative, y, below. x slope
Math 2040 Homework for Sec. 6.4: Sampling Distributions Name: 1. One day on an assembly line, three items are produced. Call the items A, B, and D. Items A & B are good but item D is defective. (Success = defective). Compute the population proportion of defective items: p =. Next complete the table of all samples of size two (with replacement) and list their corresponding probabilities. For each sample, find the proportion of defective items in the sample. Then use this as the probability distribution to find the mean of all the sample proportions (of defective). Compare this mean of sample proportions to the population proportion above. What do you notice? Samples (size 2) Sample proportion Sample probability 2. On our first week survey, students reported the number of college credits they have already earned. This is a simple random sample of four students from our class: {4, 22, 43, 44}. Use these four values as the population (so our table isn t incredibly long). First, compute the mean, standard deviation, and variance for this population data set. =, =, 2 = Next, create a table of all samples of size TWO (with replacement) and list their corresponding probabilities. Then, for each sample of size two, compute the sample mean, sample median, sample standard deviation, sample variance, and sample probability. Next, for the sample means and their probabilities, find the average of the sampling distribution. Finally, do the same for the sample medians, standard deviations, and then variances. Samples (size 2) Sample Mean Sample Std dev Sample variance Sample probability How does average of sample means compare to the population mean,? How does average of sample std devs compare to the population std dev,? How does average of sample variances compare to the population variance, 2?
3. Ariel is going door to door selling Girl Scout cookies. One day, only five houses purchased cookies. The numbers of boxes of cookies purchased at each of the five houses are 2, 3, 3, 5, & 6. (Define a success, S, as they make a purchase of at least 5 boxes of cookies, F = purchase less than 5 boxes). Use these values to find population mean, = ; range = ; and population proportion who bought at least 5 boxes p =. Complete the table of all possible samples of size TWO (with replacement) of the # boxes cookies bought. For each sample, find the sample mean, sample range, sample proportion (5+ boxes), and sample probability. Samples Sample Sample Sample Sample Samples Sample Sample Sample Sample size 2 mean range proportion probability size 2 mean range proportion probability 2, 2 2 0 0 2, 3 2.5 1 0 2, 3 2.5 1 0 2, 5 3.5 3 0.5 Now compare the average of each of the sampling distribution measures to the corresponding population value. Mean Range Proportion 4. We say that an estimator (statistic) is unbiased if the average from the sampling distribution matches the true population parameter. If the average from the sampling distribution does NOT match the population parameter, we say that the estimator is biased. Which of the measures (mean, standard deviation, variance, range, proportion) are unbiased? Which are biased?
Prof. Bodrero s Guide to Properties of Linear Transformations from R n to R m (Sec. 4.3) Names: Objectives: Connect understanding of one-to-one for functions to one-to-one for transformations. Determine if an operator (transformation) is oneto-one. Find the inverse operator for one-to-one transformations. Be able to determine if a transformation is linear. Determine matrix for a transformation using elementary vectors. Use geometric properties to find eigenvalue-eigenvector pairs for basic linear operators. In College Algebra/Calculus how do we tell if a function is one-to-one? What does it mean to be one-to-one? Is the function y = f(x) = x 3 1 one-to-one? Explain your answer. Is the function y = g(x) = x 3 x one-to-one? Explain your answer. Why did we want to determine if a function was one-to-one? What special thing happens when a function is one-to-one? A transformation from R n to R n (the same size space for input and output) is called an operator. In section 4.2, we studied several basic operators: reflection, projection, rotation, dilation, etc. Let s look a little more closely at a few of them. For each operation, find the domain, range, draw a picture for action on u = (a, b), and find the matrix that does that transformation. Reflect over the y-axis in R 2 Rotation (CCW) by angle in R 2 Orth. projection onto x-y plane R 3 Dilation by factor k ( 0) in R 3 Domain: Domain: Domain: Domain: Codomain: Codomain: Codomain: Codomain: Range: Range: Range: Range: Which of the transformations above are one-to-one? Explain how you got your answers. Compare your results with your neighbors.
Because each of the (linear) transformations above is an operator, (R 2 to R 2 or R 3 to R 3 ) the corresponding matrix is. We can find the determinant of square matrices. Find the determinant of each matrix above. What do you notice? Is the transformation w 1 = 3x 1 + 4x 2 one-to-one? Explain. Is the transformation w 3 1 = x 1 + 0x 2 w 2 = x 1 + 2x 2 w 2 = 0x 1 + (x 2 ) 2 one-to-one? Explain. For each transformation above that is one-to-one, find the matrix that undoes the transformation (the inverse transformation). What do you notice? What if the transformation is not an operator (domain and codomain are not the same size space)? How do we determine if it is one-to-one? Which of the following transformations are one-to-one? If one-to-one, prove it. If it is not one-to-one, give a counterexample. T(x, y, z) = (x, x + y + z) T(x, y) = (x 2, x + y, x y) T(x, y) = (x 3, 2x + y, x y)
Bodrero s Math 2270 Section 4.3: Properties of Linear Transformations from R n to R m (continued) Another property of transformations (independent of one-to-one-ness) is whether or not they are linear. If T is a transformation T: R n R m, what TWO things do we need to check to see if it is a linear transformation? Is w 1 = 3x 1 + 4x 2 a linear transformation? Explain. Is the transformation w 3 1 = x 1 + 0x 2 w 2 = x 1 + 2x 2 w 2 = 0x 1 + (x 2 ) 2 linear? Explain. Create a linear transformation from R 3 R 3 and write it down but don t tell Prof. Bodrero. He will ask you a few questions and then see if he can come up with your linear transformation. (How does he do it?) In chapter 2, we learned about eigenvalues and eigenvectors. What was the special equation (that should be seared into your mind) for finding them? Since linear operators behave like (square) matrices, they have the potential to have eigenvalues & eigenvectors. Without going through all the work of chapter 2, let s use the geometry of the transformations to find the eigenvalue and eigenvectors (if they exist) of the linear operators from page 1. Reflect over the y-axis in R 2 Rotation (CCW) by angle in R 2 Orth. projection onto x-y plane R 3 Dilation by factor k ( 0) in R 3
Mathematics 2040 Statistics Chapter 9 Activity 1 (Sec. 9.1 & 9.2) SCATTER DIAGRAMS & LINEAR CORRELATION Names: Objectives: Understand meaning of correlation and estimate correlation coefficient for a scatterplot. Understand how r is calculated from a data set. Know that correlation is not causation. Use technology to compute the regression equation for a data set and use it to make predictions. Understand interpolation vs extrapolation. Understand the meaning of the coefficient of determination and interpret it for a given context. A correlation is an association between two variables (typically with different units). In the next few sections, we are going to study linear correlation. For positive linear correlation, as one variable increases, the other variable tends to (at a corresponding fixed rate). For negative linear correlation, as one variable increases, the other variable tends to. There are other correlations (quadratic, exponential, logarithmic, ) but we won t study them here. For the figures below, determine the type of correlation between x and y. (Hint: think slope of a line.) r r r When dealing with scatter diagrams, we often think that one variable is more likely to depend on the other. We call the dependent variable, y, the and the independent variable, x, the We now introduce one more descriptor called the Sample Correlation Coefficient, r. r is a number from 1 to 1 that tells us how well the given (sample) data fit a line. If the data is approximately linear and goes up from left to right, r is positive (think positive slope). This means that if one variable increases so does the other one. If the data is fairly linear but goes down from left to right, r is negative (think negative slope). This means that as one variable increases the other decreases. If r = 1, we have a perfect linear fit for a positive correlation. An r value of 0.9 is considered a very high (but not perfect) positive linear correlation. If r = 1, we have a perfect linear fit for a negative correlation. An r value around 0.9 is considered a very good (but not perfect) negative linear correlation. If r is close to 0, then we have no linear correlation. Estimate the r value in each of the first three graphs above. (Fill in the blanks next to r above).
As a group, come up with a real life example of two variables that have a positive linear correlation. Write it here. Note: I typically have each group share an example and have other groups determine if it is positive or negative linear correlation. Think of a real life example of two variables that have a negative linear correlation. Write it here. Let s review some notation in preparation for the formula to calculate r. What does x 2 mean? What does x mean? What does x 2 mean? What does ( x ) 2 mean? For suitable data points, (x, y), the value of r is calculated by the formula r = n xy ( x)( y) n x 2 ( x) 2 n y 2 ( y) 2 Let s look at an example to find out more about correlation. Example # 1. Is the ACT a good predictor of college success? Prof. Bodrero looked up the ACT and Snow GPA for a random sample of 8 of his statistics students (have completed at least 1 semester). ACT score (composite) 24 17 20 15 24 18 27 23 Snow College GPA 3.71 3.56 3.31 3.04 3.82 3.38 4.00 3.26 Based on the statement of the problem, ACT score is the variable and Snow GPA is the variable. First, let s make a scatter plot of the data points to the right. Decide on a horizontal scale and vertical scale. Mark them on the axes. Then plot each corresponding point, remember -ing to label your axes. Look at your plot. Does the data appear to have a linear correlation? If so, it is positive or negative?
For our given data, find x = ; x 2 = ; ( x ) 2 =. Then find y = ; y 2 = ; ( y ) 2 = ; and xy =. (Hint: use your calculator!) Use these values and the formula to compute the value of the sample correlation coefficient, r. Write out the formula and calculate: Just like s is the sample standard deviation and is the population standard deviation, r is the sample correlation coefficient. If Prof. Bodrero had a lot of free time, he could look up every student s ACT and GPA and get the population correlation coefficient, (called rho, pronounced row ). However, since he doesn t have that much time, he will need to do a random sample and go from there (like we did in chapters 7 & 8). Look at the sample data. For what range of ACT scores do you think that a linear model would be applicable? (Hint: look at our computed r value.) Are there any ACT scores that this model wouldn t be applicable for? If so, which? Be careful with your conclusions. Just because there is a correlation between two variables does NOT mean that one causes the other. Correlation is not the same as causation! There may be a factor that influences both variables. This is sometimes referred to as a lurking variable. Also, there may be a relationship between the two variables but it may not be linear (see p. 1 image 4). As a group, come up with a real-world example of two variables being correlated but not have a cause-and-effect relationship. Is there a lurking variable? If so, what is it? Section 9.2: Linear Regression and the Coefficient of Determination When we have a situation where the scatterplot looks (somewhat) like a line, we want to find the line that best fits the data (called the regression line). But what do we mean by best fit? The most accepted answer is what we call the least-squares criterion. What does the least-squares criterion say? Go back to the scatterplot of ACT vs GPA (page 2) and lightly sketch in what you think the leastsquares line should look like.
Remember back to your prior math classes. What was the equation of a line? What does each letter represent? Putting things together, and giving it a statistics spin, we are looking for the line y = a + bx that best fits the data. Fortunately, there are lots of smart people who have already figured out how to find a and b in this formula based on sample data. Write out the formulas for computing b and a (p. 522). b = a = Use our ACT and GPA data to find the precise equation of the least squares line, y = a + bx. What are the meanings of a and b? Discuss with your group and summarize your thoughts below. What is marginal change? [try to put it in your own words so it sticks in your brain longer.] What is the value of the marginal change of the ACT vs GPA problem? How do we interpret this for our problem? Note: Because they are used so often, the calculator has a program to compute a, b, and r. Put the x- data in List 1, the y-data in List 2. Then go to STATS CALC, and choose option 8: LinReg(a + bx) then L1, L2. Hint Enter and write what you see below. Note: if you do NOT see r and r 2, do the following: 2 nd 0 (to get to catalog) and then scroll down to DiagnosticOn. Hit enter to put this on the home screen and then press enter again to run it. Now, rerun the LinReg(a+bx) command. The most useful thing about a least-squares line (regression line) is the ability to make predictions. Use the equation found above to predict the Then use the equation to predict the GPA of a GPA for someone with a composite ACT student with an ACT score of 31. score of 22.
Which prediction do you think is more accurate, the GPA for ACT of 22 or 31? Why? What is the difference between interpolation and extrapolation? In statistical correlation, what does residual mean? Find the residual for the data point (23, 3.26). Find the residual for the data point (24, 3.71). The last thing to discuss in this section is the coefficient of determination. The symbol for coefficient of determination is. What is most important is the interpretation of the coefficient of determination. In general, what does the coefficient of determination tell us? (Hint: p. 531) What is the coefficient of determination for the ACT vs GPA problem? Interpret the coefficient of determination in the context of ACT vs GPA. Could there be a lurking variable in the ACT vs GPA problem? If so, what might it be? Example # 2: Prof. Bodrero is trying to determine the effectiveness of the pre-reading quizzes at predicting final grade (percentage of total points earned) in his Math 1040 classes. He randomly selected 10 students and their pre-reading score (out of 100%) and their overall grade in the class (out of 100%) are given below. Pre-Read Quiz 80.8 69.5 84.4 81.8 58.5 58.1 88.1 72.8 60 85.1 Overall Grade 84.8 80.5 91.3 83 63.1 82.1 94.5 86.3 66.9 86.2 Showing work on the next page, do the following: a). Determine which variable is x and which is y and then make a scatterplot of the data. b). Does the scatterplot appear to have linear correlation? If so describe it. (ex: positive linear ) c). Find the regression equation and then graph it on the scatter plot of the data. d). Use the regression equation to predict the overall grade of a student with a quiz average of 80. e). Would it be good to use our model to predict a grade for a student whose quiz average is 40? Why? f). Find and interpret the marginal change (in y as x changes by one unit). g). Find and interpret the coefficient of determination for this problem.
Mathematics 1040 Statistics Chapter 3 Activity 1 (Sec. 3.2) MEASURES OF CENTER Names: Objectives: Be able to compute and understand the meaning of each of the mean, median, mode, and midrange for a data set. Be able to compute the mean of a grouped data set. When we have a (large) data set, we often need to describe or summarize it. One way important characteristic of a data set is its center or middle. However there are several different ways to compute / talk about this center or middle, each with its own strengths and weaknesses. This activity will discuss the most common measures of center and ask you to apply what you learned. Let s first review two important terms: Population: Sample: Mean The most commonly used measure of center is the mean (technically the arithmetic mean but often referred to as the average ). How do you find the mean? Write out the steps in the space below: The symbol for population mean is ( mu ) and for sample mean it is x ( x-bar ). Using the correct symbol, find the mean for the following data set, (a random sample of reported amount students paid for their stats book this semester): {35, 40, 60, 15, 15}. Your answer should be = # OR x = #. The next randomly selected student paid $50 for the text. Find the mean for the new data set {35, 40, 60, 15, 15, 50}. How did the mean change? Why did it change that way? The next randomly selected student paid $130 for his/her book. What is the mean of this new dataset: {35, 40, 60, 15, 15, 50, 130} How did the mean change? Why did it change that way? What does the mean signify? In other words, what does the mean tell us? In general, what are some advantages of the mean? Disadvantage(s) of the mean?
Median Another commonly used measure of center is the median. How do you find the median for a data set? Write out the steps: Find the median for each of the data sets. {35, 40, 60, 15, 15} {35, 40, 60, 15, 15, 50} {35, 40, 60, 15, 15, 50, 130} Compared to the mean, did the median change more or less as we added additional values? What does the median signify? In other words, what does the median tell us? In general, what are some advantages of the median? Disadvantage(s) of the median? Mode The next measure of center we ll study is the mode. How do you find the mode for a data set? Write out the steps: Find the mode for each of the data sets. {35, 40, 60, 15, 15} {35, 40, 60, 15, 15, 50} {35, 40, 60, 15, 15, 50, 130} Find the mode for the following data sets: {35, 40, 60, 15} {35, 40, 60, 15, 15, 60} What does the mode signify? In other words, what does the mode tell us? In general, what are some advantages of the mode? Disadvantage(s) of the mode?
Midrange The last measure of center that we will discuss (briefly) is the midrange (although it is rarely used). How do you find the midrange for a data set? Write out the steps: Find the midrange for each of the data sets. {35, 40, 60, 15, 15} {35, 40, 60, 15, 15, 50} {35, 40, 60, 15, 15, 50, 130} What does the midrange signify? In other words, what does the midrange tell us? In general, what is the advantage of the midrange? Disadvantages of the midrange? In real life, we generally have data sets that are larger than 5 or 7 points. The calculator can help us find some of these measures of middle. Below is a random sample of 25 cost of stats textbooks (rounded to the nearest dollar) reported from Prof. Bodrero s students this semester. 50, 6, 20, 82, 30, 30, 5, 30, 0, 8, 20, 12, 17, 50, 40, 40, 0, 5, 6, 15, 82, 130, 20, 20, 37 Enter this data into your calculator by pushing the STAT button and choosing the EDIT option. Go to List 1 (L1) and enter each of the values. [If there is already data in L1, clear it out by using arrows to go up to the L1 icon and pushing the CLEAR button, not the DEL button.] Then push the colored 2 nd button and the MODE button next to it to QUIT out to the home screen. Next, push the STAT button and move over to the CALC option. Run option 1: 1-Var Stats and tell it to use List 1 (2 nd button then 1 key is a shortcut for L1 ). Write what you see below. Then find each of the four measures of middle (mean, median, mode, midrange) for this sample of data. Mean of Grouped Data Sometimes we have data reported in a frequency distribution and we need to find the mean. Even if the data is grouped into classes, (so we don t know each original data value) we can still approximate the mean to a fairly high degree of accuracy. Let s look at an important topic: US unemployment. Unemployment rates for the 50 states and DC. (Source US Bureau of Labor Statistics, 2015) Rate (%) 2.0 2.9 3.0 3.9 4.0 4.9 5.0 5.9 6.0 6.9 7.0 7.9 Frequency 1 7 14 17 11 1 Class midpt. (Note: North Dakota = 2.7%, Utah = 3.6%, California = 6.2%, Nevada = 6.8%, W. Virginia = 7.5%)
To compute the mean on data that has already been grouped, we assume that each value is at the midpoint of its class. (This is a pretty safe assumption as the higher values and the lower values within each category tend to average out to the class midpoint). Fill in the class midpoints in the table above. Next we do the average on all 51 data points. We could do 2.45 + 3.45 + 3.45 + 3.45 + 3.45 + and then divide by 51. But since many of them are repeated, we can save a lot of time another way. Take the first class midpoint, 2.45, and multiply by its frequency, 1, to get 2.45. Do the same for each of the classes and add them all up. Then divide by 51 (not 6) to get the mean. This is shown by the formula x = (f x) where x represents each class midpoint and f is the corresponding frequency. Use this f method to compute the mean unemployment rate for states (round to 2 decimal places). How does your computed value compare with the mean (using every actual data value) of 5.13? The unemployment rate in the US (as a whole) is 5.3%. Why are the numbers computed above off? The mean of a frequency distribution can be calculated with your TI-83/84. First, go to the STATS menu and EDIT. Put the class midpoints in the first list, L1 and the corresponding frequencies in L2. Then QUIT and do STAT then CALC and 1:1-Var Stats L1, L2. Write what you see below. Skewness A distribution of data is skewed if it is not symmetric and extends more to one side than the other. Left skewed (negatively skewed) Symmetric Right skewed (positively skewed) Which shape does the unemployment data most closely resemble, left skewed, symmetric, or right skewed?