Could you check all of these questions for me as soon as possible, see if they are correct and I need help on the question 9 and the last page of question 4D. Thank you
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8. As another check, perform a quadratic regression on your data which will allow your calculator to find the "best-fitting" quadratic function (in general form) through the set of data. (This can be done by pressing the STAT key, go to CALC, and select 5:QuadReg.) Again, round all values to the nearest hundredth and record this equation below. Write your equation in terms of the variable t, not x. y ( t ) = _ 6 72 x 2 - 2.52 x + 3.7] Compare these values of a, b, and c to the values found in part #6 above. (They should be close!) Clear the functions in Y1 and Y2, and enter this function in Y1 of your calculator. For the remainder of this problem, use the regression equation ( now in Y1 ) found in part #8 above. (Note: Do not use the values in the chart. You may want to turn off the scatterplot.) 9. Find the average rate of change (average velocity) of the ball over the first two seconds. (Remember, you are using the regression equation, not the data from the chart. And also remember that there is an easy way of doing this with your calculator because your function is stored in Y1!) Average velocity over the first two seconds is approximately ft/sec. How does this answer compare to your answer to question #la on the first page? 10. We want to find the instantaneous rate of change (instantaneous velocity) of the ball at the time t = 2.0. From our discussion in class, we know that we can estimate this value by calculating the average rate of change (average velocity) of the position of the ball over a "small time interval" that includes t = 2.0. Estimate the instantaneous velocity of the ball at t = 2.0 by calculating tha average velocity from t = 2.0 to t = 2.001. Round your answer to 2 decimal places. Instantaneous velocity at t = 2.0, y' ( 2), is Q . 73 1 1. By editing the expression that we used in #10 above we can easily calculate the instantaneous velocity of the ball at any time t. Complete the table below that shows the velocity (instantaneous rate of change) of the ball at the times indicated. (Round answers to two decimal places.) Time (sec) 0.4 1.2 2.0 2.8 Velocity (ft/sec) 2. 32 0.96 0.78 1.78 12. Create a scatterplot of this data. (Time in LI and Velocity in L2. You can Delete the "time" and "distance" data that are in L1 and L2)13. Since the scatterplot of the velocity vs. time data appears to be linear, choose two points from the chart above and find a linear function that models the velocity of the ball at any time t. Show your work below, and write the function that models the data in the form v(t) = m . t + b . (Round the values of m and b to two decimal places.) v (t ) = - 1.7+ +3 On your calculator, sketch the graph of your v(t) function. Does it match the graph of your "derivative function" in part #5 of this worksheet? _ Yes 4. Use your velocity function v(t) from #13 above and answer the following. a. What was the velocity of the ball when we began collecting data? 3 b.. Where can we find this value in the position/distance function? The y - intercept c. Use the velocity function to determine the time that the ball reached it's maximum height? (Hint: Your graphs from problem #5 might help you determine how to answer this question.) t = 1.77 d. Where did we see this value before?Calculus Application Problem Name The Ball Goes Up, the Ball Comes Down Due date: Introduction: The objective of this activity is to collect data from pushing a ball up a ramp and, with a motion detector placed at the top of the ramp, record the distance vs. time data of the ball as it moves up and then back down the ramp. We will then analyze the data by finding a model for the distance vs. time data, and a model for the velocity vs. time data. Complete the chart below from the data that was collected from the experiment. Time (sec) 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 Distance (feet) 3.77 2.87 2.21 1.78 1.58 1.60 1.86 / 2.35 3.07 1. Using the chart above, a. Calculate the average velocity of the ball over the first two seconds. Show the expression you are using, and include units in your answer. Average velocity =_ b. Estimate the instantaneous velocity of the ball at 3.2 seconds. Show the expression you are using and include units in your answer. Instantaneous velocity = 16 2. Create a scatterplot of the data on your calculator (Time in LI and Distance in L2). Note: Instructions for creating a scatterplot are located on D2L in the Miscellaneous Worksheets section.) 3. We want to find a function, y(t) , that will model the position/distance data in the chart. From the scatterplot it appears that the data is "parabolic" and, therefore, a quadratic function should be used. So we will attempt to fit our data with a quadratic function in its vertex form: y (t) = a(t - h) +k Recall, from algebra, that the values of h and k represent the vertex ( h , k ) of the parabola. By tracing on the scatterplot, you should be able to obtain values for h and k. (Note: You will probably need to "approximate" the values of h and k because the vertex might be located between two data points, not exactly on a data point.) It appears that h = . 8 and k =_ 1.6 The values of h and k mean something with respect to the activity! Explain what the values of h and k represent with respect to the data collected. Be specific! h represents The time takes for the ball to reach the minimum actual prom the top of the k represents The minimun distance to the top of the ramp in feet. in secon4. We can find the value of a for our equation using algebra. Select a point (that is not close to the vertex) from the chart, say (2.4 , 1.86 ). Substitute this point for t and y into the quadratic equation, along with the values of h and k. The only unknown is a, so solve the equation for a. Round a to the nearest hundredth. Show your work below . a = _ 0.72 Record your final function. y(t) = 0 72 (x-).8) +1.6 Enter this equation into Y1 of your calculator and graph it to see how it fits the data. If it doesn't fit well, find your mistake! (Hint: If the data does not fit well, you may need to try a different vertex and recalculate the a value.) 5. Sketch a graph of the distance vs. time function below that you found in part #4. Then, beside it, sketch a graph of the derivative of the function, which will show a graph of the velocity vs. time of the ball as it goes up, then down the ramp, since a velocity function is the derivative of a distance (position) function. Distance Velocity 1 2 3 2 3 (Note: We will come back to the velocity graph later.) 6. It is also possible to express any quadratic function in the general form: y (t ) = at+ bt + c To determine the values of b and c (since a is identical to the a value previously found), expand your equation in part #4 and collect like terms. Round all values to the nearest hundredth. Show this work below. y= 0.72 ( X - 1. 8) 2 +1. 6 Y = 0.72x2 - 2:59 x + 2:33 + 1. 6 * = 0 . 72 x2 - 2: 59 x + 3.93 y ( t ) = 0 . 72 x 2 - 2.59 x +3. 93 7. To check your work, enter this general form of the quadratic function into Y2 of your calculator, and see if it fits the data as well as your function in Y1. If not, find your mistake
