Unit Activity Unit: Limits: Introduction to Calculus This activity will help you meet these educational goals: Mathematical PracticesYou will make sense of problems and solve them, use mathematics to model real-world situations, use appropriate tools strategically, and attend to precision. Introduction In this unit, you were introduced to the concept of limits and learned how to evaluate limits using different methods, such as building a table of values, graphing the function, and using algebra. You also explored how limits can be used to find the slope of a tangent to a function and the instantaneous rate of change of a function at a given point. In addition, you learned how to approximate the area under a curve by using limits. This is done using rectangles whose widths decreased as the number of rectangles increased to give successively better approximations of the actual area. In this activity, you will apply these concepts to enhance your understanding of how you can use limits to find the instantaneous velocity of an object when you know the distance function of the object. You'll also use limits to find the area under a curve. The three tasks in this activity include: computing the limit of a difference quotient finding the area under a curve finding the area under a curve using technology __________________________________________________________________________ Directions and Analysis Task 1: Computing the Limit of a Difference Quotient In this activity, you will make compute the limit of a difference quotient to determine the rate of change of a function at different points along its curve. a. Consider the function difference quotient at . Calculate the limit of the for . Type your response here: 1 2014 EDMENTUM, INC. b. Find the equation of line passing through the same point that passes through at with a slope equal to the limit you found in part a. Type your response here: c. Graph the function and the line you identified in part b. How do the graphs relate to each other? What does the slope of the line represent about the function at the given point? Type your response here: Task 2: Finding the Area Under a Curve In this task, you will practice finding the area under a nonlinear function by using rectangles. You will use graphing skills in addition to the knowledge gathered in this unit. Sketch the graph of the function y = 20x x2, and approximate the area under the curve in the interval [0, 20] by dividing the area into the given numbers of rectangles. a. Use five rectangles to approximate the area under the curve. Type your response here: b. Use 10 rectangles to approximate the area under the curve. Type your response here: c. Calculate the area under the curve using rectangles as their number becomes arbitrarily large (tends to infinity). You do not need to sketch the rectangles. Type your response here: Task 3: Finding the Area Under a Curve Using Technology As an apprentice working for 1 plus 1 Landscaping, you're learning the tricks of the trade. Your boss wants you to calculate the area between the edge of a garden bed and the side of a house. Using the area, she can calculate the amount of mulch she will need for this job. The edge is defined by the equation y = x + 1. 1. You will use a spreadsheet program to estimate the area under the curve y = x + 1 between x = 0 and x = 10. Let the number of subintervals be n = 5. Follow these steps to calculate the area using right endpoints: Start a blank spreadsheet in Excel, Google Docs, or another spreadsheet program. 2 2. Define a field in the spreadsheet where you're going to put the values for a, b, n, and delta x (x). It's a good practice to label these cells. You can start with something like this: 3. Refer to the problem description to fill in the values for a, b, and n by placing the numbers in cells B1, B2, and B3, respectively. 4. Create a formula that will calculate x in cell B4. 5. Label the cells D1 and E1 x sub i and f(x sub i), respectively: 6. Starting at cell D2, fill in the correct values for xi by using the right endpoints. (Note: If you'd like to reference the cell that contains x, simply type a dollar sign before the column letter and another dollar sign before the row number, like this: $B$4. This will allow you to copy and paste the formula to other cells, but the formula will always use, for example, cell B4.) 7. Create a formula that will calculate f(xi) in cell E2. (Note: If you'd like to reference the cell that contains x, simply type dollar signs before the column letter and the row number, like this: $B$4. This will allow you to copy and paste the formula to other cells, but the formula will always use, for example, cell B4.) 8. Copy the formula to calculate the rest of the values for f(xi). 9. Label cell F1 Area of rectangles. 10.In cell F2, create a formula that will calculate the area of a single strip or rectangle. Copy the formula to other cells to find the area of each rectangle. 11.Add the areas found in the previous step to get your estimated area in cell F7 by using endpoints from the right. Type your response here: 3 __________________________________________________________________________ 4 Evaluation Your teacher will use this rubric to evaluate the completeness of your work as well as the clarity of thinking you exhibit. Total Points: 10 Task 1: Computing the Limit of a Difference Quotient a. Find the limit of the difference quotient. Task points: 4 1.5 b. Find the equation of the line through the point. 1 c. Describe the relationship between the function and the line. 1.5 Task 2: Finding the Area Under a Curve a. Approximate the area under a curve using five rectangles. Task points: 3 1 b. Approximate the area under a curve using 10 rectangles. 1 c. Find the area under a curve using an infinite number of rectangles. 1 Task 3: Finding the Area Under a Curve Using Technology a. Calculate the width of each rectangle. Task points: 3 0.5 b. Find the right endpoint of each subinterval. 0.5 c. Calculate the height of each rectangle. 0.5 d. Calculate the area of each rectangle. 0.5 e. Find the sum of the areas of the rectangles. 1 5