Precalculus, Semester B Unit Activity: Limits: Introduction to Calculus Alternate Assignment Student Name 0000 Student Number Fully document your algebraic work where applicable. Task 1: Computing the Limit of a Difference Quotient In this activity, you will compute the limit of a difference quotient to determine the rate of change ofa function at different points along its curve. PartA Consider the function g(x) = 2x* + 3x* - 5. Calculate the limit of the difference quotient at x, = 1 for g(x). PartB Find the equation of a line passing through the same point that g(x) passes through at x = 1 witha slope equal to the limit you found in part A. PartC Graph the function g(x) 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? continued 1 Precalculus, Semester B Unit Activity: Limits: Introduction to Calculus Alternate Assignment 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 = 144x x*, and approximate the area under the curve in the interval [0, 12] by dividing the area into the given numbers of rectangles. PartA Use four rectangles to approximate the area under the curve. PartB Use six rectangles to approximate the area under the curve. Part C Calculate the area under the curve using rectangles as their number becomes arbitrarily large (tends to infinity). o, nd(n+ 1) Hint: i = 2 3 Precalculus, Semester B Unit Activity: Limits: Introduction to Calculus Alternate Assignment Task 3: Finding the Area lInder a Curve Using Technology Use the method and setup outlined in your original Limits Unit Activity (unit 4), but evaluate the interval [2, 20] under the curve of y = ;11 x + 15 with six rectangles. Upload and attach your supporting spreadsheet file (.xIs or similar) to your submission for evaluation, then complete the following statement. With technology, I have estimated this area to be units Given the shape of the graph and position of the end points, | expect this estimate to be the actual area. continued 3