1. Express the repeating decimal 0.5757... as the ratio of two integers without your calculator. Problem of the Day

2. Express the repeating decimal 0.5757... as the ratio of two integers without your calculator. Problem of the Day x = 0.57 100x = 57.57 99x = 57 x = 57/99

3. What is Calculus? Calculus is the mathematics of change - velocities and accelerations. Calculus is the mathematics of - tangent lines, slopes, areas, volumes, arc lengths, centroids, curvatures, and other concepts that enable scientists, engineers, and economists to model real-life situations

4. Precalculus Calculus constant velocity slope of a line tangent line to a circle area of a rectangle velocity of an accelerating object slope of a curve tangent line to a curve area under a curve

5. Precalculus Limit Process Calculus What is Calculus? Calculus is a limit machine (for more examples see page 43 in textbook)

7. Without Calculus With Calculus

9. Without Calculus With Calculus

10. The tangent line problem A Classic Calculus Problem http://illuminations.nctm.org/ActivityDetail.aspx?ID=221

11. As you observed in the previous demonstration, as the secant lines approach the tangent line a "limiting position" exists. The slope of the tanget line is said to be the limit of the slope of the secant line as x decreases without bounds. x = (c, f(c)) (c + x, f(c + x)) y = f(c) - f(c + x) c - (c + x) m=

12. x = (c, f(c)) (c + x, f(c + x)) y = f(c) - f(c + x) c - (c + x) m = f(c + x) - f(c) x We can talk about the slope of the tangent line as Δx approaches 0.

13. A Classic Calculus Problem Time acceleration The area under the curve is the velocity of the object. The area under the curve is the limit of the sum of the areas of the rectangles as the number of rectangles increases without bounds.

14. Page 47 Problems 1, 3, 4, 5, 6, 7 Begin Lab 2