1. Warm-up 1-1 1)Find the equation of a line that is tangent to the equation y = -2x 3 + 3x and passes through (1, 1) Select (1, 1) and another point on the curve to estimate the tangent. Use (1, 1) and the new point to calculate a slope. Use either point and the slope to find the equation y = mx + b Y = -6x + 7 HINT 1 HINT 2 HINT 3 ANSWER

2. 2)Use the same function, y = -2x 3 + 3x, to estimate the area under the curve in the 1 st quadrant.

3. Lesson 1.1 A Preview of Calculus

4. There are two fundamental questions that underlie the study of calculus 1) The tangent line problem 2) The area problem The Tangent Line Problem: Find the equation of a line along a curve at a certain point Method: Find the slope of secant lines at smaller and smaller intervals away from the tangent point

5. Copyright © Houghton Mifflin Company. All rights reserved. Secant Lines

6. Copyright © Houghton Mifflin Company. All rights reserved. Tangent Lines

7. Copyright © Houghton Mifflin Company. All rights reserved. Slope of a secant

8. Slope of any Secant: Start with slope Change to function notation Simplify Remember f(x) is another way to write “y” Δx means “change in x” – think of it as the difference from one value to another

9. The Area Problem: Find the area of a region bounded by curves Method: Use smaller and smaller rectangles to approximate the area.

10. Copyright © Houghton Mifflin Company. All rights reserved. Using Rectangles to Find the Area

11. Copyright © Houghton Mifflin Company. All rights reserved.

12. What is Calculus? Curves Area How do we get there?

13. Both problems will require a limiting process we will talk about later. Problem Set 1.1