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)Use the same function, y = -2x 3 + 3x, to estimate the area under the curve in the 1 st quadrant.
Lesson 1.1 A Preview of Calculus
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
Copyright © Houghton Mifflin Company. All rights reserved. Secant Lines
Copyright © Houghton Mifflin Company. All rights reserved. Tangent Lines
Copyright © Houghton Mifflin Company. All rights reserved. Slope of a secant
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
The Area Problem: Find the area of a region bounded by curves Method: Use smaller and smaller rectangles to approximate the area.
Copyright © Houghton Mifflin Company. All rights reserved. Using Rectangles to Find the Area
Copyright © Houghton Mifflin Company. All rights reserved.
What is Calculus? Curves Area How do we get there?
Both problems will require a limiting process we will talk about later. Problem Set 1.1