Warm-up 1-1

1) Find the equation of a line that is tangent to the equation y = -2x3 + 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 = -2x3 + 3x, to estimate the area under the curve in the 1st quadrant.

Lesson 1.1A 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

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Secant Lines

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Tangent Lines

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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

12

12

xx

yy

cxc

cfxcf

)()(

x

cfxcf

)()(

The Area Problem:

Find the area of a region bounded by curves

Method:

Use smaller and smaller rectangles to approximate

the area.

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Using Rectangles to Find the Area

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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