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Section 1.1

Two Classic Calculus Problems

In this section, we will discuss the following topics: The tangent line problem The area problem

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Calculus is the field of mathematics that deals with rate of change, tangent lines, slopes, areas, volumes, …

It rests upon the concept of limit.

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Precalculus mathematics also deals with these concepts, but the major difference is that precalculus mathematics is more static, while calculus is more dynamic.

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Without Calculus With Calculus

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The concept of limit is fundamental to the study of calculus. To give you an idea of how the limit is used, we will look at two classic calculus problems.

The first problem comes from differential calculus, which deals with the rate of change of a function. Its applications include velocity, acceleration, and maxima and minima of functions.

The second problem comes from integral calculus, which deals with the idea of area. Its applications include length, area, and volume of geometric figures.

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In this problem, we are given a function f and a point P on its graph and asked to find the equation of the tangent line to the graph at point P.

The tangent line to a curve at point P (informal def):

The Tangent Line Problem

f

x

y

P

To find the equation of the tangent line at P, we need to find the slope of the tangent line at P.

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We will approximate the slope of the tangent line at P using the secant line.

 A secant line to a curve (informal def):

The Tangent Line Problem

x

y

P

Q

Draw a line through the point of tangency P and a second point on the curve Q.

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If we know two points on a line, we can find its slope:

If is the point of tangency and Q ( ) is a second point on the curve, the slope of the secant line through these points is given by

The Tangent Line Problem

2 1

2 1

y yym

x x x

, ( )P c f c

secm

, ( )c f c

x

y

PQ

f

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We can get more accurate approximations to the slope of the tangent by choosing points closer to P, the point of tangency.

Check this out: http://www.math.hmc.edu/calculus/tutorials/tangent_line/

 

P

x

y

Q

f

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As point Q approaches point P, the slope of the secant line approaches the slope of the tangent line.

The slope of the tangent line is said to be the limit of the slope of the secant line.

mx

cfxcfx

)()(lim

0

The slope m of the tangent line to the graph of f at x = c:

Now that is cool!

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  The slope of the tangent line to the graph at the point is also called the slope of the curve of f at x = c.

, ( )c f c

And why is this so very cool, you ask. Well…

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The limit used to define the slope of the tangent line is also used to define one of the two fundamental operations in calculus—differentiation.

The derivative of f at x is defined as

for all x for which the limit exists.

x

xfxxfxf

x

)()(lim)(

0

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A second classic problem involves finding the area of a plane region that is bounded by the graphs of functions.

The Area Problem

( )y f x

xa b

Consider the region bound by the function y = f(x), the x-axis, and the vertical lines x = a and x = b.

We can approximate the area of this region with several rectangular regions.

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The Area Problem

As you increase the number of rectangles, the approximation tends to become better because the amount of ‘missed area’ decreases.

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To find the area of the region, we want to find the limit of the sums of the areas of the rectangles as the number of rectangles increase without bound.

Check this out:

http://xanadu.math.utah.edu/java/ApproxArea.html

The Area Problem

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The area of a region bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b is given by the definite integral:

( )b

a

Area f x dx

The symbol was derived from the letter S, for summation.

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We have seen just a brief introduction to the two major branches of calculus—differential calculus (the tangent line problem) and integral calculus (the area problem).

As unrelated as they may seem, it was discovered (independently, by our heroes Isaac Newton and Gottfried Leibniz) that the two branches are actually connected.

Differentiation and definite integration are inverses.Differentiation and definite integration are inverses.

Awesome!

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End of section 1.1