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RATES OF CHANGE TANGENTS A tangent can be more precisely defined as follows:

Given a curve y = f(x), and a point P(a, f(a)), then the tangent line to the curve is the line through P with slope

m

=

lim x→a

f(x) – f(a)

1

x–a

(provided this limit exists)

In the earlier section Introduction to Calculus – The Limit Of A Function, we tried to guess the slopes of tangents by picking nearby points to P (a, f(a)) and compute the slope of the secant line PQ, which is given by

mPQ

=

f(x) – f(a) x–a

Then we let Q → P by letting x → a. By doing so, we discovered that the slope of the secant line approached a certain value m. Thus, in terms of limits, we say that ➔ the tangent line is the limiting position of the secant line PQ as Q → P, OR ➔ the slope m of the tangent line PQ is the limiting value of the slope of the secant lines PQ as Q → P. which leads to the mathematical expression

m

=

lim x→a

f(x) – f(a) x–a

Q(x, f(a))

f(x) – f(a) P(a, f(a)) x–a

a

The formula for m can be expressed in another way: Let h = x – a which gives m

=

lim x→a

f(x) – f(a) h

x


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[Title will be auto-generated] by Timothy Adu - Issuu