Secant line between a, fa and c, fc

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Introduction to Derivatives
Tangent line at (c, f(c))
y
Secant line between (a, f(a)) and (c, f(c))
(a, f(a))
Definitions (informal) 1. A secant line is a
line connecting two points of a curve. 2. A
tangent line at a point of a curve is a line that
only touches that point of the curve, locally
speaking.
(c, f(c))
x
y f(x)
We can always calculate the slope between two
points (recall ). Therefore, we
can always calculate the slope of a secant line
because it connects two points of the curve. For
example the slope of the secant line shown above
is m . On the other hand,
we cant find the slope of the tangent line since
even though it also touches two points of the
curve, only (c, f(c)) is given, the other one is
not! Let alone those tangent lines that really
touch exactly one point of the curve (see diagram
on the right)!
Can we find the slope of the tangent line at a
point given that we dont know any other points?
The answer is YES. This is when we need to use
the idea called limiting processto find the
slope of a tangent line at a given point (c,
f(c)) of a function, we can find the slope
between the fixed point (c, f(c)) and some
movable point, say, (x, f(x)). As (x, f(x))
moving closer and closer to (c, f(c)), we
calculate the slopes of the secant lines between
(c, f(c)) and (x, f(x)), and see what the limit
is, i.e., what the values of the slopes of the
secant lines are getting closer to?
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Let (1, 1) be fixed and let the other points
vary, what are slopes of the followingsecant
lines Shown Between (4, 16) and (1,
1) Between (3, 9) and (1, 1) Between (2, 4)
and (1, 1) Between (0, 0) and (1, 1) Not
Shown Between (1.1, 1.21) and (1, 1) Between
(1.01, 1.0201) and (1, 1) Conclusion The slope
of the tangent line must be What is the
equation of the tangent line at (1, 1)? (Hint
use y y1 m(x x1))
y x2
(1, 1)
Now, let (2, 4) be fixed and let the other points
vary, what are slopes of the following secant
lines Shown Between (4, 16) and (2,
4) Between (3, 9) and (2, 4) Between (1, 1)
and (2, 4) Between (0, 0) and (2, 4) Not
Shown Between (2.1, 4.41) and (2, 4) Between
(2.01, 4.0401) and (2, 4) Conclusion The slope
of the tangent line at (2, 4) must be What is
the equation of the tangent line at (2, 4)?
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Definition (Formal)
The tangent line to the graph of a function y
f(x) at a point P (c, f(c)) is defined as the
line containing the point P whose slope is
provided that this limit exists. If mtan
exists, an equation of the tangent line at (c,
f(c)), by the point-slope form (i.e., y y1
m(x x1)), is y f(c) mtan(x
c) Therefore, without calculating the slopes of
the secant lines of the points near (2, 4), we
can just use where, in this case, f(x) x2,
c 2, and f(c)) 4.
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Let (1, 1) be fixed and let the other points
vary, what are slopes of the following secant
lines Shown Between (2, 8) and (1, 1) Between
(0, 0) and (1, 1) Not Shown Between (1.1,
1.331) and (1, 1) Between (1.01, 1.030301) and
(1, 1) Conclusion The slope of the tangent line
must be What is the equation of the tangent
line at (1, 1)? (Hint use y y1 m(x x1))
Using the definition
Q What does this have to do with derivative? A
What we did is the derivative of the function at
the point!
Definition
The Derivative of a Function f at a Number c Let
y f(x) denote a function f, and if a real
number c is in the domain of f, the derivative of
f at c. denoted by f?(c), read as f prime of c,
is defined as Provided that this limit exists.
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As we can see, the formula
is really the same as the
formula
. So, Furthermore, if an equation of the
tangent line at (c, f(c)) can be written as y
f(c) mtan(x c) it can be written as How to
find the derivative of a function f at c, i.e.,
f?(c)? Example If f(x) x2 2x 4, find
f?(3). Step 1 Find f(c) if its not given. Step
2 Use . Step 3 Find
the limit Whatever you obtain in step 3 is
f?(c). Note when you try to find the
, it will be a 0/0
indeterminate form, in almost every
case. Examples Find the derivative at the
indicated x-value or at the indicated point. 1.
f(x) x2 2 at x 1 2. f(x) x2 3 at (2,
1)
mtan
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How to find an equation of the tangent line of a
function f(x) at a given x-value c or a given
point (c, f(c))? Step 0 Find f(c) if its not
given. Step 1 Find the slope of the tangent line
using the formula Step 2 Use point-slope form of
a line y y1 m(x x1) by substituting c into
x1, f(c) into y1 and f? (c) into m.
Examples Find an equation of the tangent line of
the given function at the x-value or at the given
point. 1. f(x) 2x2 x 2 at x 1 2. f(x)
x3 2 at (2, 6)
Determining whether the derivative of f at a
number should be positive, negative or zero when
a graph is given
When we are given a graph, we know that to
determine whether a function value f(c) is
positive, negative, or zero, all we need to do is
to ask whether the point (c, f(c)) is above,
below or on the x-axis, respectively. For
example, f(3) is positive because the point (3,
f(3)) is above the x-axis f(4) is negative
because the point (4, f(4)) is below the x-axis,
f(7) is zero because the point (7, f(7)) is on
the x-axis.
However, to determine the derivative of f at a
number c, i.e., f? (c), is positive, negative,
or zero, we need to ask ourselves this question
If we can construct a tangent line at (c, f(c)),
what is the slope of this tangent line? For
example, if we construct a tangent line at (3,
f(3)), what is the slopepositive, negative, or
zero? Obviously its positive! Therefore, f? (3)
is positive. Determine whether the following
derivatives are positive (), negative () or
zero (0) f? (3) () f? (1) ___ f? (1)
___ f? (3) ___ f? (4) ___ f? (7) ___