1.8. THE TANGENT LINE APPROXIMATION
83
algorithm to estimate the zeroes of a function. Local linearity also helps us to make further
sense of certain challenging limits. For instance, we have seen that a limit such as
lim
x→0
sin(x)
x
is indeterminate because both its numerator and denominator tend to 0. While there is no
algebra that we can do to simplify
sin(x)
x , it is straightforward to show that the linearization
of f (x) = sin(x) at the point (0, 0) is given by L(x) = x. Hence, for values of x near 0,
sin(x) ≈ x. As such, for values of x near 0,
sin(x)
x
≈
x
x
= 1,
which makes plausible the fact that
lim
x→0
sin(x)
x
= 1.
These ideas and other applications of local linearity will be explored later on in our work.
Summary
In this section, we encountered the following important ideas:
• The tangent line to a differentiable function y = f (x) at the point (a, f (a)) is given in
point-slope form by the equation
y − f (a) = f
′ (a)(x − a).
• The principle of local linearity tells us that if we zoom in on a point where a function
y = f (x) is differentiable, the function should become indistinguishable from its tangent
line. That is, a differentiable function looks linear when viewed up close. We rename
the tangent line to be the function y = L(x) where L(x) = f (a) + f ′ (a)(x − a) and note
that f (x) ≈ L(x) for all x near x = a.
• If we know the tangent line approximation L(x) = f (a) + f ′ (a)(x − a), then because
L(a) = f (a) and L ′ (a) = f ′ (a), we also know both the value and the derivative of the
function y = f (x) at the point where x = a. In other words, the linear approximation
tells us the height and slope of the original function. If, in addition, we know the value
of f ′′ (a), we then know whether the tangent line lies above or below the graph of
y = f (x) depending on the concavity of f .
Exercises
1. A certain function y = p(x) has its local linearization at a = 3 given by L(x) = −2x + 5.
83
algorithm to estimate the zeroes of a function. Local linearity also helps us to make further
sense of certain challenging limits. For instance, we have seen that a limit such as
lim
x→0
sin(x)
x
is indeterminate because both its numerator and denominator tend to 0. While there is no
algebra that we can do to simplify
sin(x)
x , it is straightforward to show that the linearization
of f (x) = sin(x) at the point (0, 0) is given by L(x) = x. Hence, for values of x near 0,
sin(x) ≈ x. As such, for values of x near 0,
sin(x)
x
≈
x
x
= 1,
which makes plausible the fact that
lim
x→0
sin(x)
x
= 1.
These ideas and other applications of local linearity will be explored later on in our work.
Summary
In this section, we encountered the following important ideas:
• The tangent line to a differentiable function y = f (x) at the point (a, f (a)) is given in
point-slope form by the equation
y − f (a) = f
′ (a)(x − a).
• The principle of local linearity tells us that if we zoom in on a point where a function
y = f (x) is differentiable, the function should become indistinguishable from its tangent
line. That is, a differentiable function looks linear when viewed up close. We rename
the tangent line to be the function y = L(x) where L(x) = f (a) + f ′ (a)(x − a) and note
that f (x) ≈ L(x) for all x near x = a.
• If we know the tangent line approximation L(x) = f (a) + f ′ (a)(x − a), then because
L(a) = f (a) and L ′ (a) = f ′ (a), we also know both the value and the derivative of the
function y = f (x) at the point where x = a. In other words, the linear approximation
tells us the height and slope of the original function. If, in addition, we know the value
of f ′′ (a), we then know whether the tangent line lies above or below the graph of
y = f (x) depending on the concavity of f .
Exercises
1. A certain function y = p(x) has its local linearization at a = 3 given by L(x) = −2x + 5.