1.8. THE TANGENT LINE APPROXIMATION
77
1.8 The Tangent Line Approximation
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is the formula for the general tangent line approximation to a differentiable
function y = f (x) at the point (a, f (a))?
• What is the principle of local linearity and what is the local linearization of a
differentiable function f at a point (a, f (a))?
• How does knowing just the tangent line approximation tell us information about
the behavior of the original function itself near the point of approximation? How
does knowing the second derivative’s value at this point provide us additional
knowledge of the original function’s behavior?
Introduction
Among all functions, linear functions are simplest. One of the powerful consequences
of a function y = f (x) being differentiable at a point (a, f (a)) is that, up close, the
function y = f (x) is locally linear and looks like its tangent line at that point. In certain
circumstances, this allows us to approximate the original function f with a simpler function
L that is linear: this can be advantageous when we have limited information about f
or when f is computationally or algebraically complicated. We will explore all of these
situations in what follows.
It is essential to recall that when f is differentiable at x = a, the value of f ′ (a) provides
the slope of the tangent line to y = f (x) at the point (a, f (a)). By knowing both a point
on the line and the slope of the line we are thus able to find the equation of the tangent
line. Preview Activity 1.8 will refresh these concepts through a key example and set the
stage for further study.
Preview Activity 1.8. Consider the function y = g(x) = −x 2 + 3x + 2.
(a) Use the limit definition of the derivative to compute a formula for y = g ′ (x).
(b) Determine the slope of the tangent line to y = g(x) at the value x = 2.
(c) Compute g(2).
(d) Find an equation for the tangent line to y = g(x) at the point (2, g(2)). Write your
result in point-slope form 8 .
8 Recall that a line with slope m that passes through (x 0 , y 0 ) has equation y − y 0 = m(x − x 0 ), and this is
77
1.8 The Tangent Line Approximation
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is the formula for the general tangent line approximation to a differentiable
function y = f (x) at the point (a, f (a))?
• What is the principle of local linearity and what is the local linearization of a
differentiable function f at a point (a, f (a))?
• How does knowing just the tangent line approximation tell us information about
the behavior of the original function itself near the point of approximation? How
does knowing the second derivative’s value at this point provide us additional
knowledge of the original function’s behavior?
Introduction
Among all functions, linear functions are simplest. One of the powerful consequences
of a function y = f (x) being differentiable at a point (a, f (a)) is that, up close, the
function y = f (x) is locally linear and looks like its tangent line at that point. In certain
circumstances, this allows us to approximate the original function f with a simpler function
L that is linear: this can be advantageous when we have limited information about f
or when f is computationally or algebraically complicated. We will explore all of these
situations in what follows.
It is essential to recall that when f is differentiable at x = a, the value of f ′ (a) provides
the slope of the tangent line to y = f (x) at the point (a, f (a)). By knowing both a point
on the line and the slope of the line we are thus able to find the equation of the tangent
line. Preview Activity 1.8 will refresh these concepts through a key example and set the
stage for further study.
Preview Activity 1.8. Consider the function y = g(x) = −x 2 + 3x + 2.
(a) Use the limit definition of the derivative to compute a formula for y = g ′ (x).
(b) Determine the slope of the tangent line to y = g(x) at the value x = 2.
(c) Compute g(2).
(d) Find an equation for the tangent line to y = g(x) at the point (2, g(2)). Write your
result in point-slope form 8 .
8 Recall that a line with slope m that passes through (x 0 , y 0 ) has equation y − y 0 = m(x − x 0 ), and this is