1.8: The Tangent Line Approximation

The tangent line

Given a function $f$ that is differentiable at $x = a\text{,}$ we know that we can determine the slope of the tangent line to $y = f(x)$ at $(a,f(a))$ by computing $f'(a)\text{.}$ The equation of the resulting tangent line is given in point-slope form by

$$y - f(a) = f'(a)(x-a) \ \ \text{or} \ \ y = f'(a)(x-a) + f(a)\text{.}$$

Note well: there is a major difference between $f(a)$ and $f(x)$ in this context. The former is a constant that results from using the given fixed value of $a\text{,}$ while the latter is the general expression for the rule that defines the function. The same is true for $f'(a)$ and $f'(x)\text{:}$ we must carefully distinguish between these expressions. Each time we find the tangent line, we need to evaluate the function and its derivative at a fixed $a$-value.

In Figure $1.8.2$, we see the graph of a function $f$ and its tangent line at the point $(a,f(a))\text{.}$ Notice how when we zoom in we see the local linearity of $f$ more clearly highlighted. The function and its tangent line are nearly indistinguishable up close. Local linearity can also be seen dynamically in the java applet at http://gvsu.edu/s/6J.

The local linearization

A slight change in perspective and notation will enable us to be more precise in discussing how the tangent line approximates $f$ near $x = a\text{.}$ By solving for $y\text{,}$ we can write the equation for the tangent line as

This line is itself a function of $x\text{.}$ Replacing the variable $y$ with the expression $L(x)\text{,}$ we call

the local linearization of $f$ at the point $(a,f(a))\text{.}$ In this notation, $L(x)$ is nothing more than a new name for the tangent line. As we saw above, for $x$ close to $a\text{,}$ $f(x) \approx L(x)\text{.}$

We emphasize that $y = L(x)$ is simply a new name for the tangent line function. Using this new notation and our observation that $L(x) \approx f(x)$ for $x$ near $a\text{,}$ it follows that we can write

From Activity $1.8.2$, we see that the local linearization $y = L(x)$ is a linear function that shares two important values with the function $y = f(x)$ that it is derived from. In particular,

• because $L(x) = f(a) + f'(a)(x-a)\text{,}$ it follows that $L(a) = f(a)\text{;}$ and
• because $L$ is a linear function, its derivative is its slope.

Hence, $L'(x) = f'(a)$ for every value of $x\text{,}$ and specifically $L'(a) = f'(a)\text{.}$ Therefore, we see that $L$ is a linear function that has both the same value and the same slope as the function $f$ at the point $(a,f(a))\text{.}$

Thus, if we know the linear approximation $y = L(x)$ for a function, we know the original function's value and its slope at the point of tangency. What remains unknown, however, is the shape of the function $f$ at the point of tangency. There are essentially four possibilities, as shown in Figure $1.8.4$.

These possible shapes result from the fact that there are three options for the value of the second derivative: either $f''(a) \lt 0\text{,}$ $f''(a) = 0\text{,}$ or $f''(a) \gt 0\text{.}$

• If $f''(a) \gt 0\text{,}$ then we know the graph of $f$ is concave up, and we see the first possibility on the left, where the tangent line lies entirely below the curve.
• If $f''(a) \lt 0\text{,}$ then $f$ is concave down and the tangent line lies above the curve, as shown in the second figure.
• If $f''(a) = 0$ and $f''$ changes sign at $x = a\text{,}$ the concavity of the graph will change, and we will see either the third or fourth figure.².
• A fifth option (which is not very interesting) can occur if the function $f$ itself is linear, so that $f(x) = L(x)$ for all values of $x\text{.}$

The plots in Figure $1.8.4$ highlight yet another important thing that we can learn from the concavity of the graph near the point of tangency: whether the tangent line lies above or below the curve itself. This is key because it tells us whether or not the tangent line approximation's values will be too large or too small in comparison to the true value of $f\text{.}$ For instance, in the first situation in the leftmost plot in Figure $1.8.4$ where $f''(a) > 0\text{,}$ because the tangent line falls below the curve, we know that $L(x) \le f(x)$ for all values of $x$ near $a\text{.}$

The idea that a differentiable function looks linear and can be well-approximated by a linear function is an important one that finds wide application in calculus. For example, by approximating a function with its local linearization, it is possible to develop an effective 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 the limit

is indeterminate, because both its numerator and denominator tend to 0. While there is no algebra that we can do to simplify \\(\frac{\sin(x)}{x}\text{,}\) it is straightforward to show that the linearization of $f(x) = \sin(x)$ at the point $(0,0)$ is given by $L(x) = x\text{.}$ Hence, for values of $x$ near 0, \\(\sin(x) \approx x\text{,}\) and therefore

which makes plausible the fact that

Summary

• 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
  $$y - f(a) = f'(a)(x-a)\text{.}$$
• 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 will be 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 $y = L(x)\text{,}$ where $L(x) = f(a) + f'(a)(x-a)\text{.}$ Thus, $f(x) \approx L(x)$ for all $x$ near $x = a\text{.}$
• If we know the tangent line approximation $L(x) = f(a) + f'(a)(x-a)$ to a function $y=f(x)\text{,}$ then because $L(a) = f(a)$ and $L'(a) = f'(a)\text{,}$ we also know the values of both the function and its derivative at the point where $x = a\text{.}$ In other words, the linear approximation tells us the height and slope of the original function. If, in addition, we know the value of $f''(a)\text{,}$ we then know whether the tangent line lies above or below the graph of $y = f(x)\text{,}$ depending on the concavity of $f\text{.}$