 Let's recap the major ideas from section 1.8 of active calculus on the tangent line approximation. The section focuses on the tangent line to the graph of a function at a point, and how we might use this to get more information about the function to whose graph it's attached. First, let's recall that if f is differentiable at x equals a, then f prime of a, the derivative, tells us the slope of the tangent line to the graph of f at x equals a. Specifically, the point to which the tangent line is attached has an x coordinate of a and a y coordinate of f of a, so we can use the point slope form for the equation of a line to write the equation of the tangent line, which gives us the tangent line itself. Solving for y would just give us y equals f prime of a times x minus a plus f of a. This is just the equation for the tangent line to the graph of f at x equals a in slope intercept form. Graphically, the tangent line at x equals a is an approximation to the graph of f at x equals a. If we zoom far enough in on the graph of f at x equals a, and if f is differentiable, then the graph of f will look just like the graph of the tangent line. In a previous section, we referred to this phenomenon of f looking like its tangent line when we zoom in by saying that f is locally linear at x equals a. So we often refer to the tangent line to the graph of f at x equals a as the local linearization of f at x equals a. The local linearization of f at x equals a, in other words, is just another term for the tangent line to f at x equals a. The reason this matters is that we can use the local linearization to approximate values of f that are near x equals a if we don't have a formula for f. Sometimes we may not know a formula for f, but rather we just know a single value of f at x equals a and its local linearization at x equals a. We could then use the local linearization to approximate values of f near x equals a. Since f looks like its tangent line near x equals a, the approximate values of f that are near x equals a should be close to the actual values of f as long as we don't move too far away from the point where we're linearizing.