When we write a critical appreciation, also called a critical analysis, we deeply analyze the multiple parts of a work to draw a conclusion and prove that conclusion.
Extrapolation is most certainly one main step in critical analysis because, when we extrapolate, we infer the deeper meaning of something, like the deeper meaning of an entire work, a scene, a sentence, a symbol, an image, or even a word choice. To infer means to draw conclusions based on evidence and sound premises. Even more specifically, when we write a critical analysis or critical appreciation, we "break a subject ... into its constituent parts, examine these components, and offer a meaning--or alternative meanings--about each" (University of Minnesota, "Chapter 46: Critical or Interpretive Analysis of Literature").
When beginning a critical analysis/appreciation, we want to start with a question. For example, we might start by asking, what is Charles Dickens trying to convey about society through his usage and descriptions of hard times?
Next, we try to answer our analytical question by breaking a work into individual parts and then analyzing those parts to extrapolate further meaning. Parts we can look at can include "a scene, character, activity, line, or some other segment" (University of Minnesota). We would analyze each part for theme, "social, psychological, or historical context," figurative language, structure, and other literary or rhetorical devices (Georgia Perimeter College, "Critical Analysis of Literature"). We can also apply a "literary theory or other point of view" (University of Minnesota). Research of published critical analyses on the topic is also usual essential for this type of paper. Once we have extrapolated meaning from each individual part, we should be able to synthesize our ideas into one solid conclusion that answers our analysis question. For example, using the same analysis question above, we might conclude that Dickens frequently describes multiple hard times in order to convey the types of changes society is crying out for.
We would then write paragraphs about each part of our analysis to answer our question and prove our conclusion.