"Define 'design,' 'culture,' and 'language' separately. Describe how you think they influence each other. Give an example of an intersection, and describe."

It's difficult to define design concisely enough in just a sentence or two. There are two separate facets--theory and the physical application of design principles. As best I can, without the use of a book. . .

Design: The process whereby visual artifacts (meaning every "thing" in any conceivable medium that exists to be viewed) are fashioned in a way that helps in the dissemination of information. Color, form, space between and around shapes (text included) are considered in a way that either ensures ease in comprehension (although this is not always the case), or suggests a particular mood. 

Culture, in its broadest sense, refers to a set of like goals, interests, concerns, practices, etc., that are shared by a given population of individuals. 

Another meaning refers to the appreciation of arts and humanities, based upon the ability to reason cogently, with respect to societal standards of life and learning. To say that someone is "cultured," means to say that the individual is well versed in the pursuit of the "finer" things in life.

Language refers to the mechanisms behind person-to-person communication, and the systems that enable these exchanges.

Languages refer not only to letters that make up words, but actions (gesticulations) and images that enable communication through symbols. 

Language informs the understanding of design, whether through the application of images, colors, and typefaces that communicate more easily in conjunction with one another, or through the creation of imagery that speaks easily symbolically.

Design and Language influence culture in obvious ways. Culture is established, possibly most efficiently, through the establishment of visual artifact.