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Additional resources for CK-12 Calculus, Volume 1
C. because you would expect (1) a decline as soon as you bought the car, and (2) the value to be declining more gradually after the initial drop. The pool's surface area as a function of width: The pool has area when The rate of change will be The formula will be At the time or equivalently in the year , the car will be valued at . A linear model may not be the best function to model depreciation because the graph of the function decreases as time increases; hence at some point the value will take on negative real number values, an impossible situation for the value of real goods and products.
Understand properties of continuous functions. Solve problems using the Min-Max theorem. Solve problems using the Intermediate Value Theorem. Introduction In this lesson we will discuss the property of continuity of functions and examine some very important implications. Let舗s start with an example of a rational function and observe its graph. Consider the following function: We know from our study of domains that in order for the function to be defined, we must use Yet when we generate the graph of the function (using the standard viewing window), we get the following picture that appears to be defined at : The seeming contradiction is due to the fact that our original function had a common factor in the numerator and denominator, that cancelled out and gave us a picture that appears to be the graph of But what we actually have is the original function, that we know is not defined at At we have a hole in the graph, or a discontinuity of the function at That is, the function is defined for all other values close to Loosely speaking, if we were to hand-draw the graph, we would need to take our pencil off the page when we got to this hole, leaving a gap in the graph as indicated: Now we will formalize the property of continuity of a function and provide a test for determining when we have continuous functions.
Velocity of a Falling Object We can use differential calculus to investigate the velocity of a falling object. Galileo found that the distance traveled by a falling object was proportional to the square of the time it has been falling: The average velocity of a falling object from to is given by HW Problem #10 will give you an opportunity to explore this relationship. In our discussion, we saw how the study of tangent lines to functions yields rich information about functions. We now consider the second situation that arises in Calculus, the central problem of finding the area under the curve of a function .