By Marcel B. Finan
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Additional info for A Reform Approach to Business Calculus
Reflections about the y-axis: For a given function f (x), the graph of f (−x) is a reflection of the graph of f (x) about the y-axis. Vertical Stretches and Compressions: If a function f (x) is given, then the graph of kf (x) is a vertical stretch of the graph of f (x) by a factor of k for k > 1, and a vertical compression for 0 < k < 1. What about k < 0? If |k| > 1 then the graph of kf (x) is a vertical stretch of the graph of f (x) followed by a reflection about the x-axis. If 0 < |k| < 1 then the graph of kf (x) is a vertical compression of the graph of f (x) by a factor of k followed by a reflection about the x-axis.
The final amount coming out is given by g(f (x)) = f (x) + 5 = 2x + 5. Now, if x dollars enters the second machine first, then the amount that comes out is g(x) = x + 5. If this amount enters the second machine then the final amount coming out is f (g(x)) = 2(x + 5) = 2x + 10. The function f (g(x)) is called the composition of the functions f and g; the function g(f (x)) is called the composition of the functions g and f. In general, suppose we are given two functions f and g such that the range of g is contained in the domain of f so that the output of g can be used as input for f.
For examdy = 2x for y = 2x. ple, we can write dx When using Leibniz notation to denote the value of the derivative at a point a we will write dy dx x=a Thus, to evaluate dy dx = 2x at x = 2 we would write dy dx = 2x|x=2 = 2(2) = 4. 1 dy Even though dx appears as a fraction but it is not. It is just an alternative notation for the derivative. A concept called differential will provide meaning to symbols like dy and dx. One of the advantages of Leibniz notation is the recognition of the units of the derivative.
A Reform Approach to Business Calculus by Marcel B. Finan