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Unformatted text preview: (3/31/07) Section 0.6 Linear combinations, products, quotients, and compositions Overview: Many funcions that are used in calculus are constructed from constant, power, exponential, logarithmic, trigonometric, and inverse trigonometric functions by taking linear combinations, products, quotients, and compositions of these basic functions. In this section we discuss these procedures and look at instances where the graphs of such functions can be readily explained by their formulas. Topics: Linear combinations Products and quotients Polynomials and rational functions Composite functions Linear combinations A linear combination of functions is formed by multiplying each of the functions by a constant and adding the results. Example 1 The functions (a) A ( x ) = 1 8 x 2 + 3 cos x , (b) B ( x ) = 1 8 x 2 3cos x , and (c) C ( x ) = 1 8 x 2 + 3 cos x are linear combinations of y = x 2 and y = cos x . Match them with their graphs in Figures 1 through 3. x y 10 10 2  2 x y 10 10 2  2 x y 10 10 2  2 FIGURE 1 FIGURE 2 FIGURE 3 Solution (a) The curves y = 1 8 x 2 and y = 3 cos x are shown in Figure 4. The first is y = x 2 contracted vertically by a factor of 8, and the second is y = cos x magnified vertically by a factor of 3. Because y = 3cos x oscillates between 3 and 3, the graph of A ( x ) = 1 8 x 2 + 3 cos x oscillates from 3 units above y = 1 8 x 2 to 3 units below it, and since A (0) = 1 8 (0) 2 + 3 cos(0) = 3, the graph is in Figure 3. (b) The graph of B ( x ) = 1 8 x 2 3cos x oscillates from 3 units above to 3 units below y = 1 8 x 2 and B (0) = 1 8 (0) 2 3cos(0) = 3. The graph is in Figure 2. (c) Figure 5 shows the curves y = 1 8 x 2 and y = 3 cos x . The graph of C ( x ) = 1 8 x 2 + 3 cos x oscillates from 3 units above to 3 units below y = 1 8 x 2 . It is in Figure 1. square 1 p. 2 (3/31/07) Section 0.6, Linear combinations, products, quotients, and compositions x y 10 10 2  2 y = 1 8 x 2 y = 3cos x x y 10 10 y = 1 8 x 2 y = 3 cos x FIGURE 4 FIGURE 5 C Question 1 Generate y = 1 8 x 2 + 10cos x in the window 4 x 4 , 15 y 20 with xscale = and yscale = 5 and explain how this curve differs from y = 1 8 x 2 + 3cos x in Figure 3. The domain of a linear combination of functions consists of those values of the variable x where all of the functions involved are defined. Example 2 What is the domain of g ( x ) = 2 + 3 x 1 / 2 5 x ? Solution Because the constant function y = 2 and the exponential function y = 5 x are defined for all x , the domain of g ( x ) = 2 + 3 x 1 / 2 5 x is the interval [0 , ) where the square root function y = x 1 / 2 is defined. square Products and quotients The product of two functions f and g is the function fg whose value at x is the product f ( x ) g ( x ) of the values of the two functions. Its domain consists of all values of x where f ( x ) and g ( x ) are defined....
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 Summer '08
 Eggers
 Math, Calculus

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