By Marcel B. Finan
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Extra info for A Reform Approach to Business Calculus
Sample text
3 (a) Find f (1) for f (x) = x2 . (b) Find the equation of the tangent line to the graph of f (x) at the point (1, f (1)). Solution. 1 we see that f (1) = 2. (b) The equation of the tangent line is y − f (1) = f (1)(x − 1) or y − 1 = 2(x − 1) In point-intercept form, we have y = 2x − 1. 4 (Numerical Estimation of the Derivative) Find approximate values for f (x) at each of the x-values given in the following table x 0 5 10 15 20 f(x) 100 70 55 46 40 Solution. The derivative can be estimated by using the average rate of change or the difference quotient f (a + h) − f (a) .
If 0 < |k| < 1 then the graph of f (kx) is a horizontal stretch of the graph of 47 f (x) by a factor of 1 k followed by a reflection about the y-axis. 8 Write an equation for a graph obtained by vertically stretching the graph of f (x) = x2 by a factor of 2, followed by a vertical upward shift of 1 unit. Solution. The function is given by the formula y = 2f (x) + 1 = 2x2 + 1. Recommended Problems (pp. 54 - 56): 1, 5, 7, 13, 14, 15, 16, 17, 18, 19, 31. 48 9 Power and Polynomial Functions A function f (x) is a power function of x if there is a constant k such that f (x) = kxn If n > 0, then we say that f (x) is proportional to the nth power of x.
By solving for x we find the interval of one period to be [0, 2π ]. Thus, the above mentioned functions have a period b 2π of b . The number b tells you the number of cylces in the interval [0, 2π]. Graphically,b either stretches (if b < 1) or compresses (if b > 1) the graph horizontally. Figure 21 shows the function y = sin x with period 2π and the function 54 y = sin (2x) with period π. Figure 21 Recommended Problems (pp. 67 - 9): 3, 5, 7, 9, 10, 11, 13, 14, 17, 19, 20, 24, 25. 55 11 Instantaneous Rate of Change In this section, we discuss the concept of the instantaneous rate of change of a given function.
