By Jon Rogawski (Jonathan D.)

Unmarried Variable Calculus: Early Transcendentals (ch 1-11), 2d version (c2012), textbook via Jonanthan D. Rogawski (d.)

**Read or Download Calculus: Single Variable (ch. 1-11), Early Transcendentals, 2nd Edition (c2012) PDF**

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**Extra info for Calculus: Single Variable (ch. 1-11), Early Transcendentals, 2nd Edition (c2012)**

**Example text**

Surface area of a sphere as a function of its radius Temperature at a point on the equator as a function of time Price of an airline ticket as a function of the price of oil Pressure of the gas in a piston as a function of volume y 3 2 1 3 2 1 x 1 2 3 x −3 −2 −1 −1 −2 −3 (iv) 1 2 3 (v) FIGURE 28 65. Find the domain and range of f (x)? 66. Sketch the graphs of f (x + 2) and f (x) + 2. 67. Sketch the graphs of f (2x), f 12 x , and 2f (x). 68. Sketch the graphs of f (−x) and −f (−x). 69. Extend the graph of f (x) to [−4, 4] so that it is an even function.

The graph of y = 3f (x) = 3 sin(π x) differs from y = f (x) only in amplitude: It is expanded in the vertical direction by a factor of 3 [Figure 25(C)]. y 3 y 2 y 1 1 1 x 1 2 3 x 4 1 −1 2 3 −1 One cycle x 4 1 2 3 −1 Three cycles −2 −3 (A) y = f (x) = sin (πx) FIGURE 25 Horizontal and vertical scaling (B) Horizontal compression: y = f (3x) = sin (3πx) of f (x) = sin(πx). 1 SUMMARY a −a if a ≥ 0 if a < 0 • Absolute value: |a| = • Triangle inequality: |a + b| ≤ |a| + |b| Four intervals with endpoints a and b: • [a, b], (a, b), • [a, b), (a, b] Writing open and closed intervals using inequalities: (a, b) = {x : |x − c| < r}, [a, b] = {x : |x − c| ≤ r} where c = 12 (a + b) is the midpoint and r = 21 (b − a) is the radius.

58. Show, by completing the square, that the parabola 55. Show that y/ x for the function f (x) = x 2 over the interval [x1 , x2 ] is not a constant, but depends on the interval. Determine the exact dependence of y/ x on x1 and x2 . 56. Use Eq. (2) to derive the quadratic formula for the roots of ax 2 + bx + c = 0. y = ax 2 + bx + c is congruent to y = ax 2 by a vertical and horizontal translation. 59. Prove Viète’s Formulas: The quadratic polynomial with α and β as roots is x 2 + bx + c, where b = −α − β and c = αβ.