By S.L. Salas, Einar Hille

A revised and up to date presentation of calculus with purposes to engineering and the sciences. alterations comprise an early remedy of the calculus of the trigonometric features, an elevated use of Riemann definition of the imperative, the advent of a number of numerical options, an early bankruptcy on mathematical modeling, multiplied and balanced workout units, prompt tactics for challenge fixing, revised proofs, and extra examples. bankruptcy thirteen is contained in either half I and half II.

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**Example text**

This product is zero at −2, 1, 3. It is negative on (−∞, −2), positive on (−2, 1), negative on (1, 3), positive on (3, ∞). 2 Example 4 Solve the inequality (x + 3)5 (x − 1)(x − 4)2 < 0. SOLUTION We view (x + 3)5 (x − 1)(x − 4)2 as the product of three factors: (x + 3)5 , (x − 1), (x − 4)2 . The product is zero at −3, 1, 4. These points separate the intervals (−∞, −3), (−3, 1), (1, 4), (4, ∞). On each of these intervals the product keeps a constant sign: positive on negative on positive on positive on (−∞, −3), (−3, 1), (1, 4), (4, ∞).

The point on the y-axis with line coordinate b is assigned rectangular coordinates (0, b). Thus the origin is assigned coordinates (0, 0). A point P not on one of the coordinate axes is assigned coordinates (a, b) provided that the line l1 that passes through P and is parallel to the y-axis intersects the x-axis at the point with coordinates (a, 0), and the l2 that passes through P and is parallel to the x-axis intersects the y-axis at the point with coordinates (0, b). This procedure assigns an ordered pair of real numbers to each point of the plane.

The slope is 0 and the y-intercept is − 54 . 8. 8 Example 2 Write an equation for the line l2 that is parallel to l1 : 3x − 5y + 8 = 0 and passes through the point P(−3, 2). SOLUTION The equation for l1 can be written y = 35 x + 85 . The slope of l1 is 35 . The slope of l2 must also be 35 . ) Since l2 passes through (−3, 2) with slope 35 , we can use the point-slope formula and write the equation as y − 2 = 35 (x + 3). ❏ Example 3 Write an equation for the line that is perpendicular to l1 : x − 4y + 8 = 0 and passes through the point P(2, −4).