By Angus E. Taylor, W. Robert Mann
Outlines thought and methods of calculus, emphasizing powerful figuring out of innovations, and the elemental ideas of study. studies trouble-free and intermediate calculus and lines discussions of elementary-point set concept, and houses of continuing features.
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Extra info for Advanced Calculus (3rd Edition)
If x ≥ 0, then f (x) = x2 − 4x + 3 = |(x − 1)(x − 3)|. Case (i): If 0 < x ≤ 1, then f (x) = x2 − 4x + 3. Case (ii): If 1 < x ≤ 3, then f (x) = −(x2 − 4x + 3) = −x2 + 4x − 3. Case (iii): If x > 3, then f (x) = x2 − 4x + 3. This enables us to sketch the graph for x ≥ 0. Then we use the fact that f is an even function to reflect this part of the graph about the y-axis to obtain the entire graph. Or, we could consider also the cases x < −3, −3 ≤ x < −1, and −1 ≤ x < 0. 7. Remember that |a| = a if a ≥ 0 and that |a| = −a if a < 0.
H→0 lim cos(a + h) = lim (cos a cos h − sin a sin h) = lim (cos a cos h) − lim (sin a sin h) h→0 h→0 = 59. f (x) = lim cos a h→0 0 if x is rational 1 if x is irrational h→0 lim cos h − lim sin a h→0 h→0 h→0 lim sin h = (cos a)(1) − (sin a)(0) = cos a h→0 is continuous nowhere. For, given any number a and any δ > 0, the interval (a − δ, a + δ) contains both infinitely many rational and infinitely many irrational numbers. Since f (a) = 0 or 1, there are infinitely many numbers x with 0 < |x − a| < δ and |f (x) − f (a)| = 1.
The tangent is so steep at P that we need to take x-values much closer to 1 in order to get accurate estimates of its slope. 3794. 4422. 4108. 4. 2 The Limit of a Function 1. As x approaches 2, f (x) approaches 5. ] Yes, the graph could have a hole at (2, 5) and be defined such that f (2) = 3. 3. (a) lim f (x) = ∞ means that the values of f (x) can be made arbitrarily large (as large as we please) by taking x x→−3 sufficiently close to −3 (but not equal to −3). (b) lim f(x) = −∞ means that the values of f (x) can be made arbitrarily large negative by taking x sufficiently close to 4 x→4+ through values larger than 4.