By Lynn H. Loomis, Shlomo Sternberg

ISBN-10: 0867201223

ISBN-13: 9780867201222

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Extra resources for Advanced Calculus, Revised Edition

Example text

As a result of this you should be able to see that f (x) = |x| + x = 2x 0 if x ≥ 0, if x < 0. Like |x| itself, we have here another example of a function deﬁned in two pieces, which can also be represented as a single formula. What happens if you subtract |x| from x? ) See if you can construct a single formula involving |x| to represent the following function. 2x if x ≥ 0, h(x) = 3x if x < 0. A function such as this can be plotted using MAPLE, so you can check your own formula. We use the following command.

29 gives (a cosh t)2 (b sinh t)2 − = cosh2 t − sinh2 t = 1. a2 b2 1. 7 Trigonometric and Hyperbolic Identities This section contains a list of identities, for reference. 23. 7 Inverse Functions The equation y = f (x) gives y in terms of x. To solve it for x we need to rearrange it to get x in terms of y. This occurs in many problems where we need to change the subject of a formula. For example consider the formula for the volume of a cylinder V = πr2 h. If we are given the values of r and h 30 Calculus of One Variable then of course we can calculate V .

Can you explain why? 39 Find a formula for tan sin−1 x . This is an example of a number of similar relationships involving trigonometric functions and their inverses. It can be derived algebraically using trigonometric identities as follows. tan sin−1 x = sin sin−1 x cos sin−1 x = sin sin−1 x 1 − sin 2 sin −1 =√ x x . 1 − x2 A more transparent method is to use a right-angled triangle. 24, where sin θ = BC/CA = x. 39 We then calculate the third side of the triangle (AB in the ﬁgure) using Pythagoras’ Theorem.