By Laura Taalman, Peter Kohn
Many calculus textbooks glance to have interaction scholars with margin notes, anecdotes, and different units. yet many teachers locate those distracting, who prefer to captivate their technology and engineering scholars with the wonderful thing about the calculus itself. Taalman and Kohn’s fresh new textbook is designed to aid teachers just do that.
Taalman and Kohn’s Calculus bargains a streamlined, based exposition of calculus that mixes the readability of vintage textbooks with a latest viewpoint on innovations, abilities, purposes, and concept. Its smooth, uncluttered layout removes sidebars, historic biographies, and asides to maintain scholars enthusiastic about what’s such a lot important—the foundational options of calculus which are so very important to their destiny educational careers.
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5 to prove that a function is one-to-one if and only if its graph passes the horizontal line test. 92. 5 to prove that the function f (x) = 3x + 1 is one-to-one. 93. Use the definition of decreasing to prove that the function f (x) = 1 − 3x is decreasing on (−∞, ∞). 94. Use the definition of increasing to prove that the function f (x) = 1 is increasing on (−∞, 3). 3−x 95. Prove that the average rate of change of the linear function f (x) = −2x + 4 on any interval I is always equal to −2. 96. Show that the average rate of change of every linear function f (x) = mx + b is constant, that is, the same over any choice of interval.
Y = f (x) A transformation of y = f (x) y y 2 1 1 2 3 x Ϫ3 Ϫ2 Ϫ1 Ϫ1 Ϫ2 Ϫ2 Ϫ3 Ϫ3 Ϫ4 Ϫ4 1 2 3 Ϫ3 Ϫ2 Ϫ1 Ϫ1 1 2 3 x Ϫ2 15. Suppose f is an invertible function with inverse f −1 . What is ( f −1 )−1 ? Explain your answer. 16. Given that f is an invertible function, fill in the blanks. 2 1 Ϫ3 Ϫ2 Ϫ1 Ϫ1 y (b) 4 13. Complete the entries in the following table two ways: (a) to make an even function and (b) to make an odd function: 1 , what is f (x)? x2 + 1 1 (b) If h(x) = x 2 − 1 and h(l(x)) = 4 − 1, what is l(x)?
The resulting graph will be the tion f x + 2 graph of cos x, so this problem illustrates the identity sin x + π 2 stricted function, and then sketch its inverse f −1 (x). 3 35 Algebraic Functions ALGEBRAIC FUNCTIONS Power functions, polynomial functions, and rational functions Functions that involve absolute values Domains, properties, and graphs of algebraic functions Power Functions A function is algebraic if it can be expressed in terms of constants and a variable x by using only arithmetic operations (+, −, ×, and ÷) and rational constant powers of the variable.