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 clean new textbook is designed to assist teachers do exactly that.
Taalman and Kohn’s Calculus deals a streamlined, established exposition of calculus that mixes the readability of vintage textbooks with a contemporary viewpoint on ideas, abilities, purposes, and thought. Its smooth, uncluttered layout removes sidebars, old biographies, and asides to maintain scholars curious about what’s so much important—the foundational thoughts of calculus which are so very important to their destiny educational careers.
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Extra info for Calculus
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. ) Thinking Forward Evaluations for slopes and derivatives: Evaluate each function at the values indicated. Simplify your answers if possible.
We can answer that question by just looking at what happens when we multiply by −1. Changing f (x) to −f (x) transforms all positive outputs into negative outputs, and vice versa. The graph of y = f (x) is then reflected across the x-axis to become the graph of y = −f (x), as shown in the red graph in the figure that follows. If we instead multiply the independent variable by −1, then we obtain a reflection across the y-axis, as shown in the green graph. −f (x) reflects across the x-axis f (−x) reflects across the y-axis y 6 4 2 Ϫ6 Ϫ4 Ϫ2 2 4 6 x Ϫ2 Ϫ4 Ϫ6 Now if we want to transform f (x) to f (−2x), for example, we can transform f (x) first to f (2x) and then by reflection to f (−2x).
If f (0) = 2, can f be an odd function? What if f (0) is undefined? Explain your answers. 12. Determine graphically whether each of the following four functions is even, odd, or neither. 1 f +g 7. Suppose f is a function with domain [2, ∞) and range [−3, 3], and let g be a function with domain [−10, ∞) and range [0, ∞). (a) What is the domain of the composition g ◦ f ? Justify your answer. (b) It is not possible to determine the domain of f ◦ g in this example; explain why not. What extra information would you have to know to be able to determine the domain of f ◦ g?