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**Graph of a Curve Continuity This curve is continuous**

These curves are discontinuous y x y x y x hole gap GAP Smoothness This curve is smooth These curves are not smooth y x y x y x corner cusp

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**Graph of a Curve (cont’d)**

Increasing Decreasing Constant A function f is increasing on an open interval I if, for any x1 and x2 in I, with x1 < x2, we have f(x1) < f(x2). A function f is decreasing on an open interval I if, for any x1 and x2 in I, with x1 < x2, we have f(x1) > f(x2). A function f is constant on an open interval I if, for any x1 and x2 in I, we have f(x1) = f(x2). x y x1 x2 I f(x1) f(x2) y x x1 x2 I f(x1) f(x2) y x x1 x2 I f(x1) f(x2) Local (or Relative) Extrema f(x1) is a local max and also a global max A function f has a local maximum at x = x0 if locally, f(x0) is greater than all the surrounding values of f(x). We call this f(x0) a local maximum of f. A function f has a local minimum at x = x0 if locally, f(x0) is less than all the surrounding values of f(x). We call this f(x0) a local minimum of f. f(x3) is a local max x1 x2 x3 Global (or Absolute) Extrema A function f has a global maximum at x = x0 if f(x0) is greater than or equal to all values of f(x). We call this f(x0) the global maximum of f. A function f has a global minimum at x = x0 if f(x0) is less than or equal to all values of f(x). We call this f(x0) the global minimum of f. f(x2) is a local min

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**Use each graph to find the domain and range of the function:**

-5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -2 -1 1 2 3 4 5 Domain = [–5, –2) (–1, 5] Domain = [–4, 4) Range = (–2, 2] Range = [–3, 1) [2,4) -5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -2 -1 1 2 3 4 5 Domain = (–∞, –3) [–1, 4) Domain = (–, 5) (5, ) Range = (–, –2) (–1, 3] Range = {–3} [–2, 1) (1, )

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**1) Find f(5), f(2), and f(–6): f(5) = 3, f(2) = 1, f(–6) = –1 **

Use the graph of the function f given below to answer the following questions: -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 1) Find f(5), f(2), and f(–6): f(5) = 3, f(2) = 1, f(–6) = –1 12) List the interval(s) on which f is decreasing? (–9, –2) (0, 1) (3, 4) 2) Is f(–1) positive? No f(5)? Yes 13) List the interval(s) on which f is increasing? (–2, 0) (1, 3) (4, 5) 3) Is f(2) negative? No f(–4)? Yes 14) List the interval(s) on which f constant? (5, 6) 4) What is the domain of f ? (–9,6] 15) If any, list all the local maxima? 1, 4 At which x values? 0, 3 5) What is the range of f ? [–3,4] 6) What are the x-intercepts? –7.5, –1, 1 16) If any, list all the local minima? –3, 0, 1 At which x values? –2, 1, 4 7) What is the y-intercept? 1 8) Is f continuous? Yes 17) Is there any global maximum? Yes, 4 global minimum? Yes, –3 9) Is f smooth? No 10) For what value(s) of x does f(x) = 0? –7.5, –1, 1 f(x) = 1? 0, 2, 4 11) How often does the line y = 1 intersect the graph? 3 times y = ½? 4 times y = –2? 2 times

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Copyright © Cengage Learning. All rights reserved. Applications of Differentiation.

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