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Graph of a Curve Continuity This curve is continuousThese curves are discontinuous Smoothness This curve is smoothThese curves are not smooth y x y x y.

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Presentation on theme: "Graph of a Curve Continuity This curve is continuousThese curves are discontinuous Smoothness This curve is smoothThese curves are not smooth y x y x y."— Presentation transcript:

1 Graph of a Curve Continuity This curve is continuousThese curves are discontinuous Smoothness This curve is smoothThese curves are not smooth y x y x y x y x y x y x gap hole GAP corner cusp

2 Graph of a Curve (contd) Increasing A function f is increasing on an open interval I if, for any x 1 and x 2 in I, with x 1 < x 2, we have f(x 1 ) < f(x 2 ). Decreasing A function f is decreasing on an open interval I if, for any x 1 and x 2 in I, with x 1 f(x 2 ). Constant A function f is constant on an open interval I if, for any x 1 and x 2 in I, we have f(x 1 ) = f(x 2 ). x y x1x1 x2x2 I f(x1)f(x1) f(x2)f(x2) y x x1x1 x2x2 I f(x1)f(x1) f(x2)f(x2) y x x1x1 x2x2 I f(x1)f(x1)f(x2)f(x2) Local (or Relative) Extrema A function f has a local maximum at x = x 0 if locally, f(x 0 ) is greater than all the surrounding values of f(x). We call this f(x 0 ) a local maximum of f. A function f has a local minimum at x = x 0 if locally, f(x 0 ) is less than all the surrounding values of f(x). We call this f(x 0 ) a local minimum of f. x1x1 x2x2 x3x3 f(x 3 ) is a local max f(x 1 ) is a local max and also a global max f(x 2 ) is a local min Global (or Absolute) Extrema A function f has a global maximum at x = x 0 if f(x 0 ) is greater than or equal to all values of f(x). We call this f(x 0 ) the global maximum of f. A function f has a global minimum at x = x 0 if f(x 0 ) is less than or equal to all values of f(x). We call this f(x 0 ) the global minimum of f.

3 Use each graph to find the domain and range of the function: Domain = [–5, –2) (–1, 5] Range = (–2, 2] Domain = [–4, 4) Range = [–3, 1) [2,4) Domain = (–, –3) [–1, 4) Range = (–, –2) (–1, 3] Domain = (–, 5) (5, ) Range = {–3} [–2, 1) (1, )

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


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