Curve Sketching 2.7 Geometrical Application of Calculus

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Curve Sketching 2.7 Geometrical Application of Calculus a) Find stationary points. (f’(x)=0) b) Find points of inflection. (f”(x)=0) c) Find intercepts on axes. (y=0 and x=0) d) Find domain and range. e) Find the asymptotes and limits. f) Use symmetry, odd and even functions.

Curve Sketching 2.7 Geometrical Application of Calculus 1. Sketch the curve y = 2x3 + 1. a) Find the stationary point f(x) = 2x3 + 1 f’(x) = 6x2 Stationary point when 6x2 = 0 i.e. x = 0 f(0) = 2(0)3 + 1 = 1 Stationary point at (0, 1)

Curve Sketching 2.7 Geometrical Application of Calculus 1. Sketch the curve y = 2x3 + 1. b) Determine its nature f”(x) = 12x f”(0) = 12(0) = 0 (Check concavity) x -1 0 +1 f”(x) -12 0 +12 Concavity changes Horizontal line of inflection at (0, 1)

Curve Sketching 2.7 Geometrical Application of Calculus 1. Sketch the curve y = 2x3 + 1. c) Find intercepts on axes. (y=0 and x=0) y-intercept(s) when x = 0 y = 2(0)3 + 1 = 1 x-intercept(s) when y = 0 0 = 2x3 + 1 2x3 = -1 x3 = -0.5 x = -0.8

Curve Sketching 2.7 Geometrical Application of Calculus 1. Sketch the curve y = 2x3 + 1. d) Find domain and range. Domain and range are the Reals. e) Find the asymptotes and limits. An asymptote is line that the graph approaches but never reaches. We use limits to show this but it is not applicable here.

Curve Sketching 2.7 Geometrical Application of Calculus 1. Sketch the curve y = 2x3 + 1. f) Use symmetry, odd and even functions. For even functions f(x) = f(-x) Symmetrical about y-axis For odd functions f(-x) = -f(x) Rotational symmetry (180o) f(x) = 2x3 + 1 f(-x) = -2x3 + 1 (Not even) f(-x) = -2x3 + 1 -f(x) = 2x3 - 1 (Not odd)

Curve Sketching 2.7 Geometrical Application of Calculus 1. Sketch the curve y = 2x3 + 1.