Curve Sketching2.7 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 Sketching2.7 Geometrical Application of Calculus

1. Sketch the curve y = 2x3 + 1.

f(x) = 2x3 + 1

f’(x) = 6x2

Stationary point when 6x2 = 0i.e. x = 0f(0) = 2(0)3 + 1 = 1

Stationary point at (0, 1)

a) Find the stationary point

Curve Sketching2.7 Geometrical Application of Calculus

1. Sketch the curve y = 2x3 + 1.

f”(x) = 12x

f”(0) = 12(0) = 0 (Check concavity)

x -1 0 +1f”(x) -12 0 +12

Concavity changes

Horizontal line of inflection at (0, 1)

b) Determine its nature

Curve Sketching2.7 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 = 0y = 2(0)3 + 1 = 1

x-intercept(s) when y = 00 = 2x3 + 1

2x3 = -1

x3 = -0.5

x = -0.8

Curve Sketching2.7 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 Sketching2.7 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)

For odd functions f(-x) = -f(x)

f(x) = 2x3 + 1 f(-x) = -2x3 + 1 (Not even)

Symmetrical about y-axis

Rotational symmetry (180o)

f(-x) = -2x3 + 1 -f(x) = 2x3 - 1 (Not odd)

Curve Sketching2.7 Geometrical Application of Calculus

1. Sketch the curve y = 2x3 + 1.