The curve 4 y^3=a^2(x+3 y) can be parameterised as x=a cos3 θ, y=a cosθ. (a) Obtain expressions

step 1 The curve 4y cubed is equal to a square times x plus 3y can be parameterized as x is equal to a

The curve 4 y^3=a^2(x+3 y) can be parameterised as x=a cos3...

The curve $4 y^{3}=a^{2}(x+3 y)$ can be parameterised as $x=a \cos 3 \theta, y=a \cos \theta$. (a) Obtain expressions for $d y / d x$ (i) by implicit differentiation and (ii) in parameterised form. Verify that they are equivalent. (b) Show that the only point of inflection occurs at the origin. Is it a stationary point of inflection? (c) Use the information gained in (a) and (b) to sketch the curve, paying particular attention to its shape near the points $(-a, a / 2)$ and $(a,-a / 2)$ and to its slope at the 'end points' $(a, a)$ and $(-a,-a)$.

The curve $4 y^{3}=a^{2}(x+3 y)$ can be parameterised as $x=a \cos 3 \theta, y=a \cos \theta$.
(a) Obtain expressions for $d y / d x$ (i) by implicit differentiation and (ii) in parameterised form. Verify that they are equivalent.
(b) Show that the only point of inflection occurs at the origin. Is it a stationary point of inflection?
(c) Use the information gained in (a) and (b) to sketch the curve, paying particular attention to its shape near the points $(-a, a / 2)$ and $(a,-a / 2)$ and to its slope at the 'end points' $(a, a)$ and $(-a,-a)$.

Key Concepts

Implicit Differentiation

This concept involves differentiating both sides of an equation with respect to one variable while treating other variables as implicit functions of that variable. It is used to find derivatives of equations that are not solved explicitly for one variable, making it crucial when dealing with curves defined by relations rather than functions.

Parameterisation of Curves

Parameterisation represents curves by expressing both x and y in terms of a third variable, typically denoted as a parameter such as ? or t. This approach simplifies differentiation and integration, and is especially useful when investigating curves with complex relationships between x and y or when the curve has symmetry that can be exploited via the chosen parameter.

Inflection Points

An inflection point on a curve is where the curvature changes sign, which means the curve changes from concave upward to concave downward or vice versa. Identifying these points is essential for understanding the geometric properties of the curve and for analyzing its behavior.

Stationary Points

A stationary point on a curve is a point where the first derivative (the slope of the tangent) is zero. In the context of inflection points, a stationary point of inflection is one where not only is there a change in concavity, but the derivative is zero as well, indicating a horizontal tangent at that inflection point.

Curve Sketching

Curve sketching involves using derivative information, intercepts, critical points, and points of inflection to construct an accurate image of the curve's shape. This technique helps in visualizing the behavior of the curve, especially near important points such as maxima, minima, points of inflection, and boundary or 'end' points.

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