TAILIEUCHUNG - Ebook Higher engineering mathematics (5th edition): Part 2
(BQ) Part 2 book "Higher engineering mathematics" has contents: Differentiation of parametric equations, differentiation of implicit functions, partial differentiation, standard integration, integration using partial fractions, some applications of integration,.and other contents. | Differential calculus 29 Differentiation of parametric equations Introduction to parametric equations Certain mathematical functions can be expressed more simply by expressing, say, x and y separately in terms of a third variable. For example, y = r sin θ, x = r cos θ. Then, any value given to θ will produce a pair of values for x and y, which may be plotted to provide a curve of y = f (x). The third variable, θ, is called a parameter and the two expressions for y and x are called parametric equations. The above example of y = r sin θ and x = r cos θ are the parametric equations for a circle. The equation of any point on a circle, centre at the origin and of radius r is given by: x 2 + y2 = r 2 , as shown in Chapter 14. To show that y = r sin θ and x = r cos θ are suitable parametric equations for such a circle: (e) Cardioid x = a (2 cos θ − cos 2θ), y = a (2 sin θ − sin 2θ) (f) Astroid (g) Cycloid x = a cos3 θ, y = a sin3 θ x = a (θ − sin θ) , y = a (1− cos θ) (a) Ellipse (b) Parabola (c) Hyperbola (d) Rectangular hyperbola (e) Cardioid (f) Astroid Left hand side of equation = x 2 + y2 = (r cos θ)2 + (r sin θ)2 = r 2 cos2 θ + r 2 sin2 θ = r 2 cos2 θ + sin2 θ = r 2 = right hand side (since cos2 θ + sin2 θ = 1, as shown in Chapter 16) Some common parametric equations The following are some of the most common parametric equations, and Figure shows typical shapes of these curves. x = a cos θ, y = b sin θ x = a t 2 , y = 2a t x = a sec θ, y = b tan θ c (d) Rectangular x = c t, y = t hyperbola (a) Ellipse (b) Parabola (c) Hyperbola (g) Cycloid Figure Differentiation in parameters When x and y are given in terms of a parameter, say θ, then by the function of a function rule of DIFFERENTIATION OF PARAMETRIC EQUATIONS dx = 2 cos t dt From equation (1), differentiation (from Chapter 27): x = 2 sin t, hence dy dθ dy = × dx dθ dx It may be shown that this can be written as: dy dy dθ = dx dx dθ (1) For the second .
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