Theory question 112
Differentiation(a) Find the equation of the tangent to curve x2/4 + y2 = 1 at point (1, √3/2).
(b) Express (3x + 2)/(x2 + x − 2) in partial fractions.
Show the model answer
Model answer
(a) x/2 + 2y(dy/dx) = 0, so dy/dx = −x/(4y)
At (1, √3/2): dy/dx = −1/(2√3) = −√3/6
y − √3/2 = −(√3/6)(x − 1)
√3x + 6y − 4√3 = 0 (or x + 2√3y − 4 = 0)
(b) x2 + x − 2 = (x − 1)(x + 2); 3x + 2 = A(x + 2) + B(x − 1)
x = 1: 5 = 3A, A = 5/3; x = −2: −4 = −3B, B = 4/3
(3x + 2)/(x2 + x − 2) = 5/[3(x − 1)] + 4/[3(x + 2)]
Explanation
Differentiate implicitly for the gradient; factorise the denominator for the partial fractions.