IB Maths AA HL 5.6 Differentiation rules

[Maximum number: 5]

The following question explores features of a family of curves. The family is then linked to a homogeneous differential equation.
Consider the curve given by $y=\frac{x\left(x^{2}-16\right)}{x^{2}+16}$ .

(a)

Given $f(x)=\frac{x\left(x^{2}-A\right)}{x^{2}+A}$ , prove that $f^{\prime}(\sqrt{A})$ is independent of A.

[ 4 ]

attempt to use quotient rule or product rule
$f^{\prime}(x)=\frac{\left(x^{2}+A\right)\left(3 x^{2}-A\right)-2 x\left(x^{3}-A x\right)}{\left(x^{2}+A\right)^{2}} \quad\left(f^{\prime}(x)=\frac{2 x^{2}+\left(x^{2}-A\right)}{x^{2}+A}-\frac{2 x^{2}\left(x^{2}-A\right)}{\left(x^{2}+A\right)^{2}}\right)$
(or equivalent)

Note: Award A1 for numerator, A1 for denominator.
Note: If product rule used, award A1 for each correct term.
Note: Award (M1) for an attempt to use quotient rule or product rule with A=16 or any other constant. Award no further marks.

f^{\prime}(\sqrt{A})\left(=\frac{(2 A)(2 A)-0}{(2 A)^{2}}\right)=1

hence independent of A

(b)

Write down the coordinates of a point on the curve where the oblique asymptote is parallel to the tangent to the curve at that point.

Now consider the differential equation $x^{2} \frac{\mathrm{~d} y}{\mathrm{~d} x}=x(x+y)-y^{2}$ , where $x \neq 0, y \neq \pm x$ .
Using the substitution y=v x, the differential equation can be written as $x \frac{\mathrm{~d} v}{\mathrm{~d} x}=1-v^{2}$ .

[ 1 ]

$(\sqrt{A}, 0) \mathrm{OR}(-\sqrt{A}, 0)$

IB Maths AA HL Sl 5 6 Differentiation Rules Questions

Exam points

IB Maths AA HL Sl 5 6 Differentiation Rules Questions question 1

Question (a)

Question (b)