Review practice · Section 2 of 2

· Techniques of Differentiation

$C:y=(x^2+a)/(x-a)$, x≠a, where a is a non-zero constant.

(i) Given the oblique asymptote y=x+2, find a.1 mark

(ii) Find the set of values of a if C has two turning points.3 marks

(iii) Using a from (i) and an algebraic method, show C cannot exist for y₁<y<y₂, determining y₁,y₂ exactly.3 marks

(iv) Hence sketch C, clearly showing the asymptotes and stationary-point coordinates.3 marks

$C:y=f(x)$ where $f(x)=(4x^2+bx+c)/(4x+8)$ and b,c are real constants. C has oblique asymptote y=x−5.

(i) Find b.2 marks

(ii) Given c>9, show C does not cut the x-axis.1 mark

(iii) For c=13 sketch C, clearly indicating asymptotes, axial intercepts and turning points.3 marks

$C:y=(-2x^2-2x+a)/(x+1)$, where a is a non-zero constant. C exists for y≤−2 or y≥b, where b is positive.

(i) Using an analytical method, show a=−2 and hence find b.5 marks

(ii) Sketch C, clearly showing turning-point coordinates and asymptotes.3 marks

(iii) Show(−3,6) lies on y=px+3 p+6 for every real p. Hence use the sketch in(ii) to find p for which $px+3p+6=(-2x^2-2x-2)/(x+1)$ has two distinct real roots.2 marks