IBDP Past Year Exam Questions – Application of Differentiation

Q1.  [N08.P1]- 7 marks

Find the equation of the normal to the curve 5xy22y2=18 at the point (1, 2) .

Q2.   [N09.P1]- 7 marks

A certain population can be modelled by the differential equation dydx=k y coskt ,  where y is the population at time t hours and k is a positive constant.
(a)  Given that y=y0 when t=0 , express y in terms of k , t and y0 .
(b) Find the ratio of the minimum size of the population to the maximum size of the population.

Q3.   [M09.P1]-5 marks

The diagram below shows a curve with equation y=1+k sinx , defined for 0x3π .
The point lies on the curve and B a,b is the maximum point.
(a) Show that k=6 .
(b) Hence, find the values of a and b .                                  [3 marks]

Q4.   [N10.P1]- 8 marks

Consider the curve y=xex and the line y=kx,kR (a) Let k=0 .

(l)Show that the curve and the line intersect once.

(ll)Find the angle between the tangent to the curve and the line at the point of intersection.

(b)Let k=1 . Show that the line is a tangent to the curve.

Q5.   [M10.P1]- 8 marks

The function f is defined by fx=ex22x1.5 .

(a) Find fx .

(b) You are given that y=fxx1 has a local minimum at x=a , a>1 . Find the value of a .

Q6.   [M14.P1]- 9 marks

A curve has equation arctanx2+arctany2=π4.

(a) Find dydx in terms of x  and y .

(b) Find the gradient of the curve at the point where x=12 and

y<0 .

Q7.   [N14.P1]- 6 marks

A tranquilizer is injected into a muscle from which it enters the bloodstream. The concentration   C in mgl1 , of tranquilizer in the bloodstream can be modelled by the function Ct=2t3+t2 , t0 where t is the number of minutes after the injection. Find the maximum concentration of tranquilizer in the bloodstream.

Q8.   [M15.P1]- 8 marks

In triangle ABC ,   BC=3cm , AB^C=θ and BC^A=π3 .

(a) Show that length AB=33 cosθ+sinθ .

(b) Given that AB has a minimum value, determine the value of θ for which this occurs.

Q9.   [N15.P1]- 6 marks

Consider the curve y=11x , x , x1 .

(a)Find dydx .

(b) Determine the equation of the normal to the curve at the point x=3 in the form ax+by+c=0 where a,b,c .

Q10.   [M16.P1]- 7 marks

A curve is given by the equation y=sinπcosx .
Find the coordinates of all the points on the curve for which dydx=0 ,   0xπ .