Exam-style Questions: Differentiation

1. A curve C has the equation

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a) (i) Show that:

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(ii) Hence find the coordinates of the stationary point on the curve C

(iii) Show that this stationary point is a point of inflection.

b) (i) Show that:

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where a and b are constants to be determined.

(ii) Deduce that the curve has another point of inflection.

c) Sketch the curve C, indicating the two points of inflection.

(Marks available: 11)

Answer

Answer outline and marking scheme for question: 1

Give yourself marks for mentioning any of the points below:

a) (i) Using the product rule gives:

dy/dx = 2xe-x - (x2 +1)e-x

= -e-x (x -1)2

(ii) dy/dx = 0 at stationary points, thus:

0 = 2xe-x - (x2 +1)e-x

This gives stationary point at (1, 2e-1)

(iii) dy/dx (i.e. the gradient) remains the same sign either side of the stationary point. Therefore the stationary point must be a point of inflection.

Mathematically shown below:

dy/dx is less than zero before the stationary point (i.e. at x less than one)

dy/dx equals zero at the stationary point

dy/dx is less than zero after the stationary point (i.e. at x more than one)

(5 marks)

b) (i) Using the product rule on dy/dx gives:

d2y/dx2 = 2e-x - 2xe-x - 2xe-x + (x2 +1)e-x

= e-x (x -1)(x -3)

Therefore a = 1 and b = 3.

(4 marks)

The gradient is always less than zero, so the curve slopes downwards.

From the above there are two points of inflections at x = 1 and x = 3.

The equation never becomes negative, therefore the curve does not cross the x axis.

Therefore the curve looks like:

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(2 marks)

(Marks available: 11)

2. A piece of wire, of length 20cm, is to be cut into two parts. One of the parts, of length x cm, is to be formed into a circle and the other part into a square.

a) Show that the sum, A cm2, of the areas of the circle and the square is given by

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b) Show that A has a stationary value when

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(Marks available: 8)

Answer

Answer outline and marking scheme for question: 2

Give yourself marks for mentioning any of the points below:

a) Area of circle

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Side of square

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Entering these to find a total area, gives:

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(3 marks)

b) Differentiating the area equation above, gives:

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Solving this equation when dA/dx = 0, gives:

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(5 marks)

(Marks available: 8)

3. The variables x and y are related by

y = 4x.

a) Find the value of x when y = 12, giving your answer to two decimal places

b) Show that y = ekx, where k = In 4.

c) Hence find dy/dx.

d) Given that x is a function of a third variable t and that

dy/dx = x, deduce that dy/dx = 12 ln 12, when y = 12.

(Marks available: 7)

Answer

Answer outline and marking scheme for question: 3

Give yourself marks for mentioning any of the points below:

a) 4x = 12, using logs on either side, gives:

x log 4 = log 12

Solving this for x gives, x = 1.79 to 2d.p.

(2 marks)

b) We know that 4 = eln4 therefore, multiplying both side by the power of x, gives:

4x = (eln4)x

From this you can deduce:

y = ekx where k = ln 4.

(1 mark)

c) dy/dx = kekx = (ln 4)exln4 = (ln 4).4x = yln4.

(1 mark)

d) dy/dt = dy/dx . dx/dt

= (y ln 4).x

= (y ln 4).(ln 12/ ln 4)

When y = 12, from (a)

= 12 ln 12

(3 marks)

(Marks available: 7)