2u 25 may revision

7/28/2019 2u 25 May Revision

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Integration

1. A bowl was designed by rotating the section of the curve y = 14 x2 between x = 2 and

x = 12 centimetres, about the y axis

(a) Calculate the volume of the bowl, leaving your answer in terms of

Ans: 2590m 3

(b) Hence calculate the capacity of the bowl, correct to the nearest litre (1 litre=1000cm3 )

Ans: 8L

2. Use Simpsons Rule with ve function values to nd an approximation for the value of

1

010x . Give your answer correct to three decimal places.

Ans: 3.911

3. Consider the region bounded by the curve y = ( x 1)3 , the x and y axes and the linex = 4. Calculate the area of that region.

Ans:412

u2

4. The line y = 2 x + 5 cuts the parabola y = x 2 2x at A and B . The parabola intersectsat the x axis at the origin O and at N .

(a) Find the coordinates of A, B and N .

Ans: A(5,15), B(-1,3) and N(2,0)

(b) Calculate the area contained by the line y = 2 x + 5 and the parabola y = x 2 2x .

Ans: 36 u2

5. The portion of the curve y = x 3 between x = 0 and x = 2 is rotated about the y axis.Find the volume of the solid of revolution formed.

Ans:965

u3

Probability6. The probability that the school bus runs late on any particular day is 1 in 8. Find the

probability that on three successive days, the bus is:

(a) late on all three days

Ans: 1512


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(b) late on exactly two days

Ans: 21512(c) late on at least two days

Ans: 11512

(d) on time on all three days

Ans: 343512

Series and Sequences

7. Pauline wishes to invest in a superannuation fund. She decides to invest $2400 in thefund at the beginning of each year. The fund is paying interest at 9% per annum,compounded annually.

(a) Show that the value of the rst $2400 invested when she retires after working for30 years will be $2400(1.09)30 .

(b) Write down similar expressions for the value of the second and third $2400 amountsinvested, at the end of the thirty year period.

(c) Calculate the total value of her investment when she reaches retirement.

Ans: $356,580.52

8. The geometric series 1 x + x 2 ... has a limiting sum of 4. Find the value of x .

Ans: x = 3

49. Farmer Brown has hired a driller to drill a borehole to enable her to have access to the

underground water in her property. The driller quotes a price of $260 for the rst 3metres drilled, $280 for the next 2 metres, $300 for the next 2 metres and so on. Theprice increase by the same amount for each successive 2 metres of borehole drilled.

(a) Show that the cost of drilling the portion from a depth of 25 metres to 27 metresis $500.

(b) Calculate the cost of drilling to a depth of 27 metres.

Ans: $4940

(c) The cost of drilling the borehole to reach water was $12500. nd the total depthdrilled to give access to the water.

Ans: 51m

The Exponential Function

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10. Given thatd

dx(ex

2

) = 2 xe x2

, evaluate 1

0

xe x2

dx .

Ans: 12 (e 1)

11. The region beneath the curve y = e x which is above the x axis and between the lines

x = 0 and x = 1 is rotated about the x axis.(a) Sketch the region.

(b) Find the volume of the resulting solid of revolution.

Ans: 2 (1 e 2 ) u3

12. Find the equation of the tangent to the curve y = 3 x + e x at the point where x = 0.

Ans: y = 2 x + 1

13. The line y = mx is a tangent to the curve y = e3 x . Find m .

Ans: m = 3 e14. Consider the function y = 1

xe x .

(a) For what values of x is this function dened?

Ans: All real numbers except x = 0.

(b) Describe the behaviour of the function as x:

i) approaches zero

ii) increases indenitely

Ans:i) function approaches ii) function approaches (c) Find any stationary points and determine their nature.

Ans: x = 1 is a maximum

(d) Sketch the curve of this function

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2u 25 may revision

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