Given the data in the table above. Use TWO different approximation methods to estimate one value of f(x).

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Question 1:

Given the data in the table above. Use TWO different approximation methods to estimate one value of f(x). Where x  is a decimal number of your choice that satisfies  1≤x≤4. Compare and comment on the efficiency of your chosen methods. (20 marks)

 

Question 2: 

a) Evaluate:

〖lim┬(x→∞)     〗⁡〖(x^2+x+6)/(2+x^2 )〗

 

∫_0^∞▒〖 3+6x dx〗 (10 marks)

b) i) Show that 〖cosh〗^3 x+ 〖sinh〗^3 x=  1/4 e^mx+  3/4  e^nx, where m and n are integers. ii) Hence find 〖cosh〗^6 x- 〖sinh〗^6 x in the form of 

(a cosh⁡(kx)+b)/8

Where a, b and k are integers. (20 marks)

 

Question 3:      

The equation f(x)=3x^5-8 x^2+4 has 3 real roots, 2 of which are positive.

Use TWO different numerical methods taught in lectures to find the negative root starting from (-1) correct to 3 decimal places.

 

It is known that the other 2 positive roots lie between (0.75 and 1.25). Find these roots using any numerical method and confirm your answer using ICT. By the use of diagrams and ICT discuss if any of the methods taught in lectures may be difficult in determining these roots. (50 marks)

[Total 100 marks] 

 

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