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Math 211: Calculus Ii Question Paper
Math 211: Calculus Ii
Course:Bachelor Of Education (Arts )/ Bachelor Of Education (Science)/ Bachelor Of Science (Computer Science)/ Bachelor Of Science (Economic & Statistics) Bachelor Of Science (General)/ Bachelor Of Arts (Economic & Maths)
Institution: Chuka University question papers
Exam Year:2013
CHUKA
UNIVERSITY
UNIVERSITY EXAMINATIONS
SECOND YEAR EXAMINATIONS FOR THE AWARD OF DEGREE IN BACHELOR OF EDUCATION (ARTS )/ BACHELOR OF EDUCATION (SCIENCE)/ BACHELOR OF SCIENCE (COMPUTER SCIENCE)/ BACHELOR OF SCIENCE (ECONOMIC & STATISTICS) BACHELOR OF SCIENCE (GENERAL)/ BACHELOR OF ARTS (ECONOMIC & MATHS)
MATH 211: CALCULUS II
STREAMS:BED(ARTS&SCI);BSC(COMP.SC);BSC(ECON&STAT);BSC(GEN)BA(ECON&ECON&MATHS Y2S1 TIME: 2 HOURS
DAY/DATE: MONDAY 22/4/2013 8.30 AM – 10.30 AM
INSTRUCTIONS:
ANSWER QUESTION ONE (COMPULSORY) AND ANY OTHER TWO QUESTIONS.
QUESTION ONE (COMPULSORY) (30 MARKS)
(a) Evaluate the following integrals.
(i) ?¦sin??6x cos??7x dx? ? [3 Marks]
(ii) ?_0^6¦?x/(x^2+10) dx? [4 Marks]
(iii) ?¦?xln x dx? [3 Marks]
(b) A particle starts from rest at time t = 0 and moves so that at any time t its acceleration
is t ( 8-3t) units. Find at what time it again comes to rest and the distance it ahs then moved. [6 Marks]
(c) Find the area enclosed by the curve y=x^3-x,the x-axis between x=0 and x=1 [3 Marks]
(d) Given that z=sin??3xy,show that (?^2 z)/?x?y=(?^2 z)/?y?x? [3 Marks]
(e) (i) State the fundamental theorem of integral calculus. [2 Marks]
Hence evaluate
(ii) ?_0^(p/2)¦??cos?^2 dx? [3 Marks]
(iii) ?_(-1)^1¦(x^3-?2x?^2-x+2)dx [3 Marks]
QUESTION TWO (20 MARKS)
(a) Use the trapezoidal rule to approximate ?_0^(p/2)¦?v(sin??x ? ) dx with n=6 where x? is measured in radius and find the error in the approximation. [6 Marks]
(b) (i) State the mean value Theorem. [2 Marks]
(ii) Verify the mean value theorem for the function y=x^2-2x on the interval [0 ,2] [4 Marks]
(c) (i) Find the second order partial derivatives of the function
f(x)=x^4-?3x?^2 y^2+y^4 [4 Marks]
(ii) Evaluate the integral ?¦(?6x?^2+?3y?^2+2)dxdy
R.
Where R is the region bounded by the lines 0=x=1 and 0=y=2.
[4 Marks]
QUESTION THREE (20 MARKS)
(a) (i) Find the 5th degree Taylor series for f(x)=ln??x at x=1? [6 Marks]
(ii) Use the answer in part a(i) above to approximate In 0.6. [2 Marks]
(b) Find the area enclosed by the curves y=x^2-4x+2 and y=2-x^2 [5 Marks]
(c) Evaluate the following
(i) ?¦?1/(3+x^2 ) dx? [3 Marks]
(ii) ?¦??sin?^3 x ?cos?^2 dx? [4 Marks]
QUESTION FOUR (20 MARKS)
(a) The velocity of a train after learning from a station is as follows:
Time in min 0 2 4 6 8 10 12 14 16
Speed in m/min 0 50 110 160 230 290 360 410 470
Use Sampson’s rule to find the distance travelled in the first 16 minutes. [7 Marks]
(b) Find the length of the arc of the curve y=v(x^3 ) from x=1 to x=4 [5 Marks]
(c) Express (?3x?^2-2x-7)/(x^2-x-2) into partial fractions and hence evaluate ?¦(?3x?^2-2x-7)/(x^2-x-2) dx. [8 Marks]
QUESTION FIVE (20 MARKS)
(a) (i) Explain the meaning of a parametric equation. [2 Marks]
(ii) Find the surface area of the solid generated by rotating the curve defined by
x=3 cos?,y=3sin? on the interval 0=?=p/4 about the x –axis.
[5 Marks]
(b) (i) By taking t=tanx, evaluate ?¦?1/(1+?cos?^2 x) dx? [5 Marks]
(ii) Evaluate ?_0^1¦v((1-x)/(1+x) dx) [5 Marks]
(c) The area bounded by y=x^3,x=2,x-axis and x=0 is rotated about the x-axis.
Find the volume of the solid of revolution. [3 Marks]
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