Paper Code: MATH101-09B(HAM)
2009 B SEMESTER EXAMINATIONS
DEPARTMENT: Mathematics
PAPER TITLE: Introduction to Calculus
TIME ALLOWED: Three Hours
NUMBER OF QUESTIONS
IN PAPER: Section A: SIXTEEN
Section B: SIX NUMBER OF QUESTIONS
TO BE ANSWERED: TWENTY
VALUE OF EACH QUESTION: Section A: 2.5 marks per question Section B: 15 marks per question
GENERAL INSTRUCTIONS: Answer ALL questions in SECTION A (worth 40%) and ANY FOUR questions from SECTION B (worth 60%).
SPECIAL INSTRUCTIONS: NONE
CALCULATORS PERMITTED: NO
CONTINUED SECTION A
(Attempt ALL SIXTEEN questions - worth 40%) 1. If f(x)=eg(x) and g(x)=x!nx determine f(g(x)).
2. Differentiate y=cosx+ex2 +x12.
3. Calculate the first and second derivatives of y=excosx.
4. Differentiate y=!n
(
sin( )
x)
.5. Differentiate y= x x+1.
6. A function is defined implicitly by ex +y2+!n(xy)=5.
Determine dy
dx as a function of x and y.
7. Determine the derivative of y=tan!1x.
8. Evaluate the limit
x!3
x2"6x+9
(x"3) .
9. Evaluate the limit
x!4
(
x2"5)
.10. Find the most general antiderivative of y= f(x)=x!2+ex !xsin(x2).
11. Evaluate the integral xdx 1+x2
( )
0
!
1 .12. If 5x!10
(x!4)(x!1)= A
(x!4)+ B
(x!1), what are A & B?
13. Find
"
tan!d!.14. If
!
excosx dx=ex(Ccosx+Dsinx), find C & D.15. Find
!
xexdx.16. Solve the differential equation dy
dx =2y+3 given that y(0)=1.
SECTION B
(Attempt FOUR questions - worth 60%)
1. How is the idea of continuity defined in terms of the limit concept?
If a function y= f(x) is continuous at x= x0,y= f(x0), does it necessarily have a derivative there? Explain.
Calculate the derivative of y= x3+x from first principles.
Calculate the derivatives of the following functions using any method.
(i) sin2x (ii) xe!nx (iii) xcosx.
2. If g(x) and h(x) are differentiable functions establish the product rule for differentiation, viz d
dx(gh)= dg
dxh+gdh dx. Differentiate the following functions:
(i) y=(x2+1)sinx (ii) ex!n(x) .
Oil spilled from a ruptured tanker spreads in a circle whose area increases at a constant rate of 6 m2 hour. How fast is the radius of the spill increasing when the area is 9m2?
3. Establish the fundamental theorem of Calculus i.e.
f(x)dx=F(b)!F(a)
a
"
bwhere F!(x)= f(x) valid for a"x"b.
Establish that the following integrals are correct.
(i) 1
0 5
!2
"
sinxdx= 15 (ii) 0 sec2!3
"
xdx= 3, #$tan( )
!3 = 3%&.Calculate the derivative d
dx t3dt.
2
!
xCONTINUED 4. Establish the integration by parts formula
u dv
dx
!" #
%
$dx=uv&%
v!"dudx#$dx.Indicate any assumptions you make.
Hence or otherwise find the following indefinite integrals:
(i)
!
xexdx (ii)!
eaxsin(bx)dx.Verify the formula
sin3x
!
dx=13cos3x"cosx+c.How could this be determined by Integration by parts?
(Hint:
!
eaxsin bxdx=eax(P cosbx+Q sin bx) and find P & Q).5. The inverse function ex of the natural logarithm function !nx has a derivative d
dx(ex)=ex. Show that this result is true.
If hyperbolic cosine and sine functions are defined by coshx= 1
2(ex+e!x) sinhx = 1
2(ex !e!x) establish the identities
cosh 2x=cosh2x+sinh2x 1=cosh2x!sinh2x.
A cable is suspended between two poles as shown. Assuming that the equation describing the cable is y=acosh x
a
!"# $
%&,'b(x(b a>0, show that the length of the cable is L=2asinhb
a.
6. The area under the curve of a positive function f(x) for a!x!b can be defined as f(x)dx
a
!
b .
If this area is rotated about the x axis show that the corresponding formula for the volume V is
!(f(x))2dx
a
"
b .How would the surface area of this volume of revolution be calculated?
Find the volume generated by revolving y= 1
2+x2 about the x axis for 0!x!2.
What is the volume obtained by rotating the region bounded by x=y2, about the y axis for 0!x!2?
