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Instructions and questions for section a and three questions from section b of a january 2002 math101 examination. Topics such as finding the natural domain and range of functions, finding general solutions, finding inverse functions, finding limits, differentiating functions, implicit differentiation, finding local extrema, evaluating definite integrals, and using d'alembert's ratio test.
Typology: Exams
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Instructions to candidates
Answer all of section A and THREE questions from section B. The marks shown against questions, or
parts of questions, indicate their relative weights. Section A carries 55% of the total marks.
graphs:
(i) y = x + 2 (ii)
2 y = 1 /( 2 x − 1 ). (4 marks)
x
x f x 1 2
find the corresponding inverse function ()
1 f x
−
. Verify that
1 f f x = x
− (5 marks)
(i) 2
lim
4
→ x
x x
x
(ii) 1
lim
7
→ − x
x
x
(iii) x
x
x
2 tan
1 cos lim
→ π
. (6 marks)
2 x x dx
d −
by differentiation from first principles. (4 marks)
(i)
x y x e
33 = (ii) 2 1
x
x y
= (iii) y = sin(tan( 3 x )). (6 marks)
3 sin 5 5
2 2 x y + x − y =−
at the point (0,1). Find the equation of the tangent to the curve at this point. (6 marks)
f x = x + x − x
2 2
( ) 2 3 1. (5 marks)
x x
−
. Hence find the values of x
which satisfy the equation
7cosh x - 3sinh x = 7.
The hyperbolic tangent function is defined by tanh x = sinh x / cosh x. Show that its inverse function
tanh
−
x
x x 1
ln 2
tanh
1 .
Use this formula to show that
− −
1 1
−
1
2
. (15 marks)
15 Obtain approximate values for the integral
x x dx
1
0
2 1
using (i) the trapezoidal rule and (ii) Simpson’s rule with the interval [0,1] subdivided into eight
equal parts in each case. Give your answers correct to five decimal places. Verify that (ii) is the more
accurate method by evaluating the integral directly and comparing the results. (15 marks)
16 (i) Find the following indefinite integrals
(a) e xdx
x sin
2
(b) , ( 2 2 ).
4
2
3
− < <
−
dx x
x
x
(ii) Find the arc length of the curve
3 / 2 y = x from x = 1 to x = 2 correct to 3 decimal places.
(15 marks)
