DEDAN KIMATHI UNIVERSITY OF TECHNOLOGY
UNIVERSITY EXAMINATION 2019/2020 ACADEMIC YEAR
FIRST YEAR FIRST SEMESTER EXAMINATION FOR THE DEGREE OF
BACHEROR OF SCIENCE IN INDUSTRIAL CHEMISTRY, BACHEROR OF
SCIENCE ENGINEERING (CIVIL, ELECTRICAL AND ELECTRONICS,
MECHANICAL,
CHEMICAL,
GEOLOGY,
MECHATRONICS,
TELECOMMUNICATION AND INFORMATION, GEOMATIC AND GEOSPATIAL
INFORMATION
SYSTEMS),
COMPUTER
SCIENCE,
INFORMATION
TECHNOLOGY, BTECH CIVIL, BTECH MECHANICAL, BTECH ELECTRICAL.
SMA 1117 CALCULUS I
Answer question ONE and any other Two Questions
Question One [30mks]
a) Evaluate lim
4− 𝑥2
.
[4mks]
𝑥→2 3− √𝑥 2 +5
b) Find the domain of the function 𝑔(𝑥) = √6 + 𝑥 − 𝑥 2
c) Find the derivative of 𝑓(𝑥) =
1
√1−𝑥
[4mks]
from first principles
d) Find the valuesof c and d given that the function
3𝑥 2 − 1 𝑖𝑓 𝑥 < 0
𝑓(𝑥) = {𝑐𝑥 + 𝑑 𝑖𝑓 0 ≤ 𝑥 ≤ 1 is continuous everywhere
√𝑥 + 8 𝑖𝑓 𝑥 > 1
e) Use the chain rule to determine
𝑑𝑦
given that𝑦 =
𝑑𝑥
[4mks]
[4 mks]
𝑢2 − 1
and
𝑢2 + 1
3
𝑢 = √( 𝑥 2 + 2)[5mks]
f) Given that 𝑓(𝑥) =
1
3− 𝑥 2
and 𝑔 (𝑥) = √𝑥 2 − 1 find the inverse of 𝑓𝑔(𝑥).
[5mks]
g) Verify that Rolles theorem can be applied on the function
𝑓(𝑥) = 𝑥 2 − 3𝑥 + 2 on the interval [1, 2][4mks]
Question Two
[20mks]
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a) The parametric equations of a curve are 𝑥 =
3𝑡
1+𝑡
and 𝑦 =
of the normal to this curve at 𝑡 = 2
b) Give an 𝜀 − 𝛿 proof of the fact that lim 2𝑥 − 5 = 3
c) Find lim
1+𝑡
. Find the equation
𝑥 →4
4𝑥−1
[5mks]
[5mks]
[5mks]
𝑥 → ∞ √𝑥 2 +2
d) Given that 𝑓(𝑥) =
i)
ii)
𝑡2
5𝑥 2 +7𝑥−6
𝑥 2 −4
determine
The vertical asymptotes
the horizontal asymptotes
[5mks]
Question Three [20 marks]
Find the derivatives of the following functions
a) 𝑦 =
sin 2𝑥
2𝑥+5
−2𝑥
[3mks]
3
b) 𝑦 = 𝑒
ln(𝑥 + 5𝑥)
c) 𝑦 = (2𝑥 + 1)5 (𝑥 4 − 3)6
[5mks]
[4mks]
(𝑥+2)2 (√𝑥−3)
d) 𝑦 = (2𝑥−1)2 (𝑥−3)4
[5mks]
e) 𝑦 = 3tan 𝑥
[3mks]
Question Four [20mks]
a) A manufacturer needs to make a cylindrical can that can hold 1.5 litres of
liquid. Determine the dimensions of the can that will minimize the
amount of material used in its construction
[5mks]
b) Use the intermediate Value Theorem to show that 𝑔(𝑥) = 2𝑥 3 − 5𝑥 2 −
10𝑥 + 5 has a root somewhere in the interval [-1 , 2 ]
[5mks]
4
5
c) Find the stationary points of the function 𝑦 = 5 𝑥 − 𝑥 and distinguish
between them
[5mks]
d) Find the equation of the tangent line to the curve 𝑦 =
(1 ,
1
2
√𝑥
1+ 𝑥 2
at the point
[5mks]
)
Question Five [20mks]
a)
b)
Verify the Mean Value Theorem for𝑓(𝑥) = 𝑥(𝑥 2 − 𝑥 − 2) on [-1 , 1].
[5mks]
3
2
The position of a particle is given by the equation 𝑠 = 𝑓(𝑡) = 𝑡 − 6𝑡 + 9𝑡 where
𝑡 is measured in seconds and 𝑠 in meters.
i)
Find the velocity at time 𝑡
[1mk]
ii)
What is the velocity after 2 seconds
[1mk]
iii)
When is the particle at rest
[2mks]
iv)
Find the acceleration after 3 seconds
[2mks]
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c)
The side of a square of a square is 5 cm. Find the increase in the area of the
square when the side expands by 0.01 cm
[5mks]
d)
Show that lim
|𝑥|
𝑥 →0 𝑥
does not exist
[4mks]
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