W1-2-60-1-6
JOMO KENYATTA UNIVERSITY
OF
AGRICULTURE AND TECHNOLOGY
University Examinations 2014/2015
EXAMINATION FOR THE DEGREE OF BACHELOR OF SCIENCE IN ACTUARIAL
SCIENCE/ BACHELOR OF SCIENCE IN MATHEMATICS AND COMPUTER
SCIENCE/BACHELOR OF SCIENCE IN INFORMATION TECHNOLOGY
STA 2104/SMA 2101: CALCULUS FOR STATISTICS I/CALCULUS I
DATE: August 2015
TIME: 2 HOURS
Instructions: Answer Question One (Compulsory) and Any Other Two Questions.
QUESTION ONE (30 MARKS)
a) Evaluate the following limits
i) lim
4−𝑥 2
𝑥→2 3−√𝑥 2 +5
ii) lim
6𝑥 2 +2𝑥+1
𝑥→∞ 6𝑥 2 −3𝑥+4
(4marks)
(2marks)
b) Find the derivative of the function 𝑓(𝑥) = √2𝑥 + 1 from first principles
(5marks)
c) A manufacturer determines that t months after a new product is introduced to
the market, 𝑥(𝑡) = 𝑡 2 + 3𝑡 hundred units can be produced and then sold at a
3
price of 𝑝(𝑡) = −2𝑡 ⁄2 + 30 dollars per unit
i) Express the revenue (𝑅(𝑡) for this product as a function of time
ii) At what rate is revenue changing with respect to time after 4 months?
Is the revenue increasing or decreasing at this time?
(5marks)
d) Find an equation for the tangent line to the graph 𝑓(𝑥) = 𝑥 − 𝑙𝑛(√𝑥) at the
point where 𝑥 = 1
(5marks)
1|Page
2
e) Evaluate ∫ [ 3 − 6√𝑥] 𝑑𝑥
(3marks)
√𝑥
f) The total sales (in thousands of games) for a home video t months after the
125𝑡 2
game is introduced is given by 𝑠(𝑡) = √100+𝑡 2
i) Find 𝑆 ′ (𝑡) and simplify
ii) Find S(10) and 𝑆 ′ (10) and interpret the results.
QUESTION TWO (20 MARKS)
dy
a) Find
if 𝑥 5 + 4𝑥𝑦 3 − 3𝑦 5 = 2
dx
dy
b) Find given that 𝑥 = 𝑐𝑜𝑠 (3𝑡) and 𝑦 = 𝑠𝑖𝑛 (𝑡 2 + 1)
dx
dx
c) Find dt given that
(4marks)
(2marks)
(3marks)
(4marks)
dx
𝑡 𝑙𝑛 𝑥 = 𝑥𝑒 𝑡 − 1 and evaluate dt at (𝑡, 𝑥) = (0,1)
(5marks)
d) When the price of a commodity is p dollars per unit, the manufacturer is
willing to supply x thousand units where 𝑥 2 − 2𝑥 √𝑝 − 𝑝2 = 31
How fast is the supply changing when the price is 9 dollars per unit and is
increasing at the rate of 0.2 dollars per week?
(8marks)
QUESTION THREE (20 MARKS)
dy
a) Find dx if 𝑦 = 𝑙𝑛 (𝑥 + √𝑥 2 + 1)
(𝑥−1)
dy
b) Find dx given that 𝑦 = 𝑡𝑎𝑛−1 (𝑥+1)
(6marks)
(7marks)
c) Find an equation for the tangent to the curve 𝑥𝑠𝑖𝑛2𝑦 = 𝑦𝑐𝑜𝑠2𝑥 at the point
𝜋 𝜋
(4 , 2 )
(7marks)
QUESTION FOUR (20 MARKS)
a) Calculate the area bounded by the x-axis and the parabola 𝑦 = 6 − 𝑥 − 𝑥 2
(4marks)
𝑥
b) Evaluate ∫ √4−𝑥 2 𝑑𝑥
(4marks)
c) The population growth p(t) of a bacterial colony t hours after observation
𝑑𝑝
begins is found to be changing at the rate 𝑑𝑡 = 200𝑒 0.1𝑡 + 150𝑒 −0.03𝑡
If the population was 200,000 when the observations began, what will the
population be 12 hours later?
(6marks)
𝑥
d) If y=√1+𝑥 2, prove that
(1 + 𝑥 2 )
2|Page
𝑑𝑦 2
𝑑𝑥 2
𝑑𝑦
+ 3𝑥 𝑑𝑥 = 0
(6marks)