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MTA611S - MATHEMATICS FOR AGRIBUSINESS - 1ST OPP - JUNE 2023 |
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nAm I BIA un IVERSITY
OF SCIEnCE Ano TECHn OLOGY
FACULTYOF HEALTH,NATURAL RESOURCESAND APPLIEDSCIENCES
SCHOOLOF AGRICULTUREAND NATURALRESOURCESCIENCES
DEPARTMENTOF AGRICULTURALSCIENCESAND AGRIBUSINESS
QUALIFICATION: BACHELOROF SCIENCEIN AGRICULTURE
QUALIFICATIONCODE: 07BAGA
LEVEL:
7
COURSECODE:
MTA611S
COURSENAME: Mathematics for Agribusiness
SESSION:
June 2023
PAPER:
Theory
DURATION:
3 Hours
MARKS:
100
FIRSTOPPORTUNITYEXAMINATION QUESTION PAPER
EXAMINER(S) Mr. Mwala Lubinda
MODERATOR(S) Mr. Teofilus Shiimi
INSTRUCTIONS
1. ANSWERALL questions.
2. Write clearly and neatly.
3. Number the answers clearly & correctly.
PERMISSIBLEMATERIALS
1. All written work MUST be done in blue or black ink.
2. Calculators allowed.
3. The LASTPAGEhas FORMULA.
4. No books, notes and other additional aids are allowed.
THIS QUESTION PAPERCONSISTSOF 6 PAGES(including this front page).
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Mathematics for Agribusiness
MTA611S
QUESTION ONE
[MARKS]
= a. Consider a function, f(x)
x 2 - 4x - 5. Find the range when the domain is one and the
(4)
domain when the range is zero.
b. Use interval notation to express the domain of the function:
2x - 1
g(x)
x2 - 9
(4)
c. Suppose you know that an agribusiness's production can be approximated using a
= = univariate quadratic function with a maxima and roots at x -10 and x 20. Based
on this information answer the following questions below.
i. Derive the algebraic equation for the production function.
(2)
ii. Compute the production function's y-intercept.
(2)
iii. Compute the range and domain value at the maximum point.
(3)
iv. Sketch a well labelled graph to represent the production function. On your graph
(S)
show the roots, y-intercept, and maxima.
d. A vendor's total monthly revenue is from the sale of x bags potatoes is represented by a
function:
r = 150x
Furthermore, the vendor's total month costs are given by c = lOOx + 3500. Compute,
(5)
how many bags of potatoes must the vendor sale to break even? (Hint: break even means
revenue is equal to cost).
TOTAL MARKS
[25)
First Opportunity Question Paper
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June 2023
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Mathematics for Agribusiness
MTA611S
QUESTION TWO
a. Use the Newton's Difference Quotient (or first principle of differentiation) to find the
first derivative of the function:
g(x) = x 2 -4x-5
To obtain full marks, show all the critical steps in your answer.
[MARKS]
(6)
b. Find:
i.
xII'..m.o.
(2+x)2-4
X
(4)
ii.
Jim
k ....6
-2
k-6
(6)
c. Find the equation of a straight-line that is tangent to the curve:
y = In (x 2 - 2x + 24)
(9)
at X = 0.
TOTAL MARKS
[25]
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June 2023
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Mathematics for Agribusiness
MTA611S
a. Consider the functions, f(x)
QUESTION THREE
= = (3x 4 - 5) 6 and g(x) log 8 x 4 . Find:
i. f'(x)
ii. g'(x)
b. Find Zx,Zy and Zyx, given the function:
z = 3e2Xy2
[MARKS]
(3)
(4)
(6)
c. Find the critical points of the function below and test whether it is at a relative
maximum, relative minimum, inflection point, or saddle point. Show all your
calculations.
(12)
z = 3x 3 - Sy2 - 225x + 70y + 23
TOTAL MARKS
(25)
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Mathematics for Agribusiness
MTA6115
a. Find:
QUESTION FOUR
[MARKS]
i. J0\\3x 2 - x - 2) dx
(3)
ii. Jx 2 (x 3 + 2)dx
(5)
b. Suppose an agribusiness's marginal cost function of wheat production is represented
by:
MC=
de
dq
= 250
+ 30q + 9q 2
(7)
where MC is the marginal cost, c is the total cost, and q is the units of output. Find the
cost of producing 10 units of output assuming a fixed cost of N$10,000.
c. To produce 70 tonnes of wheat, an agribusiness wishes to distribute production
between its two farms, farm 1 and farm 2. The total cost of wheat production, c, is given
by the function:
c = 4qf + 2q1 q2 +Sq?+ 1000
(10)
where q1 and q2 are tonnes of wheat produced at farm 1 and farm 2, respectively. How
should the agribusiness distributed to production between the two farms to minimize
costs? Furthermore, compute and interpret lambda (A).
TOTAL MARKS
[25)
THE END
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Mathematics for Agribusiness
MTA611S
FORMULA
'sasic Derivative Rules
Cllnn1ut :tule. ~(c)- 0
ax
i[ Cons1an1 '.\\fulliple Rule
cf(:c)[ - c/'(x)
PowL•r Rull": ~(.-.:-=} - nx '·'
ax
Sum Ruk ~[/(x)-
w:
g(x)[ - / tx)- g'(x)
Di fic«ncc Rulc" ~[ f(x) - g(x)j - f '(x) - g '(x)
ox
Product Ruic· dd [/(x)g(x)j
X
/(x) g'(x)- g(x)F(x)
Chain Ruic- f(g(x)) - / '(g(x))g'(x)
ca
Derivative Rules for Exponential Functions
.:!..__(e'=) e"
dx
-ad(') =a 'I na
dx
.:!..__(e'''>=) e'<'>g '(x)
dx
-d(a,c,J)
dx
= ln(a) a ,c,J g '(x)
Derivative Rules for Logarithmic Functions
-(dIn x) = -I, x > 0
dx
x
~ln(g(x))
dx
= g '(x)
g(x)
-(dlog.
dx
-(dlog.
dx
x)= --,xI
>0
x In a
g(x))=
g1x)·
g(x)lna
Basic Integration Rules
f I. ad.x=ax+C
xr.•1
1. fx"dx= ;,+l+C.
11o0-I
f 3. ~<Lr=h+i+C
f 4. e·' ifr = e' + C
5. fa'lfr=~+C
f Ina
<>. lnxllr =xlnx-x+C
Integration by Substitution
The following are the 5 steps for using the integration by
substitution metthod:
• Step 1: Choose a new variable u
• Step 2: Determine the value dx
• Step 3: Make the substitution
• Step 4: Integrate resulting integral
• Step 5: Return to the initial variable x
Integration by Parts
The formula for the method of integration by parts is:
_/1i=d1lv• v-.lvdu
There are three steps how to use this formula:
• Step 1: identify u and dv
• Step 2: compute u and du
• Step 3: Use the integration by parts formula
Unconstrained optimization: Univariate functions
The following are the steps for finding a solution to an
unconstrained optimization problem:
• Step 1: Find the critical value(s), such that:
f'(a) = 0
• Step 2: Evaluate for relative maxima or minima
o If f "(a) > 0 minima
o If f "(a) > 0 maxima
Unconstrained optimization: Multivariate functions
The following are the steps for finding a solution to an
unconstrained optimization problem:
Condition
FOCs or ncccssarv conditions
SOCs or suflicicni conditions
A/a:rim11m
/1 =h = 0
/11 > 0. h.2 > 0, anJ
fi I :f1.2> ({12)2
Ji =h =0
/11 < 0, /22 < 0, and
/j I /22 > (/'t2) 2
Inflection point
/11 <0,/22 <0, and/11:1:!2 <(f12) 2 or
.h I < 0. /22 < 0. :md/1 J/22 < (li2) 2
S;idJle point
/11 > 0 . ./i2< 0, andf11 /22 < (/i2) 2. or
.Ill <0,.f':!2>0,and/i1/,2
<(/·12J"
lni.:onclusivc
Constrained Optimization
The following are the steps for finding a solution to a
constrained optimization problem using the Langrage
technique:
• Step 1: Set up the Langrage equation
• Step 2: Derive the First Order Equations
• Step 3: Solve the First Order Equations
• Step 4: Estimate the Langrage Multiplier
First Opportunity Question Paper
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June 2023