Question 1 Part A Teddy J is a manufacturer of dish washing liquid . If his monthly demand function for 750ml size is q = 4000 − 250p and his total cost function is C(q) = 500 + 0.2q. (i) Derive an expression, R(q) for Teddy J ′ s total revenue curve. (ii) Derive an expression, Π(q) for Teddy J ′ s profit function. (iii) Determine whether Teddy J′s profit is increasing or decreasing when he produces 5 hundred, 750ml bottles of dish washing liquid. (iv) How many 750ml bottles of dish washing liquid should Teddy J produce per month if he wishes to maximize his profits.
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
Question 1 Part A
Teddy J is a manufacturer of dish washing liquid . If his monthly demand function for 750ml
size is q = 4000 − 250p and his total cost function is C(q) = 500 + 0.2q.
(i) Derive an expression, R(q) for Teddy J
′
s total revenue curve.
(ii) Derive an expression, Π(q) for Teddy J
′
s profit function.
(iii) Determine whether Teddy J′s profit is increasing or decreasing when
he produces 5 hundred, 750ml bottles of dish washing liquid.
(iv) How many 750ml bottles of dish washing liquid should Teddy J produce
per month if he wishes to maximize his profits.
Question 1 Part B
(b) A firm has an average cost function
A(q) =125 /q + q2 /16 − 4.
where q is the firm′s output.
(i) Determine the level of output for average costs are minimum.
(ii) Hence determine the range of values for which average costs are decreasing.
(iii) What part of the decreasing range is practically feasible?
(iv) Write an equation for the total cost function.
(v) Hence calculate the level of output for which total costs are minimum.
Question 2
(a) The sales of a book publication are expected to grow according to the function
S = 300000(1 − e−0.06t), where t is the time, given in days.
(i) Show using differentiation that the sales never attains an exact maximum value.
(ii) What is the limiting value approached by the sales function?
(b) A poll commissioned by a politician estimates that "t" days after he makes a statement denegrating women,the percentage of his constituency (those who support him at the time he made the statement) that still supports him is given by S(t) =75(t2 − 3t + 25) / t2 + 3t + 25
The election is 10 days after he made the statement.
(i) If the derivative S’(t) may be thought of as an approval rate, derivate the a function for his approval rate.
(ii) When was his support at its lowest level?
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