MAE-Seminar and Homework 1

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Mathematics Applied to Economics
1st Seminar and Homework
Derivatives. Applications to Economics
(1) Suppose demand Q for a good is a linear function of its price per unit, p. When price is $10,
demand is 200 units, and when price is $15, demand is 150 units. Find the demand function.
Draw the graph of the demand function.
(2) A printing company quotes the price of $1400 for producing 100 copies of a report, and $3000 for
500 copies. Assuming a linear relation, what would be the price of printing 300 copies?
(3) Compute the derivatives of the following functions:
(a) f(x) = 5
(b) f(x) = 5x
(c) f(x) = 7x3 − 2x2 + 3x + 4
(d) f(x) = ex + 7x + lnx
(e) f(x) = e2x
2+3
(f) f(x) = (2x + 1)(5x− 2)
(g) f(x) = ln(5x2 + 3x + 7)
(h) f(x) =
√
9x2 − 1
(i) f(x) =
4x− 3
7x + 2
(4) Determine the linear approximation at a of each function below. Then use the linear approxima-
tion to estimate the value of each function at the given x-value.
(a) f(x) =
√
x; a = 4; x = 3
(b) f(x) =
1
x
; a = 5; x = 5.3
(c) f(x) = x2 + 3; a = 2; x = 2.2
(5) Use the linear approximation to estimate:
(a)
√
49.5
(b)
√
24.5
(c) 3
√
8.2
(d)
√
4.05 +
1√
4.05
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(6) Suppose that the total operating cost of relocating a car 500 km at an average speed of v km/h,
is C(v) = 150 + v +
6000
v
dollars. Find the approximate change in cost when the average speed
is increased from 80 km/h to 85 km/h.
(7) A supermarket determines that their yearly profit P (q) is related to the amount q spent on
advertising by P (q) = −1
6
q2 + 12q + 15; 0 ≤ q ≤ 73. where both P (q) and q are measured in
thousands of dollars. Approximate the change in profit when advertising expenditure is increased
from $30,000 to $32,000.
(8) Let A(x) denote the cost of building a house with a floor area of x square meters. What is the
interpretation of A′(100) = 2500?
(9) The price P per unit obtained by a firm in producing and selling Q units is P = 102 − 2Q, and
the cost of producing and selling Q units is C = 2Q+12Q2. Find the value of Q which maximizes
profits, and the corresponding maximal profit.
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