Problem Sets 7 through 14

Exercise 7 – Limits - L'Hôpital's Rule
1. Calculate the following limits (explain your answers):
A.
B.
C.
D.
E.
F.
G.
H.
I.
J.
K.
L.
M.
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2. What are the values of a and b such that the function f(x) will be continuous at
every point?
2
Exercise 8 – Maximum and Minimum and Function Review
1. Investigate and draw the following functions:
A.
B.
C.
D.
E.
2. We will define a total cost (TC) function, which is the total of the cost for
production of q products:
Fixed cost
Variable cost
Average total cost
Average fixed cost
Average variable cost
Marginal cost
Given the function of the total cost: TC(q) = q3 – 3q2 + 10q + 5
a. Find the functions: MC, AVC, AFC, TVC, TFC.
b. Find the ranges of the increase and decrease of the functions you obtained in
section a.
3. Prove:
a. If f(x) and g(x) are (real-valued) increasing functions then f + g is also realvalued increasing.
b. Prove that if f is real-valued increasing and g is (real-valued) decreasing, then
the function f – g is real-valued increasing.
4. Give an example for two functions f and g, that are real-valued decreasing, and
where fg is real-valued increasing.
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5. Prove that if f and g are (real-valued) increasing, then the compound function
g[f(x)] is also (real-valued) increasing.
6. The properties that maintain a transformation curve y = f(x) when x is the quantity
produced of one of the products and y is the quantity produced of the other
product are:
a. f(x)  0
b. f’(x)  0
c. f’’(x)  0
d. Which of the following functions can describe a transformation curve?
7. a. Prove that the sum of two convex functions is a convex function.
b. Prove that the sum of two concave functions is a concave function.
8. The elasticity of function f(x) at each point x will be labelled (x) and is defined
as
. Its meaning is the ratio between the relative change in f(x) and the
relative change in x.
(The price elasticity is defined as
).
There are 100 items in the market, where the demand curve for each of them is
D(p): q = 4 – 0.2p.
a. Find the elasticity of the demand of each item at a price of P = 3.
b. Find the elasticity of the demand of each item at a price of P = 10.
c. Find the demand function of the entire market.
d. Find the elasticity of the demand of the market at a price of P = 3.
9. We will indicate f, g, h as the elasticities of functions f, g and h. Express h
through f and g for the following cases:
A.
c is a constant
B.
C.
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10. The demand function for heaters is: q = 42 – 3p.
a. Find the quantity of q for which the producer will have the maximal revenue.
b. What is the producer’s maximal revenue if his cost function is c = 2q2?
11. Prove that the demand function
(where a, b and c are constant and
positive) is decreasing and convex. Does the marginal revenue (MR) function
have the same properties?
12. A radio producer sells x radios per week at a price of p per radio. The demand
function for the week is:
. The price of the production of x radios is:
.
a. What is the optimal quantity of radios that it is worthwhile for the producer to
sell per week?
b. Assume that the government levies a tax t on each radio that the producer
sells. The producer adds the tax to the production costs. What will now be the
optimal amount? What should the value of t be so that the state's income will
be maximal?
13. The total production price of x radios per day is 0.25x2 + 35x + 25, and the price
of one radio when it sells is 50 – 0.5x.
a. What should be the scope of the daily production in order to obtain the
maximal total profit?
b. Show that at the optimum point the price of the production of one radio is
minimal. Is this rule always true?
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Exercise 9 – Differential and Integral
1. Use a differential to estimate 522 if you know that 502 = 2500. Test the deviation
from the accurate value of 522.
2. With the help of a differential, estimate 15 and 3.0013.
3. Calculate the following integrals:
A.
B.
C.
D.
4. Use a specific integral to calculate the area bounded by the straight line y = 2x +1,
the x axis, the y axis and the straight line x = 2.
5. Calculate the area bounded by: y = 3x2, the y axis and the straight line y=3.
6. A tangent is passed at the intersection between the curve y = 2x3 + 3x2 + 4 and the
y axis. Calculate the area that is bounded between the tangent and the line of the
curve.
7. Tangents to the line of the curve y = x2 – 6x + 9 pass through the point (5,0).
Calculate the area that is bounded between the tangents and the parabola.
8. Calculate the integral
. What is the geometrical significance of your answer?
What is the area that is bounded between the y = x3 curve and the x axis in the
section [-1,1]?
9. What, in your opinion, is the value of the integral
Do the same for:
6
? Explain your answer.
10. Calculate the following integrals:
A.
K.
B.
L.
C.
M.
D.
N.
E.
O.
F.
P.
G.
Q.
H.
I.
R.
J.
S.
T.
11. Given the demand function:
And the supply function:
a. Calculate the surplus for the consumer at the equilibrium points.
b. A sales tax of NIS 3 is levied per unit. The tax is paid by the producer.
Calculate the surplus of the consumer at the equilibrium points, and show
that it is lower than the surplus of the consumer prior to the levying of the
tax.
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Exercise 10 – Multivariable Functions: Basic Concepts and Limits
1. For each of the following functions, determine the boundary of its definition and
present it graphically:
A.
B.
C.
D.
2. For each of the following functions, draw the equivalent curves for the integers
from -3 to +3:
A.
B.
C.
D.
E.
F.
G.
H.
I.
In this function, for the
values 1 and e3
J.
3. For each of the following limits, determine whether the limit exists, and calculate
it:
A.
B.
C.
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D.
E.
F.
4. For each of the following functions, determine whether it is continuous at point
(0,0):
A.
B.
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Exercise 11 – Derivatives and Differentials in Multivariable Functions
1. Calculate all of the partial first and second order derivatives of the following
functions:
A.
B.
C.
D.
E.
2. Prove that the function z = y  ln(x2 – y2) fulfils:
3. Given the function f(x,y), where
exists.
, prove that
4. Make an approximate calculation of the following expressions, using a
differential:
a. 3.013 + 3  2.98
b. (1.012  0.98)1/15
5. Make an approximate calculation, using a differential, of the change in the
hypotenuse of a right-angled triangle whose perpendiculars are 6 and 8, when the
short perpendicular is elongated by 1/4 and the long perpendicular is shortened by
1/8.
6. Given the implicit function x2y – xy2 + x2 + y2 = 0. Calculate dy/dx and dx/dy.
7. Given the curve 5x – 2y + y3 – x2y = 0. Calculate the equation of the straight line
that is tangential to the curve at the origin.
8. Given a box whose base is a square with a side x and height y. Calculate dx/dy
when the surface area remains unchanged.
9. Calculate z/x and z/y when it is given that exy + eyz + exz = 20.
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10. Prove that the following exists in every function (x,y) = 0:
And illustrate this on x3 – y3 – 3xy = 0
11. In the implicit function z3 – xy – y = 0, prove that the following exists:
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Exercise 12 – Maximum and Minimum in Multivariable Functions
1. Calculate the maximum and minimum points in the following functions:
A.
B.
C.
D.
E.
2. Prove that the function f(x,y) = ex + x  y + ey has no maximum and minimum
points.
3. A company produces two types of chocolate. The production costs are NIS 5 per
kg of the first type, and NIS 6 per kg of the second type. If the chocolates are sold
at prices P1 and P2 per kg, respectively, then the amount sold in a week are:
x1 = 5(P2 – P1) ; x2 = 40 + 5P1 – 10P2
Determine P1 and P2 such that the company’s revenue will be maximal.
4. Given the function
, subject to the constraint:
, find points that are candidates for being extrema.
5. Given the function f(x,y) = x2 + y2; subject to the constraint
,
where a2>b2.
Prove that the values of the function that are candidates for being extrema are a2
and b2.
6. Find values that are candidates for being extrema in the following functions:
A.
B.
7. Calculate values that are candidates for being extrema of Z for:
2x2 + 3y2 + 3z2 – 12xy + 4xz = 85
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8. A rectangular box that is open at the top must have a volume of 32 cm3.
What must its dimensions be so that its total area will be minimal? Solve the
question in two ways:
a. Using Lagrange multipliers.
b. By substituting the constraint in the target function.
9. Reuben and Shimon consume two products. If x and y are the quantities of the
products (respectively), then Reuben’s utility function is: U(x,y) = ln x + 2 ln y;
and Shimon’s is U(x,y) = x y2. The prices of the products per unit are: Px = 5 and
Py = 2. Reuben and Shimon have identical incomes, of NIS 90 each. Find the
optimal consumed quantity from each of the two products by each of them.
10. A firm committed to supply 10 units of product z. It produces this product using
the factors of production x and y, where the production function is: Z = 2xy (x, y
and z also represent the respective quantities of each). The price of a unit of x is
fixed and is equal to 5, whereas the price of y depends on the quantity consumed
according to Py = y/2. What are the optimal quantities of x and y which the firm
will use?
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Exercise 13 – Homogeneous Functions
1. Determine which of the following functions are homogeneous, for the
homogeneous functions find the degree of homogeneity:
A.
B.
C.
D.
E.
2. Let f and g be functions with n variables, f is homogeneous of degree r1 and g is
homogeneous of degree r2. For each of the following functions, determine
whether it is homogeneous, and at what degree:
a. f  g
b. f/g
c. f + g
3. The function f(x,y) is homogeneous of degree 2, the functions x = g(s,t) and
y = h(s,t) are homogeneous of degree 3. We will define w(s,t) = f(g(s,t),h(s,t).
Determine whether w is an homogeneous function and at what degree.
4. Let function f(x,y) be an homogeneous function of degree r, then prove that the
sum of the elasticities of f according to x and according to y is equal to r.
5. Let f(x,y) be an homogeneous function of degree 1. Prove that the following
exists:
x2  fxx + 2xy  fxy + y2  fyy  0
6. f(x,y) is homogeneous of degree 0, prove that the following exists at each point:
7. Given the function
functions in order to prove the following:
a.
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. Use the properties of homogeneous
b.
8. f(x,y) is homogeneous of degree 3 and fulfils f(6,9) = 54. g(t) is a function with
one variable which is equal at every point to f(0.5t2, 0.25t3 + 1). Calculate g’(2).
9. f(x,y) is homogeneous of degree 3 and fulfils f(2,-1) = 4.
Calculate fx(8,-4) – 2  fy (4,-2).
10. Let g be a function with one variable, and let f be a function with two variables
which fulfils:
a. Prove that f is homogeneous and determine the degree of homogeneity.
b. Given that
. Calculate fx(1,1) + fy(1,1)
11. f(x,y) is an homogeneous function that fulfils: fx(-2,1) = 30, f(4,-2) = 120,
f(-2,1) = 15. Calculate the degree of homogeneity of f and of fy(4,-2).
12. Let f(x,y) be an homogeneous function of degree r.
a. Is ln f(s,y) a homogeneous function?
b. Prove that at point (x,0), the following is fulfilled:
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Exercise 14 – The Double Integral
Calculate the following integrals:
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