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Calculus
At noon, ship A is 50km west of ship B. Ship A is sailing south at 30km/h and ship B is sailing north at 20km/h. How fast is the distance between the ships changing at 4:00pm.
Answer (in kilometers per hour):
Answer (in kilometers per hour):
Calculus
The price of the average detached single family house P = Pe + Pd consists of two components: the equity component Pe (cash down-payment) and the debt component Pd (amount financed by a loan). The equity component is constant Pe = $200000 whereas the debt component follows an exchange equation MV = PdQ
at every time. Here, Q is the number of existing houses, M denotes the total amount of
mortgages held on the books of the banks, and V denotes the velocity of credit money in the
real estate market. We assume that V = 1=3 is constant. On January 1, 2012, Q is 5 million
units, growing at a rate of 50000 per year, and Pd is $200000.
(a) If the banks are reducing their lending standards such that M increases at a rate of $300 billion per year, at what rate (in percent) does the average house price P increase?
(b) there is no new mortgage lending and the current borrowers only pay back their existing loans, M decreases at a rate of $30 billion per year. Calculate the relative rate of change of the house P.
at every time. Here, Q is the number of existing houses, M denotes the total amount of
mortgages held on the books of the banks, and V denotes the velocity of credit money in the
real estate market. We assume that V = 1=3 is constant. On January 1, 2012, Q is 5 million
units, growing at a rate of 50000 per year, and Pd is $200000.
(a) If the banks are reducing their lending standards such that M increases at a rate of $300 billion per year, at what rate (in percent) does the average house price P increase?
(b) there is no new mortgage lending and the current borrowers only pay back their existing loans, M decreases at a rate of $30 billion per year. Calculate the relative rate of change of the house P.
Calculus
Which of the following statements are true (T) and
which are false (F)? Justify if true and give a counter-example if false.
Suppose that f(x) has domain [1; 3] and is continuous, that g(x) has domain R and is
di erentiable, and that h(x) has domain (1; 3) and is differentiable.
(a) There is some a E R with 1 <= a <= 3 such that f has a global maximum at a.
(b) There is some a E R with 1 < a < 3 such that f has a global maximum at a.
(c) There is some a E R with 1 < a < 3 such that f has a local minimum at a.
(d) f cannot have in finitely many different local maximum points.
(e) There is some a E R such that g has a global maximum at a.
(f) g has at least one local extremum.
(g) If h has a local maximum at a 2 (1; 3), then h'(a) = 0.
(h) If h'(a) = 0 for some a E (1; 3), then h has a local maximum or a local minimum at a.
which are false (F)? Justify if true and give a counter-example if false.
Suppose that f(x) has domain [1; 3] and is continuous, that g(x) has domain R and is
di erentiable, and that h(x) has domain (1; 3) and is differentiable.
(a) There is some a E R with 1 <= a <= 3 such that f has a global maximum at a.
(b) There is some a E R with 1 < a < 3 such that f has a global maximum at a.
(c) There is some a E R with 1 < a < 3 such that f has a local minimum at a.
(d) f cannot have in finitely many different local maximum points.
(e) There is some a E R such that g has a global maximum at a.
(f) g has at least one local extremum.
(g) If h has a local maximum at a 2 (1; 3), then h'(a) = 0.
(h) If h'(a) = 0 for some a E (1; 3), then h has a local maximum or a local minimum at a.
Calculus
True OR False ? No justi cation required
The company Blissful Batteries Inc. sell replacement batteries for Opple's oPad. In the following, p denotes the unit price of these batteries in dollars and q the quantity sold. We assume that the function q(p) is differentiable with q'(p) < 0. Blissful Batteries Inc. sell 10000 batteries per week at $5.00 each.
(d) If the replacement battery is a price unit elastic good at a unit price of $5.00, if their total cost function is of the form C(q) = a+bq with constants a; b > 0 and management increases the price by a small amount, then their pro t decreases.
(e) If the replacement battery is a price inelastic good at every unit price and management decreases the unit price by 0.2%, then quantity demanded increases by less than 0.2%.
(f) If the replacement battery is a price inelastic good at every unit price and management decreases the unit price by 1%, then quantity demanded decreases by more than 1%.
The company Blissful Batteries Inc. sell replacement batteries for Opple's oPad. In the following, p denotes the unit price of these batteries in dollars and q the quantity sold. We assume that the function q(p) is differentiable with q'(p) < 0. Blissful Batteries Inc. sell 10000 batteries per week at $5.00 each.
(d) If the replacement battery is a price unit elastic good at a unit price of $5.00, if their total cost function is of the form C(q) = a+bq with constants a; b > 0 and management increases the price by a small amount, then their pro t decreases.
(e) If the replacement battery is a price inelastic good at every unit price and management decreases the unit price by 0.2%, then quantity demanded increases by less than 0.2%.
(f) If the replacement battery is a price inelastic good at every unit price and management decreases the unit price by 1%, then quantity demanded decreases by more than 1%.
Calculus
True OR False ? No justi cation required
The company Blissful Batteries Inc. sell replacement batteries for Opple's oPad. In the following, p denotes the unit price of these batteries in dollars and q the quantity sold. We assume that the function q(p) is differentiable with q'(p) < 0. Blissful Batteries Inc. sell 10000 batteries per week at $5.00 each.
(a) If the replacement battery is a price elastic good at a unit price of $5.00 and management increases the price by a small amount, then their revenue increases.
(b) If the replacement battery is a price elastic good at a unit price of $5.00, then it is also an elastic good at a unit price of $10.00.
(c) If the replacement battery is a price elastic good at a unit price of $5.00 and management increases the price by a small amount, then their pro t increases.
The company Blissful Batteries Inc. sell replacement batteries for Opple's oPad. In the following, p denotes the unit price of these batteries in dollars and q the quantity sold. We assume that the function q(p) is differentiable with q'(p) < 0. Blissful Batteries Inc. sell 10000 batteries per week at $5.00 each.
(a) If the replacement battery is a price elastic good at a unit price of $5.00 and management increases the price by a small amount, then their revenue increases.
(b) If the replacement battery is a price elastic good at a unit price of $5.00, then it is also an elastic good at a unit price of $10.00.
(c) If the replacement battery is a price elastic good at a unit price of $5.00 and management increases the price by a small amount, then their pro t increases.
Calculus
Prove that if a is a double zero of the polynomial p(x) then p has a critical point
at x = a (that is p′(a) = 0).
at x = a (that is p′(a) = 0).
Calculus
You work for the Department of Pathology where you study the growth of a microbial culture of Escherichia coli under very special conditions. The number of bacteria at time t [in hours] can be estimated by
f(t)=(t+1)e^(√( t+2))
Use logarithmic differentiation in order to determine the relative growth rate at time t = 3
and give your result in percent growth per hour.
f(t)=(t+1)e^(√( t+2))
Use logarithmic differentiation in order to determine the relative growth rate at time t = 3
and give your result in percent growth per hour.
Calculus
Estimate the square root of 37 using linear approximation
Calculus
A function is defined as f(x)=x(squared)+3
Find a simplified expression for f(a+2)-f(a-5)
Find a simplified expression for f(a+2)-f(a-5)
Calculus
Each curve in the figure below represents the balance in a bank account into which a single deposit was made at time zero.
http://imageshack.us/f/822/ob50k849dw033574setassi.png/
Assuming continuously compounded interest, find (enter each answer as value or a list, e.g, I or II,III): :
(a) The curves representing the same initial deposit :
(b) The curves representing the same interest rate :
(c) The curve(s) representing the largest interest rate :
(d) The curve(s) representing the largest initial deposit :
http://imageshack.us/f/822/ob50k849dw033574setassi.png/
Assuming continuously compounded interest, find (enter each answer as value or a list, e.g, I or II,III): :
(a) The curves representing the same initial deposit :
(b) The curves representing the same interest rate :
(c) The curve(s) representing the largest interest rate :
(d) The curve(s) representing the largest initial deposit :
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#339914 on Dec 2023
#339914 on Dec 2023