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Limit x approach infinity x[ ( 1 + a/x )^ 1+ 1/x - x^-1/x ( x+ a ) ]
Exercise 2. prove the statement, Let a, x and y are real numbers so that x < y and a > 0. Then ax < ay
Exercise 1. prove the statement, Every rational number can be expressed as a quotient p/q so that p and q has no common factors.
Exercise 4. Let X be a non-empty set of real numbers. If it has a supremum, then it must be unique.
A producer of perishable product offers a wage incentive to the drivers of its trucks. A
standard delivery route takes an average of 20 hours. Drivers are paid at the rate of $10 per
hour up to a maximum of 20 hours (if the trip requires 30 hours, the drivers receives payment
for only 20 hours). There is an incentive for drivers to make the trip in less than 20 hours. for
each hour under 20, the hourly wage increases by $1. Assume x equals the number of hours
required to complete the trip.
(a) Determine the function w = f(x), where w equals the hourly wage in dollars
(b) Determine the function which states the driver’s salary for the trip as a function of x.
(c) What trip time x will maximise the driver’s salary for the trip
(d) What hourly wage is associated with this trip time?
(e) What is the maximum salary?
A window consists of a rectangle surmounted by semi-circle. What shape gives the most area for a given perimeter?
Find the point of the line y=x nearest to the point (4,1)
Find the dimensions of the right circular cone of least volume that can be circumscribed about a sphere of radius a. Hint: x+2a=altitude of cone
Exercise 3. Let α and β be two real numbers so that α ≤ β and β ≤ α. Then prove α = β.
FIND THE SURFACE AREA OF THE OBJECT OBTAINED BY ROTATING Y=4+3x^2, 1<=x<=2 ABOUT THE Y-AXIS
