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For "Martin" 1. Use calculus to...

For "Martin"

1. Use calculus to find the absolute maximum and minimum values of the function

( ) for .

2. Find a parabola with equation that has slope at , slope at , and passes through the point ( ).

3. Find the derivative for each of the following:

a) ( )

b) ( ) √

c)

d)

e)

4. Find an equation of the tangent line to the curve with parametric equations , at the point ( ).

5. The equation is called a differential equation because it is an equation involving an unknown function in terms of its derivatives. Find constants and such that the function satisfies the differential equation.

6. The curve with equation is called the Tschirnhausen cubic. Find an equation of the tangent line to this curve at the point ( ).

7. When air expands adiabatically (without gaining or losing heat), its pressure and volume are related by the equation where is a constant. Suppose that at a certain instant the volume is 200cm3 and the pressure is 100kPa and is decreasing at a rate of 5kPa/minute. At what rate is the volume increasing at this instant?

8. A manufacturer produces bolts of a fabric with a fixed width. The quantity of this fabric (measured in metres) that is sold is a function of the selling price (in dollars per metre), that is ( ) for some function . The total revenue earned with selling price is ( ) ( ).

a) What does it mean to say that ( ) and ( ) ?

b) Assuming these values for ( ) and ( ), find ( ) and interpret your answer.

9. The cost of fuel per hour (in dollars/hour) of running a certain jet aircraft is a related to its cruising velocity (in km/hour) by the equation:

( )

Find the most economical cruising velocity for a journey of 6000 km (Hint: First find an expression for the total cost of the journey).

10. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1m higher than the bow of the boat (as shown below). If the rope is pulled in at a rate of , find an expression for the speed at which the boat is approaching the dock in terms of its distance from the dock. Explain whether there are points where the speed of the boat is maximum or minimum.

𝒙

𝑚

1. Use calculus to find the absolute maximum and minimum values of the function

( ) for .

2. Find a parabola with equation that has slope at , slope at , and passes through the point ( ).

3. Find the derivative for each of the following:

a) ( )

b) ( ) √

c)

d)

e)

4. Find an equation of the tangent line to the curve with parametric equations , at the point ( ).

5. The equation is called a differential equation because it is an equation involving an unknown function in terms of its derivatives. Find constants and such that the function satisfies the differential equation.

6. The curve with equation is called the Tschirnhausen cubic. Find an equation of the tangent line to this curve at the point ( ).

7. When air expands adiabatically (without gaining or losing heat), its pressure and volume are related by the equation where is a constant. Suppose that at a certain instant the volume is 200cm3 and the pressure is 100kPa and is decreasing at a rate of 5kPa/minute. At what rate is the volume increasing at this instant?

8. A manufacturer produces bolts of a fabric with a fixed width. The quantity of this fabric (measured in metres) that is sold is a function of the selling price (in dollars per metre), that is ( ) for some function . The total revenue earned with selling price is ( ) ( ).

a) What does it mean to say that ( ) and ( ) ?

b) Assuming these values for ( ) and ( ), find ( ) and interpret your answer.

9. The cost of fuel per hour (in dollars/hour) of running a certain jet aircraft is a related to its cruising velocity (in km/hour) by the equation:

( )

Find the most economical cruising velocity for a journey of 6000 km (Hint: First find an expression for the total cost of the journey).

10. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1m higher than the bow of the boat (as shown below). If the rope is pulled in at a rate of , find an expression for the speed at which the boat is approaching the dock in terms of its distance from the dock. Explain whether there are points where the speed of the boat is maximum or minimum.

𝒙

𝑚

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