Chapter 5: Applications of the Derivative Derivatives Chapter 5: Applications

Objectives: To be able to use the derivative to analyze function Draw the graph of the function based on the analysis Apply the principles learned to problem situations

Example 1: A square box is to be made from a square cardboard 30 in on one side, by cutting out equal squares from all corners and turning up the sides. Find the volume of the largest box that can be made.

Example 2: Find an equation of the tangent line to the curve y = x3 - 3x2 + 5x that has the least slope.

Example 3: A rectangular field having an area of 2700 sq m is to be enclosed with a fence and an additional fence used to divide the field down the middle. The cost of the fence down the middle is 24 pesos per meter, the fence along the sides 36 pesos per meter. Determine the dimensions of the field for the total cost of the fencing material to be minimum.

Example 4: The number of dollars in the cost per hour of a cargo ship is 0.02v2, where v is the average speed of the ship. If there is an additional cost of 400 dollars per hour find the average speed when the cost per mile is to be least.

Example 5: A shoreline goes in the east-west direction. A tiny island lies north of a point A on the shoreline. A lighthouse L is located on the island and is 10 km from A. A cable is to be laid from L to a point B on the shoreline east of A. The cable will be laid thru the water in a straight line from L to a point C on the shoreline between A and B, and from C to B along shoreline. The part of the cable lying in the water costs Php 50,000/km and the part along the shoreline costs Php 30,000/km. What must the distance AC be so as to attain the least possible total cost of the cable and what is the cost cost if B is 20 km from A.

Example 6: Find two nonnegative numbers whose sum is 12 such that their product is an absolute maximum.

Example 7: Estimate the dimensions of a right-circular cylinder of greatest lateral area that can be inscribed in a sphere with a radius of 6 in

Example 8: Given a circle having the equation x2 + y2 = 9, find: a. the shortest distance from the point (4,5) to a point on the circle, and b. the longest distance from the point (4,5) to a point on the circle.

Example 9: A piece of wire 10 ft long is cut into two pieces. One piece is a bent into the shape of a circle and the other into the shape of a square. How should the wire be cut so that: (a) the combined area of the two figures is as small as possible? (b) the combined area of the two figures is as large as possible?

Example 10: A Norman window consists of a rectangle surmounted by a semicircle. If the perimeter of a Norman window is to be 32 ft, determine what should be the radius of the semicircle and the height of the rectangle such that the window will admit the most light.

Example 11: A right circle cone is to be circumscribed about a sphere of a radius 6 units. Find the ratio of the altitude to the radius of the base of the cone of least possible volume.

Example 12: Find the points on the hyperbola, y2 - x2 = 4, that are closest to (2,0)

Example 13: Find the equation of the line through the point (3,5) that can cut off the least area from the first quadrant.

Example 14: A manufacturer of canned food packages the product in cylindrical tin cans. The volume of each can is to be equal to a given value of V cubic units. Find the ratio of the height of the can to its radius to minimize its surface (side, top and bottom) area.

Example 15: A right circular cylinder is to be inscribed in a sphere of radius R. What is its maximum volume?

Example 16: A conical drinking cup is to be made from a circular piece of paper of radius, 6”, by cutting out a sector and joining the edges CA and CB. Find the maximum capacity of the cup.

Example 18: The top and bottom margin of a poster are each 6cm and the side margins are each 4 cm. If the area of the printed region is fixed at 384 sq. cm , find the dimensions of the poster with smallest area.