Differential Calculus

The sum of two positive numbers is 50. What are the numbers if their product is to be the largest possible.

A triangle has variable sides x,y,z subject to the constraint such that the perimeter is fixed to 18 cm. What is the maximum possible area for the triangle?

A farmer has enough money to build only 100 meters of fence. What are the dimensions of the field he can enclose the maximum area?

A fencing is limited to 20 ft length. What is the maximum rectangular area that can be fenced in using two perpendicular corner sides of an existing wall?

The coordinates (x,y) in feet of a moving particle P are given by \(x=\cos t-1\) and \(y=2 \sin t+1\), where t is the time in seconds. At what extreme rates in fps is P moving along the curve?

Water is flowing into a conical cistern at the rate of 8 \(m^3/min\). If the height of the inverted cone is 12 m and the radius of its circular opening is 6 m. How fast is the water level rising when the water is 4 m deep?

Water is pouring into a conical vessel 15 cm deep and having a radius of 3.75 cm across the top. If the rate at which the water rises is 2 cm/sec how fast is the water flowing into the conical vessel when the water is 4 cm deep?

A balloon is released from the ground 100 meters from an observer. The balloon rises directly upward at the rate of 4 meters per second. How fast is the balloon receding from the observer 10 seconds later?

A balloon is rising vertically over a point A on the ground at the rate of 15 ft./sec. A point B on the ground level with and 30 ft from A. When the balloon is 40 ft. from A, at what rate is its distance from B changing?

A standard cell has an emf "E" of 1.2 volts. If the resistance "R" of the circuit is increasing at the rate of 0.03 ohm/sec, at what rate is the current "I" changing at the instant when the resistance is 6 ohms? Assume Ohm's law E = IR.

Sand is pouring to form a conical pile such that its altitude is always twice its radius. If the volume of the conical pile is increasing at the rate of 25π cu.ft/min, how fast is the radius increasing when the radius is 5 feet?

A wall 8 feet high is 3.375 feet from a house. Find the shortest ladder that will reach from the ground to the house when leaning over the wall.

A coat of paint of thickness 0.01 inch is applied to the faces of a cube whose edge is 10 inches, thereby producing a slightly larger cube. Estimate the number of cubic inches of paint used.

A particle has a position vector \((2\cos 2t,\ 1 + 3\sin t)\). What is the speed of the particle at time t = π/4?

A rectangular trough is 8 feet long, 2 feet across the top and 4 feet deep. If water flows in at a rate of 2 ft³/min, how fast is the surface rising when the water is 1 ft deep?

A spherical balloon is inflated with \(r = 3\sqrt[3]{t}\) for \(0 \lt t \le 10\). Find the rate of change of volume in cubic cm at t = 8.

Evaluate \(\displaystyle\lim_{x \to 3}\dfrac{\sqrt{x^2 - 9}}{2x - 6}\).

What is the coefficient of the \((x - 1)^3\) term in the Taylor series expansion of \(f(x) = \ln x\) expanded about x = 1?

A balloon is released at eye level and rises at the rate of 5 ft/s. An observer 50 ft away watches the balloon rise. How fast is the angle of elevation increasing 6 seconds after the moment of release?

Water is flowing into a conical vessel 10 ft high and 2 ft in radius at the rate of 50 cu. ft per minute. If the depth of the water is 6 ft, how fast is the radius increasing?