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: Find how fast the area of the square increases as the side increases.

Example 2: Find how fast the circumference of a circle increases as (a) the radius increases (b) the area increases

Example 3: Find how fast (a) the volume (b) surface area (c) the diagonal of a cube increases as the length of the edge increases.

Example 4: Find how fast (a) the volume (b) surface area of a sphere increases as the radius increases.

Example 5: A right circular cylinder has a fixed height of 6 units (a) find the rate of change of its volume with respect to the radius of its base (b) find the rate of change of the surface area with respect to the radius.

Example 6: The dimensions of a box are b, b+1, b+4. (a) Find how fast the total surface area increases as b increases (b) Find how fast the volume increases as b increases.

Example 7: If a tank holds 5000 gallons of water, which drains from the bottom of the tank in 40 minutes, then Torricelli’s Law gives the volume of the water remaining in the tank after t minutes as: Find the rate at which the water is draining from the tank after (a) 5 min (b) 10 min (c) 20 min (d) at what time is water flowing out the fastest? (e) the slowest?

Example 8: Newton’s Law of Gravitation says that the magnitude F of the force exerted by a body of mass m on a body of mass M is (a) If the bodies are moving, find dF/dr and explain its meaning, what does the minus sign indicate? (b) Suppose it is known that the earth attracts an object with a force that decreases at the rate of 2 N/km when r is 20,000 km. How fast does this force change when r = 10,000 km?

Example 9: The quantity of charge Q in coulombs (C) that has passed through a point in a wire up to time t (in seconds) is given by Q(t) = t3 – 2t2 + 6t + 2. Find the current when (a) t = 0.5 sec (b) t = 1 sec. The unit of current is an ampere, 1 ampere = 1 C/sec (c) At what time is the current lowest?

Example 10: The cost function of a certain commodity is C(x) = 84 + 0.16x – 0.0006x2 + 0.00001x3. Where: C(x) = production cost x = no. of items produced (a) Find C’(x) and interpret C ’(100). (b) Compare C ‘(x) with the cost in producing the 101 th term.

Example 11: Find the equations of the normal lines to the curve at a point where the rate of change of the slope of the tangent line is 1.