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Calculus I

Calculus I. Section 3.3 Rates of Change in the Natural and Social Sciences. Quote. The French Mathematician Joseph Fourier said “Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them.”. The Derivative.

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Calculus I

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  1. Calculus I Section 3.3 Rates of Change in the Natural and Social Sciences

  2. Quote • The French Mathematician Joseph Fourier said “Mathematics compares the most diverse phenomena and discovers the secret analogies that unite them.”

  3. The Derivative • We have seen the derivative as a tool for finding the velocity in physics, but it is used in many more applications. In any discipline that has a need for studying rates of change the derivative will occur as a most useful tool.

  4. 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 Where G is the gravitational constant and r is the distance between the bodies. Find dF/dr and explain its meaning. What does the minus sign indicate? Suppose it is known that Earth attracts an object with a force that decreases at the rate of 2 N/km when r=20,000km. How fast does this force change when r=10,000km?

  5. In a fish farm, a population of fish is introduced into a pond and harvested regularly. A model for the rate of change of the fish population is given by the equation where r0 is the birth rate of the fish, Pc is the maximum population that the pond can sustain, and b is the percentage of the population that is harvested. What value of dP/dt corresponds to a stable population? If the pond can sustain 10,000 fish, the birth rate is 5%, and the harvesting rate is 4%, find the stable population level. What happens if b is raised to 5%?

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