CHAPTER 3 DERIVATIVES

Aim #3.4 How do we apply the first and second derivative? Applications of the derivative Physician may want to know how a change in dosage affects the body’s response to a drug Economist want to study how the cost of producing steel varies with the # of tons produced

Example 1: Enlarging Circles:

Instantaneous Velocity Is the derivative of the position function s = f(t) with respect to time.

Speed is the absolute value of velocity. Example 3: Reading a Velocity Graph Insert

Velocity Tells us the direction of motion when the object is moving forward (s is increasing) the velocity is positive when the object is moving backward (when s is decreasing) the velocity is negative.

Acceleration Is the derivative of velocity with respect to time. If a body’s velocity at time t is v(t)= ds/dt, then the body’s acceleration at time t is

Acceleration When velocity and acceleration have the same sign the particle is increasing in speed. When the velocity and acceleration opposite signs the particle is slowing down. When the velocity =0 and the acceleration ≠ 0 particle is stopped momentarily or changing directions.

Example 4: Modeling Vertical Motion

Example 5: Studying Particle Motion

Summary: Answer in complete sentences How might engineers refer to the derivatives of functions describing motion? Explain how to find the velocity and acceleration given the position function. Explain how to find the displacement of a particle. Complete ticket out and turn in.

Extension: Derivatives in Economics Economists refer to rates of changes and derivatives as marginals. In manufacturing the cost of production c(x) is a function of x, the number of units produced. Marginal cost is the rate of change of cost with respect to the level of production so it is dc/dx. Sometimes marginal cost of production is loosely defined to be the extra cost of producing one more unit.

Example : Marginal Cost and Marginal Revenue