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.

AccelerationIs 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