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