chapter 3 derivatives. aim #3.4 how do we apply the first and second derivative? applications of the...
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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