Download Math 31 Ch. 3 Review notes

Chapter 3: Applications of Derivatives
3.1 Velocity
Recall:
s: position
𝑑𝑠
: velocity
𝑑𝑡
𝑑𝑣
𝑑𝑡
𝑑2 𝑠
= 𝑑𝑡 2 : acceleration (will come back to this)
Position graphs:
Step 1:
Step 2:
Step 3:
Step 4:
Find the initial and final position
Find time when v = 0.
Find position when v = 0.
Sketch the position on the number line.
Displacement: how far the object has travelled from the original
Distance: how far the object has travelled in total
3.2 Acceleration
If an object moves along a straight line, acceleration is the rate of change of velocity with
respect to time. Therefore, it is the second derivative.
Rates of Change in the Natural Sciences (3.3)
Derivatives can be interpreted as rates of change. So now we’re going to use derivatives
to find rates of change in physics, biology and chemistry.
Physics Application
average density is:
∆𝑚 𝑓(𝑥2 ) − 𝑓(𝑥1 )
=
∆𝑥
𝑥2 − 𝑥1
linear density at x will be the limit of the average density as ∆𝑥 approaches 0:
𝑝 = lim
∆𝑚
∆𝑥→0 ∆𝑥
=
𝑑𝑚
𝑑𝑥
(just find the derivative)
Biology Application
If the number of items in a population at time t is n = f(t), then:
∆𝑛
𝑓(𝑡2 )−𝑓(𝑡1 )

Average rate of growth =

Instantaneous rate of growth = lim
∆𝑡
=
𝑡2 −𝑡1
∆𝑛
∆𝑥→0 ∆𝑡
=
𝑑𝑛
𝑑𝑡
Chemistry Application
The concentration of a substance A is the number of moles (6.022𝑥1023 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠) per
litre and is denoted by [A]. During a chemical reaction the concentration will vary and
so [A] is a function of time. During a time interval 𝑡1 ≤ 𝑡 ≤ 𝑡2 , the average rate of
reaction of a reactant A is:
[𝐴](𝑡2 ) − [𝐴](𝑡1 )
∆[𝐴]
=−
∆𝑡
𝑡2 − 𝑡1
and the instantaneous rate of reaction is the rate of change of concentration with
respect to time.
Rate of reaction= lim
∆[𝐴]
∆𝑡→0 ∆𝑡
=−
𝑑[𝐴]
𝑑𝑡
Since the rate of reaction is the derivative of the concentration function, chemists often
determine the rate of change of reaction by measuring the slope of a tangent.
Rates of Change in the Social Sciences (3.4)
We will look at rates of change in business and economics.
Cost Function: [C(x)] = the cost of producing x units of a certain commodity.
Average rate of change in cost:
∆𝐶 𝐶(𝑥2 ) − 𝐶(𝑥1 ) 𝐶(𝑥1 + ∆𝑥) − 𝐶(𝑥1 )
=
=
∆𝑥
𝑥2 − 𝑥1
∆𝑥
Marginal Cost: [C’(x)] = the instantaneous rate of change of cost with respect to number
of units produced.
∆𝐶 𝑑𝐶
=
∆𝑥→0 ∆𝑥
𝑑𝑥
lim
We approximate by taking ∆𝑥 = 1, so that for large values of x,
C’(x) is approximately equal to C(x + 1) – C(x).
Demand (Price) Function [p(x)] = the price per unit that a company can charge if it sells
x units.
Revenue Function [R(x) = xp(x)] = total revenue of selling x units at a unit price of p(x).
Marginal Revenue Function [R’(x)] = the rate of change in revenue with respect to the
number of units sold.
Profit Function
[P(x) = R(x) – C(x)] = the difference between cost and revenue
Marginal Profit Function
number of units.
[P’(x)] = the rate of change in profit with respect to the
Profit will be maximized when first derivative = 0 means horizontal line, slope of 0
(maximum on a parabola)
Related Rates (3.5)
We are given the rate of change of one quantity and we are asked to find the rate of
change of a related quantity. We find an equation that relates the 2 (or more) quantities
and use the Chain Rule to differentiate BOTH sides with respect to time.