Transcript Business Calculus - Front Range Community College

Business Calculus
Other Bases & Elasticity of Demand
 3.5 The Exponential Function
Know your facts for
f ( x)  a
x
1. Know the graphs: A horizontal asymptote on one side at y = 0.
Through the point (0,1)
Domain: (-∞, ∞) Range: (0, ∞)
a>1
0<a<1
2. Evaluate exponential functions by calculator.
3. Differentiate
ya
x
:
dy
 a x (lna)
dx
4. Differentiate exponential functions using the sum/difference,
coefficient, product, quotient, or chain rule.
 Logarithmic Function
Know your facts for
f ( x)  loga x
1. Know the graph: A vertical asymptote on one side of the x axis
at x = 0.
Through the point (1,0).
Domain: (0, ∞) Range: (-∞, ∞)
a>1
0<a<1
2. Evaluate logarithmic functions by calculator.
3. Change of Base formula:
4. Differentiate
lnb
loga b 
ln a
y  loga: x
dy
1

dx x ln a
5. Differentiate exponential functions using the sum/difference,
coefficient, product, quotient, or chain rule.
 3.6 Elasticity of Demand
Given a demand function D(x) (quantity, also called q) where x
is the price of an item, we want to determine how a small
percentage increase in price affects the demand for the item.
x
Percent change in price:
x
Percent change in quantity:
(given as a decimal)
q
q
(given as a decimal)
E(x) is the negative ratio of percent change in quantity to
percent change in price.
  q 
q

E ( x) 
x
x
For example: if a small increase in price causes a larger decrease
in demand, this could result in a decrease in revenue.
(Bad for business)
  q 
q

E ( x) 
x
x
In this case, E(x) would be a number > 1, and
the demand is called elastic.
On the other hand, a small increase in price may cause a smaller
decrease in demand, resulting in an increase in revenue. Here,
E(x) would be a number < 1, and the demand is called inelastic.
If a percent change in price results in an equal percent change
in demand, E(x) = 1, and demand is unit elastic.
 Calculus & Elasticity and Revenue
Using calculus, and taking the limit as ∆x → 0, we can show that
 x  D ( x)
E ( x) 
D( x)
We will use this formula for elasticity to find the elasticity of
demand for a given demand function D(x), and to
calculate elasticity at various prices.
Revenue is related to elasticity since R(x) = price * demand.
If demand is inelastic, a small increase in price will result in an
increase in revenue.
If demand is elastic, a small increase in price will result in an
decrease in revenue.
If demand is unit elastic, the current price represents maximum
revenue.