Exam 3 Review

Exam 3 Review

Notions from Exam 2

I Cost C (x) (usually given)

I Average cost C (x)

C (x) =C (x)

x,

I Marginal Avergage Cost

d

dxC (x) =

d

dx

(C (x)

x

)

Exam 3 Review

Notions from Exam 2

I Cost C (x) (usually given)

I Average cost C (x)

C (x) =C (x)

x,

I Marginal Avergage Cost

d

dxC (x) =

d

dx

(C (x)

x

)

Exam 3 Review

Notions from Exam 2

I Cost C (x) (usually given)

I Average cost C (x)

C (x) =C (x)

x,

I Marginal Avergage Cost

d

dxC (x) =

d

dx

(C (x)

x

)

Exam 3 Review

Notions from Exam 2

I Cost C (x) (usually given)

I Average cost C (x)

C (x) =C (x)

x,

I Marginal Avergage Cost

d

dxC (x) =

d

dx

(C (x)

x

)

Exam 3 Review

Notions from Exam 2

I Cost C (x) (usually given)

I Average cost C (x)

C (x) =C (x)

x,

I Marginal Avergage Cost

d

dxC (x) =

d

dx

(C (x)

x

)

Exam 3 Review

Notions from Exam 2

I Revenue: If the price p is modeled as a function of thedemand x (p = p(x)) then the revenue is given by

R(x) = xp(x)

I If on the other hand the demand is a function of the price p,say f (p) then

R(p) = pf (p)

I Marginal revenue: derivative of the revenue R as a functioneither of p or of x .

Exam 3 Review

Notions from Exam 2

I Revenue: If the price p is modeled as a function of thedemand x (p = p(x)) then the revenue is given by

R(x) = xp(x)

I If on the other hand the demand is a function of the price p,say f (p) then

R(p) = pf (p)

I Marginal revenue: derivative of the revenue R as a functioneither of p or of x .

Exam 3 Review

Notions from Exam 2

I Revenue: If the price p is modeled as a function of thedemand x (p = p(x)) then the revenue is given by

R(x) = xp(x)

I If on the other hand the demand is a function of the price p,say f (p) then

R(p) = pf (p)

I Marginal revenue: derivative of the revenue R as a functioneither of p or of x .

Exam 3 Review

Notions from Exam 2

I Revenue: If the price p is modeled as a function of thedemand x (p = p(x)) then the revenue is given by

R(x) = xp(x)

I If on the other hand the demand is a function of the price p,say f (p) then

R(p) = pf (p)

I Marginal revenue: derivative of the revenue R as a functioneither of p or of x .

Exam 3 Review

Notions from Exam 2

I Revenue: If the price p is modeled as a function of thedemand x (p = p(x)) then the revenue is given by

R(x) = xp(x)

I If on the other hand the demand is a function of the price p,say f (p) then

R(p) = pf (p)

I Marginal revenue: derivative of the revenue R as a functioneither of p or of x .

Exam 3 Review

Notions from Exam 2

I Revenue: If the price p is modeled as a function of thedemand x (p = p(x)) then the revenue is given by

R(x) = xp(x)

I If on the other hand the demand is a function of the price p,say f (p) then

R(p) = pf (p)

I Marginal revenue: derivative of the revenue R as a functioneither of p or of x .

Exam 3 Review

Notions from Exam 2

I Profit

P(x) = R(x)− C (x)

I Marginal Profit

P′(x) = R

′(x)− C

′(x)

Exam 3 Review

Notions from Exam 2

I Profit

P(x) = R(x)− C (x)

I Marginal Profit

P′(x) = R

′(x)− C

′(x)

Exam 3 Review

Notions from Exam 2

I Profit

P(x) = R(x)− C (x)

I Marginal Profit

P′(x) = R

′(x)− C

′(x)

Exam 3 Review

Notions from Exam 2

I Profit

P(x) = R(x)− C (x)

I Marginal Profit

P′(x) = R

′(x)− C

′(x)

Exam 3 Review

Notions from Exam 2

I Profit

P(x) = R(x)− C (x)

I Marginal Profit

P′(x) = R

′(x)− C

′(x)

Exam 3 Review

Marginal Functions in Economics

I Elasticity of demand

I p price

I x demand (as a function of the price)

I If x = x(p) = f (p) then the elasticity of demand is defined as

E (p) = −pf ′(p)

f (p)

or

E (p) = −px ′(p)

x(p)

Exam 3 Review

Marginal Functions in Economics

I Elasticity of demand

I p price

I x demand (as a function of the price)

I If x = x(p) = f (p) then the elasticity of demand is defined as

E (p) = −pf ′(p)

f (p)

or

E (p) = −px ′(p)

x(p)

Exam 3 Review

Marginal Functions in Economics

I Elasticity of demand

I p price

I x demand (as a function of the price)

I If x = x(p) = f (p) then the elasticity of demand is defined as

E (p) = −pf ′(p)

f (p)

or

E (p) = −px ′(p)

x(p)

Exam 3 Review

Marginal Functions in Economics

I Elasticity of demand

I p price

I x demand (as a function of the price)

I If x = x(p) = f (p) then the elasticity of demand is defined as

E (p) = −pf ′(p)

f (p)

or

E (p) = −px ′(p)

x(p)

Exam 3 Review

Marginal Functions in Economics

I Elasticity of demand

I p price

I x demand (as a function of the price)

I If x = x(p) = f (p) then the elasticity of demand is defined as

E (p) = −pf ′(p)

f (p)

or

E (p) = −px ′(p)

x(p)

Exam 3 Review

Marginal Functions in Economics

I Elasticity of demand

I p price

I x demand (as a function of the price)

I If x = x(p) = f (p) then the elasticity of demand is defined as

E (p) = −pf ′(p)

f (p)

or

E (p) = −px ′(p)

x(p)

Exam 3 Review

Marginal Functions in Economics

I Elasticity of demand

I p price

I x demand (as a function of the price)

I If x = x(p) = f (p) then the elasticity of demand is defined as

E (p) = −pf ′(p)

f (p)

or

E (p) = −px ′(p)

x(p)

Exam 3 Review

Optimization

Suppose that a business is transitioning from using lightbulbs oftype A to lightbulbs of type B which tend to be more expensivebut are expected to save energy in the long run.

I The cost of one lightbulb of type A is $2.

I The cost of one lightbulb of type B is $3.

I The budget for buying lightbulbs for next year is $300.

I The expected cost of using x lightbulbs of type A and ylightbulbs of type B is

c(x , y) = 8x2 + 9y2 + 10x + 15y .

I Find the number of lightbulbs of type A needed to minimizethe expected cost c(x , y) if one assumes that the entirebudget for buying lightbulbs has to be exhausted.

Exam 3 Review

Optimization

Suppose that a business is transitioning from using lightbulbs oftype A to lightbulbs of type B which tend to be more expensivebut are expected to save energy in the long run.

I The cost of one lightbulb of type A is $2.

I The cost of one lightbulb of type B is $3.

I The budget for buying lightbulbs for next year is $300.

I The expected cost of using x lightbulbs of type A and ylightbulbs of type B is

c(x , y) = 8x2 + 9y2 + 10x + 15y .

I Find the number of lightbulbs of type A needed to minimizethe expected cost c(x , y) if one assumes that the entirebudget for buying lightbulbs has to be exhausted.

Exam 3 Review

Optimization

Suppose that a business is transitioning from using lightbulbs oftype A to lightbulbs of type B which tend to be more expensivebut are expected to save energy in the long run.

I The cost of one lightbulb of type A is $2.

I The cost of one lightbulb of type B is $3.

I The budget for buying lightbulbs for next year is $300.

I The expected cost of using x lightbulbs of type A and ylightbulbs of type B is

c(x , y) = 8x2 + 9y2 + 10x + 15y .

I Find the number of lightbulbs of type A needed to minimizethe expected cost c(x , y) if one assumes that the entirebudget for buying lightbulbs has to be exhausted.

Exam 3 Review

Optimization

Suppose that a business is transitioning from using lightbulbs oftype A to lightbulbs of type B which tend to be more expensivebut are expected to save energy in the long run.

I The cost of one lightbulb of type A is $2.

I The cost of one lightbulb of type B is $3.

I The budget for buying lightbulbs for next year is $300.

I The expected cost of using x lightbulbs of type A and ylightbulbs of type B is

c(x , y) = 8x2 + 9y2 + 10x + 15y .

I Find the number of lightbulbs of type A needed to minimizethe expected cost c(x , y) if one assumes that the entirebudget for buying lightbulbs has to be exhausted.

Exam 3 Review

Optimization

Suppose that a business is transitioning from using lightbulbs oftype A to lightbulbs of type B which tend to be more expensivebut are expected to save energy in the long run.

I The cost of one lightbulb of type A is $2.

I The cost of one lightbulb of type B is $3.

I The budget for buying lightbulbs for next year is $300.

I The expected cost of using x lightbulbs of type A and ylightbulbs of type B is

c(x , y) = 8x2 + 9y2 + 10x + 15y .

I Find the number of lightbulbs of type A needed to minimizethe expected cost c(x , y) if one assumes that the entirebudget for buying lightbulbs has to be exhausted.

Exam 3 Review

Optimization

Suppose that a business is transitioning from using lightbulbs oftype A to lightbulbs of type B which tend to be more expensivebut are expected to save energy in the long run.

I The cost of one lightbulb of type A is $2.

I The cost of one lightbulb of type B is $3.

I The budget for buying lightbulbs for next year is $300.

I The expected cost of using x lightbulbs of type A and ylightbulbs of type B is

c(x , y) = 8x2 + 9y2 + 10x + 15y .

I Find the number of lightbulbs of type A needed to minimizethe expected cost c(x , y) if one assumes that the entirebudget for buying lightbulbs has to be exhausted.

Exam 3 Review

Optimization

Suppose that a business is transitioning from using lightbulbs oftype A to lightbulbs of type B which tend to be more expensivebut are expected to save energy in the long run.

I The cost of one lightbulb of type A is $2.

I The cost of one lightbulb of type B is $3.

I The budget for buying lightbulbs for next year is $300.

I The expected cost of using x lightbulbs of type A and ylightbulbs of type B is

c(x , y) = 8x2 + 9y2 + 10x + 15y .

I Find the number of lightbulbs of type A needed to minimizethe expected cost c(x , y) if one assumes that the entirebudget for buying lightbulbs has to be exhausted.

Exam 3 Review

Optimization

I Constraint: 2x + 3y = 300 =⇒ y =300− 2x

3= 100− 2

3x

I Interval of definition: 0 ≤ x ≤ 150 =⇒ [0, 150]

I Minimize

f (x) = 8x2 + 9

(2

3x − 100

)2

+ 1500.

I f ′(x) = 24x − 1200 =⇒ f ′(x) = 0 only for x = 50.

I f ′′(x) = 24 > 0 relative/local minimum.

I Global minimum? Compare to the endpoints.

Exam 3 Review

Optimization

I Constraint: 2x + 3y = 300 =⇒ y =300− 2x

3= 100− 2

3x

I Interval of definition: 0 ≤ x ≤ 150 =⇒ [0, 150]

I Minimize

f (x) = 8x2 + 9

(2

3x − 100

)2

+ 1500.

I f ′(x) = 24x − 1200 =⇒ f ′(x) = 0 only for x = 50.

I f ′′(x) = 24 > 0 relative/local minimum.

I Global minimum? Compare to the endpoints.

Exam 3 Review

Optimization

I Constraint: 2x + 3y = 300 =⇒ y =300− 2x

3= 100− 2

3x

I Interval of definition: 0 ≤ x ≤ 150 =⇒ [0, 150]

I Minimize

f (x) = 8x2 + 9

(2

3x − 100

)2

+ 1500.

I f ′(x) = 24x − 1200 =⇒ f ′(x) = 0 only for x = 50.

I f ′′(x) = 24 > 0 relative/local minimum.

I Global minimum? Compare to the endpoints.

Exam 3 Review

Optimization

I Constraint: 2x + 3y = 300 =⇒ y =300− 2x

3= 100− 2

3x

I Interval of definition:

0 ≤ x ≤ 150 =⇒ [0, 150]

I Minimize

f (x) = 8x2 + 9

(2

3x − 100

)2

+ 1500.

I f ′(x) = 24x − 1200 =⇒ f ′(x) = 0 only for x = 50.

I f ′′(x) = 24 > 0 relative/local minimum.

I Global minimum? Compare to the endpoints.

Exam 3 Review

Optimization

I Constraint: 2x + 3y = 300 =⇒ y =300− 2x

3= 100− 2

3x

I Interval of definition: 0 ≤ x ≤ 150 =⇒ [0, 150]

I Minimize

f (x) = 8x2 + 9

(2

3x − 100

)2

+ 1500.

I f ′(x) = 24x − 1200 =⇒ f ′(x) = 0 only for x = 50.

I f ′′(x) = 24 > 0 relative/local minimum.

I Global minimum? Compare to the endpoints.

Exam 3 Review

Optimization

I Constraint: 2x + 3y = 300 =⇒ y =300− 2x

3= 100− 2

3x

I Interval of definition: 0 ≤ x ≤ 150 =⇒ [0, 150]

I Minimize

f (x) = 8x2 + 9

(2

3x − 100

)2

+ 1500.

I f ′(x) = 24x − 1200 =⇒ f ′(x) = 0 only for x = 50.

I f ′′(x) = 24 > 0 relative/local minimum.

I Global minimum? Compare to the endpoints.

Exam 3 Review

Optimization

I Constraint: 2x + 3y = 300 =⇒ y =300− 2x

3= 100− 2

3x

I Interval of definition: 0 ≤ x ≤ 150 =⇒ [0, 150]

I Minimize

f (x) = 8x2 + 9

(2

3x − 100

)2

+ 1500.

I f ′(x) = 24x − 1200 =⇒ f ′(x) = 0 only for x = 50.

I f ′′(x) = 24 > 0 relative/local minimum.

I Global minimum? Compare to the endpoints.

Exam 3 Review

Optimization

I Constraint: 2x + 3y = 300 =⇒ y =300− 2x

3= 100− 2

3x

I Interval of definition: 0 ≤ x ≤ 150 =⇒ [0, 150]

I Minimize

f (x) = 8x2 + 9

(2

3x − 100

)2

+ 1500.

I f ′(x) = 24x − 1200

=⇒ f ′(x) = 0 only for x = 50.

I f ′′(x) = 24 > 0 relative/local minimum.

I Global minimum? Compare to the endpoints.

Exam 3 Review

Optimization

I Constraint: 2x + 3y = 300 =⇒ y =300− 2x

3= 100− 2

3x

I Interval of definition: 0 ≤ x ≤ 150 =⇒ [0, 150]

I Minimize

f (x) = 8x2 + 9

(2

3x − 100

)2

+ 1500.

I f ′(x) = 24x − 1200 =⇒ f ′(x) = 0 only for x = 50.

I f ′′(x) = 24 > 0 relative/local minimum.

I Global minimum? Compare to the endpoints.

Exam 3 Review

Optimization

I Constraint: 2x + 3y = 300 =⇒ y =300− 2x

3= 100− 2

3x

I Interval of definition: 0 ≤ x ≤ 150 =⇒ [0, 150]

I Minimize

f (x) = 8x2 + 9

(2

3x − 100

)2

+ 1500.

I f ′(x) = 24x − 1200 =⇒ f ′(x) = 0 only for x = 50.

I f ′′(x) = 24 > 0

relative/local minimum.

I Global minimum? Compare to the endpoints.

Exam 3 Review

Optimization

I Constraint: 2x + 3y = 300 =⇒ y =300− 2x

3= 100− 2

3x

I Interval of definition: 0 ≤ x ≤ 150 =⇒ [0, 150]

I Minimize

f (x) = 8x2 + 9

(2

3x − 100

)2

+ 1500.

I f ′(x) = 24x − 1200 =⇒ f ′(x) = 0 only for x = 50.

I f ′′(x) = 24 > 0 relative/local minimum.

I Global minimum? Compare to the endpoints.

Exam 3 Review

Optimization

I Constraint: 2x + 3y = 300 =⇒ y =300− 2x

3= 100− 2

3x

I Interval of definition: 0 ≤ x ≤ 150 =⇒ [0, 150]

I Minimize

f (x) = 8x2 + 9

(2

3x − 100

)2

+ 1500.

I f ′(x) = 24x − 1200 =⇒ f ′(x) = 0 only for x = 50.

I f ′′(x) = 24 > 0 relative/local minimum.

I Global minimum?

Compare to the endpoints.

Exam 3 Review

Optimization

I Constraint: 2x + 3y = 300 =⇒ y =300− 2x

3= 100− 2

3x

I Interval of definition: 0 ≤ x ≤ 150 =⇒ [0, 150]

I Minimize

f (x) = 8x2 + 9

(2

3x − 100

)2

+ 1500.

I f ′(x) = 24x − 1200 =⇒ f ′(x) = 0 only for x = 50.

I f ′′(x) = 24 > 0 relative/local minimum.

I Global minimum? Compare to the endpoints.

Exam 3 Review

Optimization

I Look at f ′(0) = −1200 < 0 and atf ′(150) = 3600− 1200 = 2400 > 0

I The endpoints cannot be minimizers

I Evaluate f (0), f (50) and f (150) directly.

Exam 3 Review

Optimization

I Look at f ′(0) = −1200 < 0 and atf ′(150) = 3600− 1200 = 2400 > 0

I The endpoints cannot be minimizers

I Evaluate f (0), f (50) and f (150) directly.

Exam 3 Review

Optimization

I Look at f ′(0) = −1200 < 0

and atf ′(150) = 3600− 1200 = 2400 > 0

I The endpoints cannot be minimizers

I Evaluate f (0), f (50) and f (150) directly.

Exam 3 Review

Optimization

I Look at f ′(0) = −1200 < 0 and atf ′(150) = 3600− 1200 = 2400 > 0

I The endpoints cannot be minimizers

I Evaluate f (0), f (50) and f (150) directly.

Exam 3 Review

Optimization

I Look at f ′(0) = −1200 < 0 and atf ′(150) = 3600− 1200 = 2400 > 0

I The endpoints cannot be minimizers

I Evaluate f (0), f (50) and f (150) directly.

Exam 3 Review

Optimization

I Look at f ′(0) = −1200 < 0 and atf ′(150) = 3600− 1200 = 2400 > 0

I The endpoints cannot be minimizers

I Evaluate f (0), f (50) and f (150) directly.

Exam 3 Review

Optimization

I If now the expected revenue for the previous distribution oflightbulbs is

3x2 + 18y2 + 14x + 20y

find the distribution of lightbulbs that maximizes the profitusing again the assumption that the entire $300 budget isexhausted.

I Look atf (x) = revenue−cost = −5x2 + 9 (100− 2/3x)2−2/3x + 500

I f ′(x) = −2x − 1200− 2/3.

I No critical points in [0, 150] f ′(0) = −1200− 2/3 < 0,f ′(150) < 0 (or f (0) = 90500, f (150) = −110600).

Exam 3 Review

Optimization

I If now the expected revenue for the previous distribution oflightbulbs is

3x2 + 18y2 + 14x + 20y

find the distribution of lightbulbs that maximizes the profitusing again the assumption that the entire $300 budget isexhausted.

I Look atf (x) = revenue−cost = −5x2 + 9 (100− 2/3x)2−2/3x + 500

I f ′(x) = −2x − 1200− 2/3.

I No critical points in [0, 150] f ′(0) = −1200− 2/3 < 0,f ′(150) < 0 (or f (0) = 90500, f (150) = −110600).

Exam 3 Review

Optimization

I If now the expected revenue for the previous distribution oflightbulbs is

3x2 + 18y2 + 14x + 20y

find the distribution of lightbulbs that maximizes the profitusing again the assumption that the entire $300 budget isexhausted.

I Look atf (x) = revenue−cost = −5x2 + 9 (100− 2/3x)2−2/3x + 500

I f ′(x) = −2x − 1200− 2/3.

I No critical points in [0, 150] f ′(0) = −1200− 2/3 < 0,f ′(150) < 0 (or f (0) = 90500, f (150) = −110600).

Exam 3 Review

Optimization

I If now the expected revenue for the previous distribution oflightbulbs is

3x2 + 18y2 + 14x + 20y

find the distribution of lightbulbs that maximizes the profitusing again the assumption that the entire $300 budget isexhausted.

I Look atf (x) = revenue−cost = −5x2 + 9 (100− 2/3x)2−2/3x + 500

I f ′(x) = −2x − 1200− 2/3.

I No critical points in [0, 150] f ′(0) = −1200− 2/3 < 0,f ′(150) < 0 (or f (0) = 90500, f (150) = −110600).

Exam 3 Review

Optimization

I If now the expected revenue for the previous distribution oflightbulbs is

3x2 + 18y2 + 14x + 20y

find the distribution of lightbulbs that maximizes the profitusing again the assumption that the entire $300 budget isexhausted.

I Look atf (x) = revenue−cost = −5x2 + 9 (100− 2/3x)2−2/3x + 500

I f ′(x) = −2x − 1200− 2/3.

I No critical points in [0, 150] f ′(0) = −1200− 2/3 < 0,f ′(150) < 0 (or f (0) = 90500, f (150) = −110600).

Exam 3 Review

Optimization

I If now the expected revenue for the previous distribution oflightbulbs is

3x2 + 18y2 + 14x + 20y

find the distribution of lightbulbs that maximizes the profitusing again the assumption that the entire $300 budget isexhausted.

I Look atf (x) = revenue−cost = −5x2 + 9 (100− 2/3x)2−2/3x + 500

I f ′(x) = −2x − 1200− 2/3.

I No critical points in [0, 150] f ′(0) = −1200− 2/3 < 0,f ′(150) < 0 (or f (0) = 90500, f (150) = −110600).

Exam 3 Review

Optimization

I If now the expected revenue for the previous distribution oflightbulbs is

3x2 + 18y2 + 14x + 20y

find the distribution of lightbulbs that maximizes the profitusing again the assumption that the entire $300 budget isexhausted.

I Look atf (x) = revenue−cost = −5x2 + 9 (100− 2/3x)2−2/3x + 500

I f ′(x) = −2x − 1200− 2/3.

I No critical points in [0, 150]

f ′(0) = −1200− 2/3 < 0,f ′(150) < 0 (or f (0) = 90500, f (150) = −110600).

Exam 3 Review

Optimization

I If now the expected revenue for the previous distribution oflightbulbs is

3x2 + 18y2 + 14x + 20y

find the distribution of lightbulbs that maximizes the profitusing again the assumption that the entire $300 budget isexhausted.

I Look atf (x) = revenue−cost = −5x2 + 9 (100− 2/3x)2−2/3x + 500

I f ′(x) = −2x − 1200− 2/3.

I No critical points in [0, 150] f ′(0) = −1200− 2/3 < 0,

f ′(150) < 0 (or f (0) = 90500, f (150) = −110600).

Exam 3 Review

Optimization

I If now the expected revenue for the previous distribution oflightbulbs is

3x2 + 18y2 + 14x + 20y

find the distribution of lightbulbs that maximizes the profitusing again the assumption that the entire $300 budget isexhausted.

I Look atf (x) = revenue−cost = −5x2 + 9 (100− 2/3x)2−2/3x + 500

I f ′(x) = −2x − 1200− 2/3.

I No critical points in [0, 150] f ′(0) = −1200− 2/3 < 0,f ′(150) < 0

(or f (0) = 90500, f (150) = −110600).

Exam 3 Review

Optimization

I If now the expected revenue for the previous distribution oflightbulbs is

3x2 + 18y2 + 14x + 20y

find the distribution of lightbulbs that maximizes the profitusing again the assumption that the entire $300 budget isexhausted.

I Look atf (x) = revenue−cost = −5x2 + 9 (100− 2/3x)2−2/3x + 500

I f ′(x) = −2x − 1200− 2/3.

I No critical points in [0, 150] f ′(0) = −1200− 2/3 < 0,f ′(150) < 0 (or f (0) = 90500, f (150) = −110600).

Exam 3 Review

Other Optimization Problems

I Suppose that a cost function is given by

c(x) = 2x2 +3x

1 + xfor x > 0

Minimize the average cost.

I What if now

c(x) =xe−2x

1 + x2?

Exam 3 Review

Other Optimization Problems

I Suppose that a cost function is given by

c(x) = 2x2 +3x

1 + xfor x > 0

Minimize the average cost.

I What if now

c(x) =xe−2x

1 + x2?

Exam 3 Review

Other Optimization Problems

I Suppose that a cost function is given by

c(x) = 2x2 +3x

1 + xfor x > 0

Minimize the average cost.

I What if now

c(x) =xe−2x

1 + x2?

Exam 3 Review

Other Optimization Problems

I Suppose that a cost function is given by

c(x) = 2x2 +3x

1 + xfor x > 0

Minimize the average cost.

I What if now

c(x) =xe−2x

1 + x2?

Exam 3 Review

Other Optimization Problems

I Suppose that a cost function is given by

c(x) = 2x2 +3x

1 + xfor x > 0

Minimize the average cost.

I What if now

c(x) =xe−2x

1 + x2?

Exam 3 Review

Compounded Interest

I Interest compounded periodically

A︸︷︷︸Accumulated Amount

= P︸︷︷︸Principal

(1 + r/m)mt

I r Interest rate,

I Divide the year into m periods of equal duration

I t = # of years

I Continuous compounding od interest A = Pert .

I Initial capital from accumulated amount: P = A(1 + r

m

)−mt.

Exam 3 Review

Compounded Interest

I Interest compounded periodically

A︸︷︷︸Accumulated Amount

= P︸︷︷︸Principal

(1 + r/m)mt

I r Interest rate,

I Divide the year into m periods of equal duration

I t = # of years

I Continuous compounding od interest A = Pert .

I Initial capital from accumulated amount: P = A(1 + r

m

)−mt.

Exam 3 Review

Compounded Interest

I Interest compounded periodically

A︸︷︷︸Accumulated Amount

= P︸︷︷︸Principal

(1 + r/m)mt

I r Interest rate,

I Divide the year into m periods of equal duration

I t = # of years

I Continuous compounding od interest A = Pert .

I Initial capital from accumulated amount: P = A(1 + r

m

)−mt.

Exam 3 Review

Compounded Interest

I Interest compounded periodically

A︸︷︷︸Accumulated Amount

= P︸︷︷︸Principal

(1 + r/m)mt

I r Interest rate,

I Divide the year into m periods of equal duration

I t = # of years

I Continuous compounding od interest A = Pert .

I Initial capital from accumulated amount: P = A(1 + r

m

)−mt.

Exam 3 Review

Compounded Interest

I Interest compounded periodically

A︸︷︷︸Accumulated Amount

= P︸︷︷︸Principal

(1 + r/m)mt

I r Interest rate,

I Divide the year into m periods of equal duration

I t = # of years

I Continuous compounding od interest A = Pert .

I Initial capital from accumulated amount: P = A(1 + r

m

)−mt.

Exam 3 Review

Compounded Interest

I Interest compounded periodically

A︸︷︷︸Accumulated Amount

= P︸︷︷︸Principal

(1 + r/m)mt

I r Interest rate,

I Divide the year into m periods of equal duration

I t = # of years

I Continuous compounding od interest A = Pert .

I Initial capital from accumulated amount: P = A(1 + r

m

)−mt.

Exam 3 Review

Compounded Interest

I Interest compounded periodically

A︸︷︷︸Accumulated Amount

= P︸︷︷︸Principal

(1 + r/m)mt

I r Interest rate,

I Divide the year into m periods of equal duration

I t = # of years

I Continuous compounding od interest A = Pert .

I Initial capital from accumulated amount: P = A(1 + r

m

)−mt.

Exam 3 Review

Compounded Interest

I Interest compounded periodically

A︸︷︷︸Accumulated Amount

= P︸︷︷︸Principal

(1 + r/m)mt

I r Interest rate,

I Divide the year into m periods of equal duration

I t = # of years

I Continuous compounding od interest A = Pert .

I Initial capital from accumulated amount:

P = A(1 + r

m

)−mt.

Exam 3 Review

Compounded Interest

I Interest compounded periodically

A︸︷︷︸Accumulated Amount

= P︸︷︷︸Principal

(1 + r/m)mt

I r Interest rate,

I Divide the year into m periods of equal duration

I t = # of years

I Continuous compounding od interest A = Pert .

I Initial capital from accumulated amount: P = A(1 + r

m

)−mt.

Exam 3 Review

Curve sketching

I Steps

1. Find and classify critical points2. Points where the function is increasing/decreasing3. Inflection points (change in convexity/concavity).4. Horizontal Asymptotes/Vertical Asymptotes.

Iex

xI e−(x−3)(x−5)

Exam 3 Review

Curve sketching

I Steps

1. Find and classify critical points2. Points where the function is increasing/decreasing3. Inflection points (change in convexity/concavity).4. Horizontal Asymptotes/Vertical Asymptotes.

Iex

xI e−(x−3)(x−5)

Exam 3 Review

Curve sketching

I Steps

1. Find and classify critical points

2. Points where the function is increasing/decreasing3. Inflection points (change in convexity/concavity).4. Horizontal Asymptotes/Vertical Asymptotes.

Iex

xI e−(x−3)(x−5)

Exam 3 Review

Curve sketching

I Steps

1. Find and classify critical points2. Points where the function is increasing/decreasing

3. Inflection points (change in convexity/concavity).4. Horizontal Asymptotes/Vertical Asymptotes.

Iex

xI e−(x−3)(x−5)

Exam 3 Review

Curve sketching

I Steps

1. Find and classify critical points2. Points where the function is increasing/decreasing3. Inflection points (change in convexity/concavity).

4. Horizontal Asymptotes/Vertical Asymptotes.

Iex

xI e−(x−3)(x−5)

Exam 3 Review

Curve sketching

I Steps

1. Find and classify critical points2. Points where the function is increasing/decreasing3. Inflection points (change in convexity/concavity).4. Horizontal Asymptotes/Vertical Asymptotes.

Iex

xI e−(x−3)(x−5)

Exam 3 Review

Curve sketching

I Steps

1. Find and classify critical points2. Points where the function is increasing/decreasing3. Inflection points (change in convexity/concavity).4. Horizontal Asymptotes/Vertical Asymptotes.

Iex

x

I e−(x−3)(x−5)

Exam 3 Review

Curve sketching

I Steps

1. Find and classify critical points2. Points where the function is increasing/decreasing3. Inflection points (change in convexity/concavity).4. Horizontal Asymptotes/Vertical Asymptotes.

Iex

xI e−(x−3)(x−5)

Exam 3 Review