5.1 Cost, Area, and the Definite Integral

Def’n The area of the region S under the graph of a positive function f is thelimit of the sum of approximating rectangles, and is given byA=lim

n→∞Rn=lim

n→∞[ f (x1¿)∆ x+f (x2)∆ x+⋯+ f (xn)∆x ]¿ .

Def’n The definite integral of f from a to b is given by

∫a

b

f (x )dx= limn→∞

[ f (x1¿)∆ x+ f (x2)∆ x+⋯+ f (xn)∆ x ]¿ , where

x0=a , xn=b , and ∆ x=b−an , and it represents a function’s net area

above the x–axis between a and b.

Rule Definite integrals of some curves can be calculated using geometric

formulas, and others can be approximated using left or right endpoints.

C Approximate Integration

Def’n The Midpoint Rule uses the midpoint of each rectangle to estimate the

definite integral as follows:

∫a

b

f (x )dx ≈ M n=∆ x [ f ( x1)+⋯+f ( xn)] , where x i=xi−1+x i2

.

Def’n The Trapezoidal Rule uses trapezoids to estimate the definite integral as follows:

∫a

b

f (x )dx ≈Tn=∆ x [f (x0)2

+f (x1)+⋯+f (xn−1)+f (xn)2

] , where

x i=a+ i∆ x.

Rule The Trapezoidal Rule is an average of the left endpoint approximationand the right endpoint approximation.

Rule The size of the error in the Trapezoidal Rule is usually about twice the size of the error in the Midpoint Rule.

Def’n Simpson’s Rule uses parabolas to estimate the definite integral asfollows:

∫a

b

f (x )dx ≈Sn=∆ x3

¿

+2 f (xn−2)+4 f (xn−1)+ f (xn)¿ , where n is even. Rule Simpson’s Rule is a weighted average of the Midpoint Rule ( 23 ) and

the Trapezoidal Rule ( 13 ).

Rule The size of the error in Simpson’s Rule is much smaller than the sizeof the error in either the Midpoint Rule or the Trapezoidal Rule.

5.2 The Fundamental Theorem of Calculus

Def’n If ' (x)=f (x ) , then the function F is an antiderivative of f.

Rule If F is an antiderivative of f, then so is (x)+C .

Def’n The indefinite integral of f is ∫ f (x )dx=F (x ) .

Rule If '=f , then ∫a

b

f (x )dx=F (b)−F(a) .

Rule The integral of a sum or difference of functions is given by

∫ [ f (x )± g(x )]dx=∫ f ( x)dx±∫ g (x)dx .

Rule The integral of a constant times a function is given by

∫ c ∙ f (x )dx=c∫ f (x )dx .

Rule The definite integral with the endpoints of the interval reversed is given

by ∫b

a

f (x )dx=−∫a

b

f (x )dx .

Rule The definite integral over adjacent intervals is given by

∫a

c

f (x )dx+∫c

b

f (x)dx=∫a

b

f (x)dx .

5.3 The Net Change Theorem and Average Value

Rule The integral of a rate of change is the net change and is given by

∫a

b

F ' (x)dx=F (b)−F (a) .

Def’n The average value of a function on the interval [a ,b] is given by

f ave=∫a

b

f (x )dx

b−a .

5.4 The Substitution Rule

Rule If ¿ g(x ) , then the chain rule can be reversed, resulting in

∫ f (g(x )) g ' (x )=∫ f (u)du .

Rule For definite integrals, ∫a

b

f (g(x )) g ' (x )=∫g (a)

g (b)

f (u)du .

5.5 Integration by Parts

Rule The product rule can be reversed, resulting in

∫ f (x )g ' (x )dx=f (x)g (x)−∫ g (x) f '( x)dx or ∫udv=uv−∫v du .

Rule For definite integrals, ∫a

b

f (x )g '(x )dx=¿ .

6.1 Areas Between Curves

Rule If f (x)≥g (x) on the interval [a ,b], then the area between f and g is

given by A=∫a

b

[ f (x)−g (x)]dx .

6.2 Consumer Surplus and Producer Surplus

Def’n If consumers are willing to buy a quantity of q units at a unit price of p dollars per unit, then the demand function D is given by p=D (q) .

Def’n The consumer surplus is the total difference in the price all consumers

are willing to pay for a good and the actual selling price.

Rule The consumer surplus of a good when Q units are sold at price P=D (q)

is given by ∫0

q

[D (q)−P]dq .

Def’n If producers are willing to sell a quantity of q units at a unit price of p dollars per unit, then the supply function S is given by p=S (q) .

Def’n The producer surplus is the total difference in the actual selling price of a good and the price all producers are willing to sell it for.

Rule The producer surplus of a good when Q units are sold at price P=S (q)

is given by ∫0

q

[P−S(Q)]dq .

Rule Total surplus is maximized when supply equals demand, or S(q)=D(q).6.4 Differential Equations

Def’n A separable equation is a differential equation whose derivative can be

expressed as a product of functions of the input and output variables.

Rule When solving a separable equation, separate the input and output variables, then integrate the equation, and solve for the output variable.

Rule Populations with growth that is directly proportional to the population

are modeled by the differential equation dPdt =k P , resulting in

exponential

growth given by P(t)=P0 ekt .

6.5 Improper Integrals

Def’n An improper integral is an integral with one or both endpoints at infinity.

Rule Improper integrals are given by ∫a

∞

f (x )dx=¿ limt→∞

∫a

t

f (x)dx¿ or

∫−∞

b

f (x)dx=¿ limt →−∞

∫t

b

f (x )dx¿.

Def’n The improper integral is convergent if its limit exists and divergent if its

limit does not exist.

Rule Integrals over the real line are given by

∫−∞

∞

f (x)d x=∫−∞

a

f (x )dx+∫a

∞

f (x )dx .

6.2, 6.5 Income Streams

Def’n The total amount of income earned from an income stream after a given

period of time is the future value of the income stream.

Rule The future value of income earned at a rate of f (t) dollars per year andinvested at interest rate r compounded continuously for T years is

given

by FV=erT∫0

T

f ( t )e−rtdt .

Def’n The amount of money invested now at interest rate r that would provide

the same future value as an income stream is the present value.

Rule The present value of income earned at a rate of f (t) dollars per year and

invested at interest rate r compounded continuously for T years is given

by PV=∫0

T

f (t)e−rt dt .

Rule The present value of a perpetuity is given by PV ∞=∫0

∞

f (t)e−rt dt .

7.1 Functio ns of Several Variables

Def’n A bivariate function is a rule that assigns an input pair (x , y ) to anoutput value z=f (x , y ), and its graph is the surface of all points (x , y , z)in space such that z=f (x , y ).

Def’n A vertical trace is the intersection of a bivariate function with a vertical plane, and it is found by setting one of the input variables equal to a constant.

Def’n A horizontal trace is the intersection of a bivariate function with a horizontal plane, and it is found by setting the output variable equal

to a constant.

Def’n A level curve of a function f is a projection of a horizontal trace onto the xy–plane, and its equation is f (x , y )=k .

Def’n A contour map is a set of level curves with different values of the

output variable.

7.2 Partial Derivatives

Def’n The partial derivative of a bivariate function is the rate of change ofthe function with respect to one input, holding the other input

constant.

Rule The partial derivatives of f (x , y ) are given by

f x (x , y )=limh→0

f (x+h , y)−f (x , y )h

and f y (x , y )= limh→ 0

f (x , y+h)−f (x , y)h

.

Note The following notations for partial derivatives are equivalent:

f x (x , y )=f x=∂ f∂x

= ∂∂x

f (x , y )= ∂z∂ x .

Rule When finding f x , x is treated as a variable and y as a constant, andwhen finding f y , y is treated as a variable and x as a constant.

Rule The partial derivative f x is the slope of the tangent line to the surface

of f parallel to the x–axis, and the partial derivative f y is the slope of the tangent line to the surface of f parallel to the y–axis.

Rule The second partial derivatives are given by

f xx=∂∂ x ( ∂ f∂ x )= ∂2 z

∂ x2, f xy= ∂

∂ y ( ∂ f∂ x )= ∂2 z∂ y ∂ x

, and f yy= ∂∂ y ( ∂ f∂ y )= ∂2 z

∂ y2 .

7.2 Partial Derivative Applications

Rule The Cobb-Douglas production function is given byP(L , K)=b LaK 1−a , where 0<a<1 .

Def’n The marginal productivity of labor ∂P∂ L is the rate of change of

production with respect to labor.

Def’n The marginal productivity of capital ∂P∂K is the rate of change of

production with respect to capital.

Def’n Substitute products are those for which an increase in demand for one

product results in a decrease in demand for the other product.

Def’n Complementary products are those for which an increase in demand

for one product results in an increase in demand for the other product.

Rule For substitute products, ∂q2∂ p1

>0 and ∂q1∂ p2

>0 , and

for complementary products, ∂q2∂ p1

<0 and ∂q1∂ p2

<0 .

7.3 Maximum and Minimum Values

Def’n If f (a ,b)≥ f (x , y ) for all (x , y ) near (a ,b), then f (a ,b) is a local maximum.If f (a ,b)≤ f (x , y ) for all (x , y ) near (a ,b), then f (a ,b) is a local minimum.

Rule If f (a ,b) is a local maximum or local minimum, then f x (x , y )=0 andf y (x , y )=0 .

Rule Given a critical point of a multivariate function and let

D=D (a , b)= f xx(a ,b) f yy(a ,b)−¿.(1)If D>0 and f xx(a , b)>0 , then f (a ,b) is a local minimum.

(2) If D>0 and f xx(a ,b)<0 , then f (a ,b) is a local maximum. (3) If D<0 then f (a , b) is a saddle point. (4) If D=0 then the test is inconclusive.

7.4 Lagrange Multipliers

Def’n The Lagrange function L(x , y , λ)=f (x , y )− λ[g (x , y )−k ] is used to find extreme values subject to a constraint.

Rule The minimum and maximum values of f (x , y ) subject to g(x , y)=k are found by solving f x (x , y )= λgx (x , y ), f y (x , y )=λ g y(x , y), andg(x , y)=k.

Rule The Lagrange multiplier λ is the ratio of the change in the optimal value

of f to the change in the constant k.

Rule At the optimum level of a production function, PL

PK=CL

CK and PL

CL=PK

CK.

D Double Integrals

Def’n The double integral of a multivariate function over a rectangular region

R={(x , y)∨a≤x ≤b , c≤ y≤d } is given by ∫a

b

∫c

d

f (x , y )dy dx.

Rule The double integral represents a multivariate function’s volume above the xy–plane.

Def’n The double integral of a multivariate function over a non-rectangular region D={(x , y)∨a≤x ≤b ,g1(x )≤ y ≤g2(x )} is given by

∫a

b

∫g1(x)

g2(x)

f (x , y)dy dx.

Def’n The average value of a multivariate function over a region R is given by

f ave=∫a

b

∫c

d

f (x , y)dy dx

A (R) .

10.1 Geometric Series

Def’n A finite series with n terms is given by a1+a2+a3+…+an.

Def’n A series written in sigma notation is given by ∑i=1

n

ai.

Def’n An infinite series is given by a1+a2+a3+…+an+….

Def’n A finite geometric series is given by a+ar+ar2+…+a rn−1,

where r is the common ratio.

Rule The sum of a finite geometric series is given by Sn=a (1−r n)1−r

.

Def’n An infinite geometric series is given by +ar+ar2+… .

Rule The sum of an infinite geometric series is given by Sn= a1−r , for |r|<1.

Rule If |r|≥1, then the infinite geometric series is divergent.

10.2 Taylor Polynomials

Def’n Factorial notation is given by n !=n(n−1)(n−2)…(3)(2)(1).

Def’n The first Taylor polynomial at x=0 of f (x) is given by

p1(x)=f (0)+ f ' (0) ∙ x.

Def’n The n th Taylor polynomial at x=0 of f (x) is given by

pn(x )=f (0)+ f ' (0)∙ x+ f ' '(0) ∙x2

2 !+…+ f (n)(0)∙ x

n

n!.

Def’n The error of the nth Taylor polynomial at x=0 of f (x) is given by

Rn(x )=f (n+ 1)(t) ∙ xn+1

(n+1)!, where 0≤ t ≤ x.

Def’n The n th Taylor polynomial at x=a of f (x) is given by

pn(x )=f (a)+f ' (a)∙( x−a)+f ' ' (a)∙(x−a)2

2 !+…+ f (n)(a)∙ (x−a)

n

n !.

Def’n The error of the nth Taylor polynomial at x=a of f (x) is given by

Rn(x )=f (n+1)(t) ∙ (x−a)n+1

(n+1)!, where a≤ t ≤ x.

10.3 Taylor Series

Def’n A power series is an infinite geometric series with variable terms, and

it takes the form a0+a1 x+a2x2+…+an xn+… .

Def’n The radius of convergence R is a number such that a power series

converges if |x|<R and diverges if |x|>R.

Def’n The ratio of terms in a power series is given by r=limn→∞

cn+1

cn.

Rule A power series converges if |r|<1 and diverges if |r|>1.

Def’n The Taylor series at x=0 of f (x) is given by

T 0(x)=f (0)+ f '(0) ∙ x+ f ' ' (0) ∙ x2

2!+…+f (n)(0) ∙ x

n

n !+….

Def’n The Taylor series at x=a of f (x) is given by

T a ( x )= f (a )+ f ' (a ) ∙ ( x−a )+f ' ' (a ) ∙ ( x−a )2

2!+…+ f (n ) (a ) ∙¿¿..

Rule The Taylor series expansions of some common functions are given by:

ex=1+ x1 !

+ x2

2 !+ x

3

3 !+ x4

4 !+… at x=0, for −∞<x<∞

ln (x+1)=x− x2

2+ x

3

3− x4

4+…at x=0, for −1<x<1

11−x

=1+ x+x2+x3+ x4+… at x=0, for −1<x<1

10.4 Newton’s Method

Rule The solution to f (x)=0 can be approximated using the following

procedure with an initial guess x=x0:

(1) replace the function with its first Taylor polynomial at x=x0,

(2) solve p1(x0)=0 to get x=x0−f (x0)f '(x0)

,

(3) use an improved guess x=x1 and repeat steps 1 and 2.

Def’n The internal rate of return r is the interest rate for which the present

values of all payments add up to the loan amount.

Rule Newton’s Method may not work if one of the following conditions is true:

(1) the derivative is zero at any approximation

(2) an inflection point exists between two successive approximations

(3) a critical point exists near an approximation