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 the
limit of the sum of approximating rectangles, and is given by
𝒜 = lim 𝑅𝑛 = lim [𝑓(𝑥1 )∆𝑥 + 𝑓(𝑥2 )∆𝑥 + ⋯ + 𝑓(𝑥𝑛 )∆𝑥] .
𝑛→∞
𝑛→∞
Def’n The definite integral of f from a to b is given by
𝑏
∫𝑎 𝑓(𝑥)𝑑𝑥 = lim [𝑓(𝑥1 )∆𝑥 + 𝑓(𝑥2 )∆𝑥 + ⋯ + 𝑓(𝑥𝑛 )∆𝑥] , where
𝑛→∞
𝑥0 = 𝑎 , 𝑥𝑛 = 𝑏 , and ∆𝑥 =
𝑏−𝑎
𝑛
, 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:
𝑏
∫𝑎 𝑓(𝑥)𝑑𝑥 ≈ 𝑀𝑛 = ∆𝑥[𝑓(𝑥̅1 ) + ⋯ + 𝑓(𝑥̅𝑛 )] , where 𝑥̅𝑖 =
𝑥𝑖−1 +𝑥𝑖
2
.
Def’n The Trapezoidal Rule uses trapezoids to estimate the definite integral as
follows:
𝑏
∫𝑎 𝑓(𝑥)𝑑𝑥 ≈ 𝑇𝑛 = ∆𝑥[
𝑓(𝑥0 )
2
+ 𝑓(𝑥1 ) + ⋯ + 𝑓(𝑥𝑛−1 ) +
𝑓(𝑥𝑛 )
2
] , where
𝑥𝑖 = 𝑎 + 𝑖∆𝑥.
Rule The Trapezoidal Rule is an average of the left endpoint approximation
and the right endpoint approximation.
Rule The size of the error in the Trapezoidal Rule is often about twice the
size of the error in the Midpoint Rule.
Def’n Simpson’s Rule uses parabolas to estimate the definite integral as
follows:
𝑏
∫𝑎 𝑓(𝑥)𝑑𝑥 ≈ 𝑆𝑛 =
∆𝑥
3
[𝑓(𝑥0 ) + 4𝑓(𝑥1 ) + 2𝑓(𝑥2 ) + 4𝑓(𝑥3 ) + ⋯
+2𝑓(𝑥𝑛−2 ) + 4𝑓(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 size
of 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) .
𝑏
Rule If ′ = 𝑓 , then ∫𝑎 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
𝑎
𝑏
by ∫𝑏 𝑓(𝑥)𝑑𝑥 = − ∫𝑎 𝑓(𝑥)𝑑𝑥 .
Rule The definite integral over adjacent intervals is given by
𝑐
𝑏
𝑏
∫𝑎 𝑓(𝑥)𝑑𝑥 + ∫𝑐 𝑓(𝑥)𝑑𝑥 = ∫𝑎 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
𝑏
∫𝑎 𝐹′(𝑥)𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎) .
Def’n The average value of a function on the interval [𝑎, 𝑏] is given by
𝑏
𝑓𝑎𝑣𝑒 =
∫𝑎 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 .
𝑏
𝑔(𝑏)
Rule For definite integrals, ∫𝑎 𝑓(𝑔(𝑥))𝑔′(𝑥) = ∫𝑔(𝑎) 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 ∫ 𝑢 𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣 𝑑𝑢 .
𝑏
𝑏
Rule For definite integrals, ∫𝑎 𝑓(𝑥)𝑔′(𝑥)𝑑𝑥 = [𝑓(𝑥)𝑔(𝑥)]𝑏𝑎 − ∫𝑎 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
𝑏
given by 𝒜 = ∫𝑎 [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)
𝑄
is given by ∫0 [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)
𝑄
is given by ∫0 [𝑃 − 𝑆(𝑄)]𝑑𝑞 .
Rule Total surplus is maximized when supply equals demand, or 𝑆(𝑞) = 𝐷(𝑞).
6.2 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 and
invested at interest rate r compounded continuously for T years is given
𝑇
by 𝐹𝑉 = 𝑒 𝑟𝑇 ∫0 𝑓(𝑡)𝑒 −𝑟𝑡 𝑑𝑡 .
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
𝑇
by 𝑃𝑉 = ∫0 𝑓(𝑡)𝑒 −𝑟𝑡 𝑑𝑡 .
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
𝑑𝑃
𝑑𝑡
= 𝑘𝑃 , resulting in exponential
growth given by 𝑃(𝑡) = 𝑃0 𝑒 𝑘𝑡 .
Rule Populations with growth that is limited by a carrying capacity M are
modeled by the differential equation
logistic growth given by
𝑃(𝑡) =
𝑑𝑃
𝑑𝑡
𝑃
= 𝑘𝑃 (1 − ) , resulting in
𝑀
𝑀
1+𝐴𝑒 −𝑘𝑡
, where 𝐴 =
𝑀−𝑃0
𝑃0
.
6.5 Improper Integrals
Def’n An improper integral is an integral with one or both endpoints at infinity.
∞
𝑡
Rule Improper integrals are given by ∫𝑎 𝑓(𝑥)𝑑𝑥 = lim ∫𝑎 𝑓(𝑥)𝑑𝑥 or
𝑡→∞
𝑏
𝑏
∫−∞ 𝑓(𝑥)𝑑𝑥 = lim ∫𝑡 𝑓(𝑥)𝑑𝑥.
𝑡→−∞
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
∞
𝑎
∞
∫−∞ 𝑓(𝑥)𝑑𝑥 = ∫−∞ 𝑓(𝑥)𝑑𝑥 + ∫𝑎 𝑓(𝑥)𝑑𝑥 .
∞
Rule The present value of a perpetuity is given by 𝑃𝑉 = ∫0 𝑓(𝑡)𝑒 −𝑟𝑡 𝑑𝑡 .
6.6 Probability
Def’n A continuous random variable is an unknown quantity whose values
can range over an entire interval of values.
Def’n A probability density function describes the probability that a random
variable will equal a particular set of values.
Rule The probability that a random variable X lies between a and b is given by
𝑏
𝑃(𝑎 ≤ 𝑋 ≤ 𝑏) = ∫𝑎 𝑓(𝑥)𝑑𝑥.
∞
Rule For all p.d.f.’s, ∫−∞ 𝑓(𝑥)𝑑𝑥 = 1.
Def’n The mean 𝜇 of a random variable is an average of all possible values of
the random variable, weighted by their probabilities.
∞
Rule The mean of a random variable is given by 𝜇 = ∫−∞ 𝑥 ∙ 𝑓(𝑥)𝑑𝑥.
Def’n The standard deviation 𝜎 of a random variable measures the spread
between typical values of the random variable.
6.6 Exponential and Normal Distributions
1
Rule The p.d.f. of an exponential distribution is given by 𝑓(𝑡) = 𝜇 𝑒
−
𝑡
𝜇
, for 𝑡 ≥ 0,
where 𝜇 is the mean (and the standard deviation).
∞
Rule For an exponential p.d.f., ∫𝑎 𝑓(𝑥)𝑑𝑥 is given by 𝑒
−
𝑎
𝜇
.
∞
1
Rule For an exponential p.d.f., ∫𝑐 𝑓(𝑥)𝑑𝑥 = 𝑝 is solved for c by 𝜇 ∙ ln 𝑝.
1
Rule The p.d.f. of a normal distribution is given by 𝑓(𝑥) = 𝜎√2𝜋 𝑒
−
(𝑥−𝜇)2
2𝜎2
, where
𝜇 is the mean and 𝜎 is the standard deviation.
𝑏
Rule For a normal p.d.f., ∫𝑎 𝑓(𝑥)𝑑𝑥 is given by 𝑛𝑜𝑟𝑚𝑎𝑙𝑐𝑑𝑓(𝑎, 𝑏, 𝜇, 𝜎).
𝑐
Rule For a normal p.d.f., ∫−∞ 𝑓(𝑥)𝑑𝑥 = 𝑝 is solved for c by 𝑖𝑛𝑣𝑛𝑜𝑟𝑚(𝑝, 𝜇, 𝜎).
7.1 Functions of Several Variables
Def’n A bivariate function is a rule that assigns an input pair (𝑥, 𝑦) to an
output 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 of
the function with respect to one input, holding the other input constant.
Rule The partial derivatives of 𝑓(𝑥, 𝑦) are given by
𝑓𝑥 (𝑥, 𝑦) = lim
𝑓(𝑥+ℎ,𝑦)−𝑓(𝑥,𝑦)
and 𝑓𝑦 (𝑥, 𝑦) = lim
ℎ
ℎ→0
ℎ→0
𝑓(𝑥,𝑦+ℎ)−𝑓(𝑥,𝑦)
ℎ
.
Note The following notations for partial derivatives are equivalent:
𝑓𝑥 (𝑥, 𝑦) = 𝑓𝑥 =
𝜕𝑓
𝜕𝑥
=
𝜕
𝜕𝑥
𝑓(𝑥, 𝑦) =
𝜕𝑧
𝜕𝑥
.
Rule When finding 𝑓𝑥 , x is treated as a variable and y as a constant, and
when finding 𝑓𝑦 , y is treated as a variable and x as a constant.
Rule The partial derivative 𝑓𝑥 is the slope of the tangent line to the surface
of f parallel to the x–axis, and the partial derivative 𝑓𝑦 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
𝑓𝑥𝑥 =
𝜕
𝜕𝑓
𝜕2 𝑧
𝜕
𝜕𝑓
𝜕2 𝑧
𝜕
𝜕𝑓
𝜕2 𝑧
( ) = 𝜕𝑥 2 , 𝑓𝑥𝑦 = 𝜕𝑦 (𝜕𝑥 ) = 𝜕𝑦 𝜕𝑥 , and 𝑓𝑦𝑦 = 𝜕𝑦 (𝜕𝑦) = 𝜕𝑦2 .
𝜕𝑥 𝜕𝑥
7.2 Partial Derivative Applications
Rule The Cobb-Douglas production function is given by
𝑃(𝐿, 𝐾) = 𝑏𝐿𝑎 𝐾 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
for complementary products,
> 0 and
𝜕𝑞2
𝜕𝑝1
𝜕𝑞1
𝜕𝑝2
< 0 and
> 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 𝑓𝑥 (𝑥, 𝑦) = 0 and
𝑓𝑦 (𝑥, 𝑦) = 0 .
Rule Given a critical point of a multivariate function and let
𝐷 = 𝐷(𝑎, 𝑏) = 𝑓𝑥𝑥 (𝑎, 𝑏)𝑓𝑦𝑦 (𝑎, 𝑏) − [𝑓𝑥𝑦 (𝑎, 𝑏)]2 .
(1) If 𝐷 > 0 and 𝑓𝑥𝑥 (𝑎, 𝑏) > 0 , then 𝑓(𝑎, 𝑏) is a local minimum.
(2) If 𝐷 > 0 and 𝑓𝑥𝑥 (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 𝑓𝑥 (𝑥, 𝑦) = 𝜆𝑔𝑥 (𝑥, 𝑦), 𝑓𝑦 (𝑥, 𝑦) = 𝜆𝑔𝑦 (𝑥, 𝑦), and
𝑔(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
𝑏
𝑑
𝑅 = {(𝑥, 𝑦)|𝑎 ≤ 𝑥 ≤ 𝑏, 𝑐 ≤ 𝑦 ≤ 𝑑} is given by ∫𝑎 ∫𝑐 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
𝑏
𝑔2(𝑥)
∫𝑎 ∫𝑔1(𝑥) 𝑓(𝑥, 𝑦) 𝑑𝑦 𝑑𝑥.
Def’n The average value of a multivariate function over a region R is given by
𝑏 𝑑
𝑓𝑎𝑣𝑒 =
∫𝑎 ∫𝑐 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
∑𝑛𝑖=1 𝑎𝑖 .
Def’n An infinite series is given by 𝑎1 + 𝑎2 + 𝑎3 + ⋯ + 𝑎𝑛 + ⋯.
Def’n A finite geometric series is given by 𝑎 + 𝑎𝑟 + 𝑎𝑟 2 + ⋯ + 𝑎𝑟 𝑛−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 + 𝑎𝑟 2 + ⋯ .
Rule The sum of an infinite geometric series is given by 𝑆𝑛
=
Rule If |𝑟| ≥ 1, then the infinite geometric series is divergent.
𝑎
1−𝑟
, for |𝑟| < 1.
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 nth 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) ∙
𝑥𝑛
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) ∙
𝑥 𝑛+1
, where
(𝑛+1)!
0 ≤ 𝑡 ≤ 𝑥.
Def’n The nth 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 + 𝑎2 𝑥 2 + ⋯ + 𝑎𝑛 𝑥 𝑛 + ⋯ .
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 = lim
𝑐𝑛+1
𝑛→∞ 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
𝑇0 (𝑥) = 𝑓(0) + 𝑓′(0) ∙ 𝑥 + 𝑓′′(0) ∙
𝑥2
2!
+ ⋯ + 𝑓 (𝑛) (0) ∙
𝑥𝑛
𝑛!
+ ⋯.
Def’n The Taylor series at 𝑥 = 𝑎 of 𝑓(𝑥) is given by
𝑇𝑎 (𝑥) = 𝑓(𝑎) + 𝑓′(𝑎) ∙ (𝑥 − 𝑎) + 𝑓′′(𝑎) ∙
(𝑥−𝑎)2
2!
+ ⋯ + 𝑓 (𝑛) (𝑎) ∙
(𝑥−𝑎)𝑛
𝑛!
+ ⋯.
Rule The Taylor series expansions of some common functions are given by:
𝑥
𝑒 =1+
𝑥
1!
+
𝑥2
2!
ln(𝑥 + 1) = 𝑥 −
1
1−𝑥
+
𝑥2
2
𝑥3
3!
+
+
𝑥3
3
𝑥4
4!
−
+⋯
𝑥4
4
+⋯
= 1 + 𝑥 + 𝑥2 + 𝑥3 + 𝑥4 + ⋯
at 𝑥 = 0, for −∞ < 𝑥 < ∞
at 𝑥 = 0, for −1 < 𝑥 < 1
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