Calculus 8e, by James Stewart Welcome to the world inside my head. Embedded in this map is my way of connecting definitions and theorems in order to form new and original ideas. My goal is to solve the best problems I can find to the best of my abilitie

Definition 11.8.1 A power series is a series of the form Σcnxn=c0+c1x+c2x2+c3x3... where x is a variable and the cn's are the coefficients of the series.

Definition 11.8.2 More generally, a series of the form Σcn(x-a)n is called a power series centered at a or a power series about a.

Theorem 11.8.3 For a given power series Σcn(x-a)n, there are only three possibilities: (i) The series converges only when x=a. (ii) The series converges for all x. (iii) There is a positive number R s.t. the series converges if |x-a|R. Definition 11.8.4 The number R in (iii) is called the radius of convergence of the power series. By convention, the radius of convergence is R =0 in case (i) and R=∞ is case (ii). Definition 11.8.5 The interval of convergence of a power series is the interval that consists of all values of x for which the series converges. In case (i), the interval consists of just a single point a. In case (ii), the interval is (-∞,∞). Incase (iii), note that the inequality can be rewritten as a-R

Definition 11.5.1 An alternating series is a series whose terms are alternately positive and negative.

Theorem 11.5.2 If the alternating series Σ(-1)n-1bn satisfies (i) bn+1≤bn, i.e. the series is decreasing (ii) limn->∞bn=0 then the series is convergent.

Theorem 11.5.3 If s=Σ(-1)n-1bn, where bn>0, is the sum of an alternating series that satisfies (i) and (ii), then |Rn|≤bn+1

Definition 11.3.3 The remainder Rn=s-sn=an+1+an+2+an+3+... is the error made when sn is used as an approximation to the total sum.

Definition 11.1.1 A sequence can be thought of as a list of numbers written in a definite order: a1, a2, a3, ..., an,... Notation. The sequence {a1, a2, a3,...} is denoted {an}

Definition 11.2.1 If we try to add the terms of an infinite sequence {an} we get an expression of the form Σan which is called an infinite series. We consider the partials sums (n=1 to n)Σan=sn. If limn->∞sn=s exists, then we call it the sum of the infinite series.

Definition 11.2.2 Given a series Σan, let sn denote its nth partial sum. If the sequence {sn} is convergent and limn->∞sn=s exists as a real number, then the series Σan is convergent and we write Σan=s. The number is called the sum of the series. If the sequence {sn} is divergent, then the series is called divergent.

Definition 11.6.1 A series Σan is called absolutely convergent if Σ|an| is convergent.

Theorem 11.6.3 If a series Σan is absolutely convergent, then it is convergent.

Theorem 11.6.5 (The Root Test) (i) If limn->∞n√(|an|)=l<1, then Σan is absolutely convergent. (ii) (i) If limn->∞n√(|an|)=l>1, then Σan is divergent. (iii) (i) If limn->∞n√(|an|)=1, the Root Test is inconclusive.

Definition 11.6.2 A series Σan is called conditionally convergent if it is convergent but not absolutely convergent.

Theorem 11.4.1 Suppose Σan and Σbn are series with positive terms. (i) If Σbn is convergent and an≤bn, then Σan is also convergent. (ii) If Σbn is convergent and an≥bn, then Σan is also divergent.

Theorem 11.4.2 Suppose that Σan and Σbn are series with positive terms. If limn->∞an/bn=c where c is a finite number and c>0, then either both series converge or both diverge.

Theorem 11.4.3 Let Σan be a convergent series with remainder Rn=s-sn. For the comparison series Σbn, we consider the corresponding remainder Tn=b-bn. Since an≤bn for all n, we have Rn≤Tn. If Σbn is a p-series, we can estimate its remainder Tn. If Σbn is a geometric series, then Tn is the sum of a geometric series and we can sum it exactly.

Theorem 11.3.1 (The Integral Test) Suppose f is a continuous, positive, decreasing function on (1,∞] and let an=f(n). Then the series Σan is convergent iff the improper integral (a to ∞)∫f(x)dx is convergent. In other words: (i) If (1 to infinity)∫f(x)dx is convergent, then Σan is convergent. (ii) If (1 to infinity)∫f(x)dx is divergent, then Σan is divergent.

Theorem 11.3.4 Suppose f(k)=ak, where f is a continuous, positive, decreasing function for x≥n and Σan is convergent. If Rn=a-an, then ((n+1) to ∞)∫f(x)dx≤Rn≤(n to ∞)∫f(x)dx A consequence to this theorem is sn + ((n+1) to ∞)∫f(x)dx ≤ s ≤ sn + (n to ∞)∫f(x)dx

Theorem 11.3.2 The p-series Σ1/np is convergent if p>1 and divergent if p≤1.

Theorem 11.2.8 If Σan and Σbn are convergent series, then (i) The series preserves constant multiplicity. (ii) The series preserves addition. (ii) The series preserves subtraction.

Theorem 11.2.6 If the series Σan is convergent, then limn->∞an=0.

Theorem 11.2.7 (Test for Divergence) If limn->∞an does not exists of if limn->∞an≠0, then the series Σan is divergent.

Definition 11.2.5 The harmonic series Σ1/n is divergent.

Definition 11.2.3 An important example of an infinite series is the geometric series Σarn-1. Each term is obtained from the preceding one by multiplying it by the common ratio r. If r=1, then limn->∞sn does not exist. If r≠1, then sn=a(1-rn)/(1-r). If -1∞sn=a/(1-r)

Theorem 11.2.4 The geometric series Σarn-1 is convergent if |r|<1 and its sum is Σarn-1=a/(1-r). If |r|≥1, the geometric series is divergent.

Definition 11.1.11 A sequence {an} is bounded above if there is a number M s.t. an≤M for all n≥1. It is bounded below if there is a number m s.t. m≤an for all n≥1. If it is bounded above and below, then {an} is a bounded sequence.

Theorem 11.1.12 Every bounded monotonic sequence is convergent.

Definition 11.1.10 A sequence {an} is called increasing if anan+1 for all n≥1. A sequence is monotonic if it is either increasing or decreasing.

10.3

10.4

8.1,8.2

Floating topic

Gini coefficient Egalitarian society

4.7 6.4,6.5

Definition 5.1.4 We often use sigma notation to write sums with many terms more compactly. For example, (i=1 to n)Σf(xi)Δx=f(x1)Δx+f(x2)Δx+...+f(xn)Δx

Definition 5.1.1 The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles: (Left end points) A=limn->∞Rn= limn->∞[f(x1)Δx+f(x2)Δx+...+f(xn)Δx] (Right end points) A=limn->∞Ln= limn->∞[f(x0)Δx+f(x1)Δx+...+f(xn-1)Δx]

Definition 7.8.1 (Definition of an Improper Integral of Type 1) (a) If (a to t)∫f(x)dx exists for every number t≥a, then (a to ∞)∫f(x)dx=limt->∞(a to t)∫f(x)dx provided this limit exists. (b) If (t to b)∫f(x)dx exists for every number t≤b, then (-∞ to b)∫f(x)dx=limt->-∞(a to t)∫f(x)dx provided this limit exists. These improper integrals are called convergent if the corresponding limit exists and divergent if the limit does not exist. (c) If both (a to ∞)∫f(x)dx and (-∞ to a)∫f(x)dx are convergent, then we define (-∞ to ∞)∫f(x)dx=(-∞ to a)∫f(x)dx + (a to ∞)∫f(x)dx

Theorem 7.8.4 (Comparison Theorem) Suppose that f and g are continuous functions with f(x)≥g(x)≥0 for x≥a. (a) If (a to ∞)∫f(x)dx is convergent, then (a to ∞)∫g(x)dx is convergent. (b) If (a to ∞)∫g(x)dx is divergent, then (a to ∞)∫f(x)dx is divergent.

Theorem 7.8.2 (1 to ∞)∫f(x)dx is convergent if p>1 and divergent if p<1

Definition 7.8.3 (Definition of an Improper Integral of Type 2) (a) If f is continuous on [a,b) and is discontinuous at b, then (a to b)∫f(x)dx=limt->b-(a to t)∫f(x)dx provided this limit exists. (b) If f is continuous on (a,b] and is discontinuous at a, then (a to b)∫f(x)dx=limt->a+(t to b)∫f(x)dx provided this limit exists. These improper integrals are called convergent if the corresponding limit exists and divergent if the limit does not exist. (c) If f has a discontinuity at c, where a

Theorem 6.1.1 The area A of the region bounded by the curves y=f(x), y=g(x), and the lines x=a, x=b, where f and g are continuous and f(x)≥g(x) for all x in [a,b], is A=(a tob)∫[f-g]dx

Definition 6.2.1 Let S be a solid that lies between x=a and x=b. If the cross-sectional area of S in the plane Px, through x and perpendicular to the x-axis, is A(x), were A is a continuous function, then the volume of S is V=limn->∞ΣA(xi*)Δx= (a to b)∫A(x)dx

Definition 6.3.1 The method of cylindrical shells computes volume V by subtracting the volume V1 of the inner cylinder from the volume V2 of the outer cylinder: V=V2-V1 =πr22h-πr12h =π(r22-r12)h =π(r2-r1)(r2+r1)h =2π(r1+r2/h)h(r2-r1) =2πrhΔr

Theorem 6.3.2 The volume of the solid obtaining by rotating about the y-axis the region under the curve y=f(x) from a to b, is V=(a to b)∫2πxf(x)dx, where 0≤a≤b.

Definition 6.2.2 The solids of revolution is obtained by revolving a region about a line. In general, we calculate the volume of a solid of revolution by using the basic defining formula V=(a to b)∫A(x)dx=(c to d)∫A(y)dy. If the cross-section is a disk, we find the radius of the disk and use A=π(radius)2 If the cross-section is a washer, we find the inner radius r and outer radius R from a sketch and compute the area of the washer by subtracting the area of the inner disk from the area of the outer disk: A=πR2-πr2.

Definition 5.2.1 If f is a function defined for a≤x≤b, we divide the interval [a,b] into n subinterval of equal width Δx=b-a/n. We let x0=a,x1,x2,...,xn=b be the endpoints of these subintervals and we let x1,x2,...,xn* be any sample points in these subintervals, so xi* lies in the ith subinterval [xi-1,xi]. Then the definite integral of f from a to be is (a to b)∫f(x)dx=limn->∞(i=1 to n)Σf(xi*)Δx provided this limit exists and gives the same value for all possible choices of sample points. If it does exists, we say that f is integrable on [a,b]. This is called a Reimann sum after German mathematician Bernhard Riemann (1826-1866). Definition 5.2.2 f(x) is called the integrand and a and b are called the limits of integration; a is the lower limit and b is the upper limit. Definition 5.2.3 The procedure of calculating an integral is called integration. Definition 5.2.4 A definite integral can be interpreted as a net area, that is, a difference of areas (a to b)∫f(x)dx=A1-A2 where A1 is the area of the region above the x-axis and below the grpah of f, and A2 is the area of the region below the x-axis and above the graph of f.

Theorem 5.3.1 (The Fundamental Theorem of Calculus, Part 1) If f is continuous on [a,b], then the function g defined by g(x)=(a to x)∫f(t)dt is continuous on [a,b] and differentiable on (a,b) and g'(x)=f(x).

Theorem 5.3.2 (The Fundamental Theorem of Calculus, Part 2) If f is continuous on [a,b], then (a to b)∫f(x)dx=F(a)-F(b) where F is any antiderivative of f, that is, a function s.t. F'=f.

Theorem 5.2.5 (Properties of Integrals) 1. (a to b)∫cdx=c(b-a), where c is any constant. 2. (a to b)∫[f(x)+g(x)]dx=(a to b)∫f(x)dx+(a to b)(a to b)∫g(x)dx 3. (a to b)∫cf(x)dx=c(a to b)∫f(x)dx, where c is any constant. 4. (a to b)∫[f(x)-g(x)]dx=(a to b)∫f(x)dx-(a to b)∫g(x)dx 5.(a to b)∫f(x)dx=(b to a)-∫f(x)dx 6. (a to b)∫f(x)dx=0 if a=b 7.(a to c)∫f(x)dx+(c to b)∫f(x)dx=(a to b)∫f(x)dx Comparison Theorems for Integrals 8. If f(x)≥0 for a≤x≤b, then (a to b)∫f(x)dx≥0 9. If f(x)≥g(x) for a≤x≤b, then (a to b)∫f(x)dx≥(a to b)∫g(x)dx 10. If m≤f(x)≤M for a≤x≤b, then m(b-a)≤(a to b)∫f(x)dx≤M(b-a)

Theorem 5.5.3 (Integrals of Symmetric Functions) Suppose f is continuous on [-a,a]. (a) If f is even, then (a to -a)∫f(x)dx=2(a to 0)∫f(x)dx (b) If f is odd, then (a to -a)∫f(x)dx=0

Theorem 5.2.4 If f is continuous on [a,b], or if f has only a finite number of jump discontinuities, then f is integrable on [a,b]; that is, the definite integral (a to b)∫f(x)dx exists.

Definition 5.1.2 Instead of using left endpoints or right endpoints, we could take the height of the ith rectangle to be the value of f at any number xi* in the ith subinterval [xi-1,xi]. We call the numbers x1,x2,...,xn* the sample points.

Theorem 7.7.1 (Midpoint Rule) M(n)≈(a to b)∫f(x)dx ≈[f(x1+f(x2)+...+f(xn)]Δx ≈(i=1 to n)Σf(xi)Δx where Δx=(b-a)/n and xi*=(1/2)(xi-1+xi)

Theorem 7.7.4 (Simpson's Rule) (a to b)∫f(x)dx≈ S(n)= (Δx/3)[f(x0)+4f(x1)+2f(x2)+...+2f(xn-2)+4f(xn-1)+f(xn)], where n is even and Δx=(b-a)/n

Theorem 7.7.5 (Error Bounds for Simpson's Rule) Suppose that |f(4)(x)|≤K for a≤x≤b. If Es is the error involved in using Simpson's Rule, then |Es|≤K(b-a)5/180n4

Theorem 7.7.2 (Trapezoidal Rule) T(n)≈(a to b)∫f(x)dx ≈[(1/2)f(x0)+f(x1)+f(x2)+...+f(xn-1)+ (1/2)f(xn)]Δx ≈(1/2f(x0)+(i=1 to n-1)Σf(xi)+(1/2)f(xn) where Δ=(b-a)/n and xi=a+iΔx

Theorem 7.7.3 (Error Bounds for Trapezoidal Rule) absolute and relative errors

Definition 5.1.3 We form lower (and upper) sums by choosing the sample points xi* so that f(xi*) is the minimum (and maximum) value of f on the ith subinterval.

Definition 4.9.1 A function F is called an antiderivative of f on an interval I if F'(x)=f(x) for all x in I.

Theorem 4.9.2 If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is F(x)+C where C is an arbitrary constant.

Definition 5.4.1 The relation between antiderivatives and integrals given by the Fundamental Theorem of Calculus, the notation ∫f(x)dx is traditionally used for an antiderivative of f and is called an indefinite integral. Thus, ∫f(x)dx=F(x) means F'(x)=f(x)

Theorem 5.4.2 (Table of Indefinite Integrals)

Theorem 5.4.3 (Net Change Theorem) The integral of a rate of change is the net change: (a to b)∫F'(x)dx=F(b)-F(a).

Theorem 5.5.1 (The Substitution Rule) If u=g(x) isa differentiable function whose range is an interval I and f is continuous on I, then ∫f(g(x))g'(x)=∫f(u)du

Theorem 5.5.2 (The Substitution Rule for Definite Integrals) If g' is continuous on [a,b] and f is continuous on the range of u=g(x), then (a to b)∫f(g(x))g'(x)dx= (g(a) to g(b))∫f(u)du

Theorem 7.3.1 Case 1. sqrt(a2+x2). Substitute x=atan(θ) Case 2. sqrt(a2-x2). Substitute x=asin(θ) Case 3. sqrt(x2-a2). Substitute x=asec(θ)

Theorem 7.4.1 We integrate rational functions of the form f(x)=P(x)/Q(x) by expressing it as sum of simpler fractions, called partial fractions. Case 0. deg(P)≥deg(Q). Use polynomial long division and integrate. Case 1. Q(x) is a product of distinct linear factors. Case 2. Q(x) is a product of linear factors, some of which are repeated. Case 3. Q(x) contains irreducible quadratic factors, none of which is repeated. Numerator is written as linear functions with arbitrary coefficents. Case 4. Q(x) contains a repeated irreducible quadratic factor.

Theorem 7.1.1 The formula ∫udv=uv-∫vdu is called the formula for integration by parts.

Strategy 7.2.3 To evaluate the integrals ∫sin(mx)cos(nx)dx or ∫sin(mx)sin(nx)dx or ∫cos(mx)cos(nx)dx, use the corresponding identity: (a) sin(A)cos(B)=1/2[sin(A-B)+sin(A+B)] (b) sin(A)sin(B)=1/2[cos(A-B)-cos(A+B)] (c) cos(A)cos(B)=1/2[cos(A-B)+cos(A+B)]

Strategy 7.2.2 (Strategy for Evaluating ∫tanm(x)secn(x)dx) (a) If the power of secant is even, save a factor of sec2(x) and use sec2(x)=1+tan2(x) to express the remaining factors in terms of tan(x). Then substitute =tan(x). (b) If the power of tangent is odd, save a factor of sec(x)tan(x) and use tan2(x)=sec2(x)+1 to express the remaining factors in terms of sec(x). Then substitute u=sec(x).

Strategy 7.2.1 (Strategy for Evaluating ∫sin(mx)cos(nx)dx) (a) If the power of cosine is odd, save one cosine factor and use cos2(x)=1-sin2(x) to express the remaining factors in terms of sine. Then substitute u=sin(x). (b) If the power of sine is odd, save one sine factor and use sin2(x)=1-cos2(x). Then substitute u=cos(x). (c) If the powers of both sine and cosine are even, use the half angle trigonometric identities.

Definition 3.11.1 sinh(x)=[ex-e-x]/2 cosh(x)=[ex+e-x]/2 tanh(x)=sinh(x)/cosh(x) csch(x)=1/sinh(x) sech(x)=1/cosh(x) coth(x)=cosh(x)/tanh(x)

Theorem 3.11.2 sinh(-x)=-sinh(x) cosh(-x)=cosh(x) cosh2(x)-sinh2(x)=1 1-tanh2(x)=sech(x) sinh(x+y)=sinh(x)cosh(y)+cosh(x)sinh(y) cosh(x+y)=cosh(x)cosh(y)+sinh(x)sinh(y)

Theorem 3.3.1a limθ->0sin(θ)/θ=1 Theorem 3.3.1a limθ->0cos(θ)-1/θ=0

Theorem 3.3.2 (Derivatives of Trigonometric Functions) d/dx[sin(x)]=cos(x) d/dx[cos(x)]=-sin(x) d/dx[tan(x)]=sec2(x) d/dx[csc(x)]=-csc(x)tan(x) d/dx[sec(x)=sec(x)tan(x) d/dx[cot(x)]=-csc2(x)

3.7,3.8,3.9,3.10,3.11

Definition 4.1.5 A critical number of a function f is a number c in the domain of f s.t. either f'(c)=0 or f'(c) does not exists.

Theorem 4.1.6 If f has a local maximum value or minimum value at c, then c is a critical number of f.

Theorem 4.2.1 (Rolle's Theorem) Let f be a function that satisfies the following three hypothesis: 1. f is continuous on the closed interval [a,b]. 2. f is differentiable on the open interval (a,b). 3. f(a)=f(b). Then there is a number cε(a,b) s.t. f'(c)=0.

Theorem 4.2.2 (The Mean Value Theorem) Let f be a function that satisfies the following hypothesis: 1. f is continuous on the closed interval [a,b]. 2. f is differentiable on the open interval (a,b). Then there is a number cε(a,b) s.t. f'(c)=(f(b)-f(a))/(b-a) or f(b)-f(a)=f'(c)(b-a)

Theorem 4.2.3 If f'(x)=0 for all xε(a,b), then f is constant on (a,b).

Corollary 4.2.4 If f'(x)=g'(x) for all xε(a,b), then f-g is constant on (a,b); that is f(x)=g(x)+c where c is a constant.

The Closed Interval Method To find the absolute maximum and minimum values of a continuous function f on a closed interval [a,b]: 1. Find the values of f at the critical numbers of f in (a,b). 2. Find the values of f at the endpoints of the interval. 3. The largest of the values from 1. and 2. is the absolute value; the smallest of these values is the absolute minimum value.

Definition 4.4.1 In general, if we have a limit of the form limx->af(x)/g(x) where both f(x)->0 and g(x)->0, then this limit may or may not exist and is called an indeterminate form of type 0/0. Definition 4.4.2 In general, if we have a limit of the form limx->af(x)/g(x) where both f(x)->∞ and g(x)->∞, then this limit may or may not exist and is called an indeterminate form of type ∞/∞.

Theorem 4.4.3 (L'Hospital's Rule) Suppose f and g are differentiable and g'(x)≠0 on an open interval I that contains a (except possibly at a). Suppose that limx->a f(x)=0 and limx->a g(x)=0 or that limx->a f(x)=+/-∞ and limx->a g(x)=+/-∞ (i.e. we have an indeterminate form of type 0/0 or ∞/∞). Then limx->a f(x)/g(x)= limx->a f'(x)/g'(x) if the limit on the right side exists (or is ∞ or -∞)

Definition 4.4.4 If limx->a f(x)=0 and limx->a g(x)=∞, then the limit limx->a[f(x)g(x)] is called an indeterminate form of type 0*∞. We can deal with if by writing the product fg as a quotient: fg=f/(1/g) or fg=g/(1/f). This converts the given limit into an indeterminate for of type 0/0 or ∞/∞ so that we can use L'Hospital's Rule.

Definition 4.4.6 Several indeterminate forms arise from the limit limx->a f(x)=[f(x)]g(x). 1. limx->af(x)=0 and limx->ag(x)=0 is type 00 2. limx->af(x)=∞ and limx->ag(x)=0 is type ∞0 3. limx->af(x)=1 and limx->ag(x)=+/- ∞ is type 1∞ are called indeterminate powers. Each of these three cases can be treated either by taking the natural logarithm: Let y=[f(x)]g(x), then ln(y)=g(x)ln(f(x)). In either method we are let to the indeterminate product g(x)ln(fx)), which is of type 0∞.

Definition 4.4.5 If limx->a f(x)=∞ and limx->a g(x)=∞, then the limit limx->a[f(x)-g(x)] is called an indeterminate form of type ∞-∞.

Definition 2.7.5 The difference quotient Δy/Δx =f(x2)-f(x1)/x2-x1 is called the average rate of change of y with respect to x over the interval[x1,x2].

Definition 2.7.6 The limit of the average rates of change is called instantaneous rate of change of y with respect to x at x=x1, denoted limΔx->0Δy/Δx=limx2->x1f(x2)-f(x1)/x2-x1

Definition 2.4.1 Let f be a function defined on some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is l, and we write limx->af(x)=l if for every ε>0 there is a number δ>0 s.t. if 0<|x-a|<δ, then |f(x)-l)<ε.

Definition 11.1.2 A sequence {an} has the limit l and we we write limn->∞an=l or an->l as n->∞ if we can make the terms an as close to l as we like by taking n sufficiently large. If limn->∞an exists, we say the sequence converges. Otherwise, we say the sequence diverges.

Definition 11.1.3 A sequence has the limit l and we write limn->∞an=l or an->l as n->∞ if for every ε>0 there is a corresponding integer N s.t. if n>N then |an-l|<ε.

Theorem 11.1.8 If limn->∞an=l and the function f is continuous at l, then limn->∞f(an)=f(l)

Theorem 11.1.9 The sequence {rn} is convergent if <-1r≤1 and divergent for all other values of r

Theorem 11.1.6 If an≤bn≤cn for n≥n0 and limn->∞an=limn->∞cn=l, then limn->∞bn=l

Theorem 11.1.5 If {an} and {bn} are convergent sequences and c is a constant, then 1. The limit of the sums of the sequence equals the sum of the limits of the sequences. 2. The limit of the differences of the sequence equals the difference of the limits of the sequences. 3. The limit of a constant multiple of the sequence equals the constant multiple of the limit of the sequence 4. The limit of the products of the sequences is the product of the limits of the sequences. 5. The limit of the quotient of the sequences if the quotient of the limits of the sequences. 6. The limit of a power of the sequences is the power of the limit of the sequence.

Definition 11.4.5 We write limn->∞an=∞ means that for every positive number M there is an integer N s.t. if n>N then an>M

Theorem 11.1.4 If limx->∞=l and f(n)=an where n is an integer, then limn->∞an=l

Theorem 11.1.7 If limn->∞|an|=0 then limn->∞=0

Definition 2.5.1 A function f is continuous at a number a if limx->af(x)=f(a).

Theorem 2.5.9 If g is continuous at a and f is continuous at g(a), then the composite function f∘g given by (f∘g)(x)=f(g(x)) is continuous at a.

Theorem 2.5.6 If f and g are continuous at a and c is a constant, then the following functions are also continuous at a: 1. f+g 2. f-g 3. cf 4. fg 5. f/g if g(a)≠0

Theorem 2.5.7 (a) Any polynomial is continuous everywhere; that is, it is continuous on R. (b) Any rational function is continuous wherever it is defined; that is, it is continuous on its domain.

Theorem 2.5.8 The following types of functions are continuous at every number in their domains - polynomials - rationals functions - root functions - trigonometric functions - inverse trigonometric functions - exponential functions - logarithmic functions

Definition 2.5.5 A function f is continuous on an interval if it is continuous at every number in the interval. If f is defined only on one side of an endpoint of the interval, we understand continuous at the endpoint to mean continuous from the right or continuous from the left.

Theorem 2.5.10 (Intermediate Value Theorem) Suppose that f is continuous on the closed interval [a,b] and let N be any number between f(a) and f(b), where f(a)≠f(b). Then there exists a number cε(a,b) s.t. f(c)=N.

Definition 2.5.4a A function f is continuous from the right at a number a if limx->a+f(x)=f(a) Definition 2.5.4b A function f is continuous from the left at a number a if limx->a-f(x)=f(a)

Definition 2.5.2 If f is defined near a, we say that f is discontinuous at a if f is not continuous at a.

Definition 2.5.3a A discontinuity at a point is called removable if we could remove the discontinuity by redefining f at that point. Definition 2.5.3b A discontinuity at a point is called an infinite if the function approaches infinity as x approaches that point. Definition 2.5.3c A discontinuity at a point is called a jump discontinuity if the function "jumps" from one value to another.

Definition 2.4.4 Let f be a function defined on some open interval that contains the number a, except for possibly at a itself. Then limx->af(x)=∞ means for every positive number M there is a positive number δ s.t. if 0<|x-a|<δ, then f(x)>M.

Definition 2.4.2a limx->a-f(x)=l if for every number ε>0 there is a number δ>0 s.t. if a-δa+f(x)=l if for every number ε>0 there is a number ε>0 s.t. if a

Definition 2.2.1 Suppose f(x) is defined when x is near the number a. Then we write limx->af(x)=l and say "the limit of f(x), as x approaches a, equals l" if we can make the values of f(x) arbitraryily close to l (as close to l as we like) by restricting x to be sufficiently close to a (on either side of a) but does not equal a.

Definition 2.7.1 The tangent line to the curve y=f(x) at the point P(a,f(a)) is the line through P with slope m=limx->af(x)-f(a)/x-a provided the limit exists. We sometimes refer to the slope of the tangent line to a curve at a point as the slope of the curve at the point. Alternative Definition m=limh->0f(a+h)-f(a)/h

Definition 2.7.3 The derivative of a function f at a number a, denoted by f'(a), is f'(a)=limh->0f(a+h)-f(a)/h if this limit exists. Alternative definition. f'(a)=limh->0f(x)-f(a)/x-a

Definition 2.8.1 Given any number x for which the limit exists, we assign to x the number f'(x) called the derivative of f, defined by f'(x)=limh->0f(x+h)-f(x)/h

Theorem 3.4.1 (The Chain Rule) If g is differentiable at x and f is differentiable at g(x), then the composite function F=f∘g defined by F(x)= f(g(x)) is differentiable at x and F' is given by the product F'(x)=f'(g(x))g'(x)

10.2

Definition 3.5.1 The method of implicit differentiation consists of differentiating both sides of an implicit equation with respect to x and then solving the resulting equation for y'.

Theorem 3.11.2 (Derivatives of Hyperbolic Functions) d/dx[sinh(x)]=cosh(x) d/dx[cosh(x)]=sinh(x) d/dx[tanh(x)]=sech2(x) d/dx[csch(x)]=-csch(x)tanh(x) d/dx[sech(x)=sech(x)tanh(x) d/dx[coth(x)]=-csch2(x)

Theorem 3.6.1 d/dx[logb(x)=1/xln(b)

Theorem 3.6.2 d/dx[ln(x)=1/x

Definition 3.6.3 The calculation of derivatives of complicated functions involving products, quotients, or powers can be simplified by taking logarithms, called logarithmic differentiation. Steps: 1. Take natural logarithms of both sides of an equation y=f(x) and use the Laws of Logarithms to simplify. 2. Differentiate implicitly with respect to x. 3. Solve the resulting equation for y'.

Theorem 3.5.2 (Derivatives of Inverse Trigonometric Functions) d/dx[sin-1(x)]=1/sqrt(1-x2) d/dx[cos-1(x)]=-1/sqrt(1-x2) d/dx[tan-1(x)]=1/1+x2 d/dx[csc-1(x)]=-1/xsqrt(x2-1) d/dx[sec-1(x)]=1/xsqrt(x2-1) d/dx[cot-1(x)]=-1/1+x2

Theorem 3.4.2 d/dx[bx]=ln(b)bx

Theorem 3.2.4 If f and g are differentiable, then d/dx[f(x)/g(x)]={d/dx[f(x)]g(x)-f(x)d/dx[g(x)]}/{g2(x)}

Theorem 3.2.3 (Product Rule) If f and g are both differentiable, then d/dx[f(x)g(x)]=d/dx[f(x)]g(x)+f(x)d/dx[g(x)

Theorem 3.1.1 (Derivative Rules) 1. d/dx[c]=0 2. If n is a positive integer, then d/dx[xn]=nxn-1 3. If n is a real number, then d/dx[xn]=nxn-1 4. d/dx[cf(x)]=c*d/dx[f(x)] 5.d/dx[f(x)+g(x)=d/dx[f(x)+d/dx[g(x) 6. d/dx[f(x)-g(x)=d/dx[f(x]-d/dx[g(x)] 7. d/dx[ex]-ex

Definition 3.1.2 The number e is the number s.t. limh->0[eh-1]/[h]

Definition 2.8.2 A function f is differentiable at a if f'(a) exists. It is differentiable on an open interval if it is differentiable at every number in that interval.

9.1,9.3

Theorem 2.8.3 If f is differentiable at a, then f is continuous at a.

Definition 2.7.4 The tangent line to y=f(x) at (a,f(a)) is the line through (a,f(a)) whose slope is equal to f'(a), the derivative of f at a.

Definition 2.7.2a The function f that describes motion is called the position function of the object. Definition 2.7.2b We defined the velocity (or instantaneous velocity) v(a) at time t=a to be the limit of the average velocities: v(a)=limh->0f(a+h)-f(a)/h

Definition 2.6.1a Let f be a function defined on some interval (a,∞). Then limx->∞f(x)=l means the values of f(x) can be made arbitrarily close to l by requiring x to be sufficiently large. Definition 2.6.1b Let f be a function defined on some interval (-∞,a). Then limx->-∞f(x)=l means that the values of f(x) can be made arbitrarily close to l by requiring x to be sufficiently large negative.

Definition 2.6.5 Let f be a function defined on some interval (a,∞). Then limx->∞f(x)=l means for every ε>0 there is a corresponding number N s.t. if x>N, then |f(x)-l|<ε Definition 2.6.5b Let f be a function defined on some interval (-∞,a). Then limx->-∞f(x)=l means for every ε>0 there is a corresponding number N s.t. if x

Definition 2.6.6 Let f be a function defined on some interval (a,∞). Then limx->∞f(x)=∞ means for every positive number M there is a corresponding number N s.t. if x>N, then f(x)>M

Definition 2.6.4 If r>0 is a rational number, then limx->∞a/xr=0. If r>0 is a rational number s.t. xr is defined for all x, then limx->-∞1/xr=0.

Definition 2.6.2 The line y=l is called a horizontal asymptote of the curve y=f(x) if either limx->∞f(x)=l or limx->-∞f(x)=l

Theorem 2.6.3 limx->∞arctan(x)=π/2 and limx->-∞arctan(x)=-π/2

Theorem 2.3.1 Suppose that c is a constant and the limits limx->af(x) and limx->ag(x) exist. Then 1. limx->a[f(x)+g(x)]= limx->af(x)+limx->ag(x) 2. limx->a[f(x)-g(x)]= limx->af(x)-limx->ag(x) 3. limx->a[cf(x)]=climx->af(x) 4. limx->a[f(x)g(x)]=limx->af(x)*limx->ag(x) 5. limx->af(x)/g(x)=limx->af(x)/limx->ag(x) if limx->ag(x)≠0 6. limx->a[f(x)]n=[limx->af(x)]n 7. limx->ac=c 8. limx->ax=a 9. limx->axn=an, where n is a positive integer 10. limx->anthroot(x)=nthroot(a) 11.limx->anth root(f(x))=nth root(limx->af(x))

Theorem 2.3.2 (Direct Substitution Property) If f is a polynomial or a rational function and a is in the domain of f, then limx->af(x)=f(a)

Theorem 2.3.3 If f(x)=f(x) when x≠a, then limx->af(x)=limx->ag(x), provided the limits exist.

Theorem 2.3.4 If f(x)≤g(x) when x is near a (except possibly at a) and the limits of f and g both exist as x approaches a, then limx->af(x)≤limx->ag(x)

Theorem 2.3.5 (The Squeeze Theorem) If f(x)≤g(x)≤h(x) when x is near a (except possibly at a) and limx->af(x)=limx->ah(x)=l then limx->ag(x)=l.

Definition 2.2.4a Let f be a function defined on both sides of a, except possibly at a itself. Then limx->af(x)=∞ means that the values of f(x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a, but not equal a. Definition 2.2.4b Let f be a function defined on both sides of a, except possibly at a itself. Then limx->af(x)=-∞ means that the values of f(x) can be made arbitrarily large negative (as largely negative as we please) by taking x sufficiently close to a, but not equal a.

Definition 2.2.5 The vertical line x=a is called a vertical asymptote of the curve y=f(x) if at least one of the following statements is true: limx->af(x)=∞ limx->af(x)=-∞ limx->a-f(x)=∞ limx->a-f(x)=-∞ limx->a+f(x)=∞ limx->af(x)=-∞

Fact 2.2.6 limx->0+ln(x)=-∞

Definition 2.2.2 We write limx->a-=l and say the left-hand limit of f(x) as a approaches a (or the limit of f(x) as x approaches a from the left) is equal to l if we can make the value of f(x) arbitrarily close to l by taking x to be sufficiently close to a with xa, we get the "right-hand limit of f(x) as x approaches a is equal to l" and we write limx->a+f(x)=l and we consider only x>a.

Theorem 2.2.3 limx->af(x) =l iff limx->a-f(x)=limx->a+f(x)=l

Definition 2.1.1 The word tangent is derived from the latin word tangens, which means "touching." Thus a tangent to a curve is a line that touches tthe curve, that is, a tangent line should have the same direction as a curve at the point of contact. A secant line, from the Latin word secans, meaning cutting, is a line that cuts (intersects) a curve more than once.

Definition 2.1.2 The instantaneous velocity is defined to be the limiting value of average velocities over shorter and shoter time periods.

Definition 4.3.3 If the graph of f lies above all of its tangent on an interval I, then it is called concave upward on I. If the graph of f lies below all of its tangents on I, it is called concave downward on I.

Test 4.3.4 (Concavity Test) (a) If f''(x)>0 for all xεI, then the graph of f is concave upward on I. (b) If f''(x)<0 for all xεI, then the graph of f is concave downward on I.

Definition 4.5.5 A point P on a curve y=f(x) is called an inflection point if f is a continuous function there and the curve changes from concave upward to concave downward or from concave downward to concave upward at P.

Test 4.3.6 (The Second Derivative Test) Suppose f'' is continuous near c. (a) If f'(c)=0 and f''(c)>0, then f has a local minimum at c. (b) If f'(c)=0 and f''(c)<0, then f has a local maximum at c

Test 4.3.1 (increasing/Decreasing Test) (a) If f'x)>0 on an interval, then f is increasing on that interval. (b) If f'(x)<0 on an interval, then f is decreasing on that interval.

Test 4.3.2 (The First Derivative Test) Suppose that c is a critical number of a continuous function f. (a) If f' changes from positive to negative, then f has a local maximum at c. (b) If f' changes from negative to positive at c, then f has a local minimum at c. (c) If f' is positive to the left and right of c, or ngative to the left and right of c, then f has o local maximum or minimum at c.

Definition 4.1.1 Let c be a number in the domain D of a function f. Then f(c) is a the - absolute maximum value of f on D if f(c)≥f(x) for all xεD -absolute minimum value of f on D if f(c)≤f(x) for all xεD An absolute maximum of minimum is sometimes called a global maximum or minimum. The maximum or minimum values of f are called extreme values of f.

Definition 4.1.2 The number f(c) is a - local maximum value of f if f(c)≥f(x) when x is near c - local minimum value of f if f(c)≤f(x) when x is near c. If we say that something is true near c,we mean that it is true on some open interval containing c.

Theorem 4.1.3 (The Extreme value Theorem) If f is continuous on a closed interval [a,b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c,dε[a,b].

Theorem 4.1.4 If f has a local maximum or minimum at c, and if f'(c) exists, then f'(c)=0.

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