Calculus

critical point

a point in domain of f at which either the graph of f has a hor tangent line or f is not diff

1.5.9 theorem if f cont on [a, b) & if f(a) & f(b) are nonzero & have opp signs

at least 1 sol of equation f(x) = 0 in interval (a,b)

1.5.8 theorem (Intermediate-value) if f is cont on closed interval and k is any number between f(a) and (b) inclusive

at least one number of x in interval [a,b) such that f(x) = k

graph of polynomial of degree n(>2)

at most n x-intercepts, at most n-1 relative extrema, and at most n-2 inflection points

end behavior

behavior of function f(x) as x increases/decreases without bound

if denom is zero but limit of nom isn't

can prove that limit of rational function does not exist

increment

change from initial value and h is the increment

definition of tan line

exists secant line that passes through points p and q and if move Q along curve toward P: secant line will rotate toward limiting position

1.5.7 theorem if f is one-to-one function that is cont at each point of its domain

f inverse is cont at each point of domain/that is f^-1 is cont at each point in the range of f

if function f is continuous at each number in an open interval (a,b)

f is continuous on (a,b)and if continuous on (-infinity, +infinity): continuous everywhere

if function f defined on closed interval [a,b] but not outside interval

f not defined at endpoints of interval because derive are 2-sided limits

stationary point

f'(x) = 0

3.1.2 theorem

f'(x) > 0: increasing on interval f'(x) < 0: decreasing on interval f'(x) = 0: constant on interval

goal of related rates problem

find unknown rate of change by relating it to other vars whose values and rates of change at time t are known or can be found

theorem 1.2.3

for any polynomial p(x) and any real number a, the limit of p(x) as x approaches a is equal to p(a)

2.2.2 def

func f said to be differentiable at x_0 if limit f'(x_0) if lim of f(x_0+)

removable discontinuity

function defined at c and limit of f(x) as x approaches c exists but 2 values are not equal

1.5.5 theorem

if limit of g(x) as x approaches c = L and if function is cont at L then limit of f(g(x)) as x approaches c = f(c): limit of f(g(x)) as x approaches = f (limit of g(x) as x approaches c)

2.3.2 theorem (power rule)

if n is pos number and any real number: derivative of x^n = nx^n-1

relative minimum

if there is an open interval containing x_0 on which f(x_0) </= f(x) for all x in the interval

Relative maximum

if there is open interval containing x_0 on which f(x_0) is the largest value (f(x_0) >/= f(x)) for all x in the interval

limits

if values of f(x) can be made as close as we like to L by taking values of x close to a (but not equal)

2.3.4 theorem (constant multiplication)

if x is differentiable at x and c is any real number: c(f) is also diff at x and the derivative of c(f(x)) is equal to c(derivative of f(x))

newton's method

if x_0 is nth approximation: evident that improved approximation x_n+1: x_n+1 = x_n - f(x_n)/f'(x_n)

Jump discontinuity

one-sided limits exist but are not equal

infinite discontinuity

one-sided limits of f(x) as x approaches c are infinite

economics

profit function = revenue function - cost function

for two-sided limit to exist at point a

values of f(x) must approach some real # L as x approaches a and number must be the same from left & right

when one-sided or two-sided limit fails to exist

values of function increase/decrease without bound

Evaluating Definite Integrals Using Substitution

- 4.9.1 theorem: if g prime is cont on [a,b] and f is cont on int containing values of g(x) for a</=x</=b: integral from a to b of f(g(x))g'(x)dx = integral from g(a) to g(b) of f(u) du - proof: let u = g(x) - d/dx F(g(x)) = d/dx F(u) = dF/du du/dx = f(u)du/dx = f(g(x))g'(x) -therefore: integral from a to b of f(g(x))g'(x)dx = F(g(x)) from a to b = F(g(b))-F(g(a)) = integral from g(a) to g(b) of f(u) du

if particle in rectilinear motion moves along x-axi so that pos coord frame of elapsed time t: s = f(t)

- f called pos func of particle - avg velocity over time interval [t_0, t_0+h], h > 0: v_avg = delta pos/ delta t = f(x_0+h) - f(x_0)/h

3.1.4 theorem

- f''(x) > 0 for every value of x in open interval: f is concave up on that interval - f''(x) < 0 for every value of x in open interval: f is concave down on that interval

def 2.2.1

- function f prime defined by formula f'(x) = limit of f(x+h)-f(x)/h as h approaches 0 - called derivative of f with respect to x - domain of f prime: all x in domain of f for which limit exists

2.7.1 def

- given equation in x and y define function f implicitly if graph of y = f(x) coincides with portion of graph of equation - ex: x=y^2 defined implicitly by (x)^1/2 = y and -(x)^1/2 = y

3.8.3 theorem (constant difference theorem)

- if f and g are diff on an interval, and if f'(x) = g'(x) for al x in that interval, then f-g is constant on the interval - is a constant k such that f(x) - g(x) = k

1.6.2 theorem (squeezing) Let f, g, and h be functions satisfying g(x) </= f(x) </= h(x) for all x in some open interval containing number c with exception that inequalities need not to hold at c

- if g and h have same limit as x approaches c: lim of g(x) as x approaches c = lim of h(x) as x approaches c = L - then f also has limit as x approaches c: limit of f(x) as x approaches c = L

2.6.1 theorem

- if g is diff at x and f is diff at g(x): f of g is diff at x - alternate def: f'(g(x)) * g(x)

Root test

- if limit as n approaches infinity of abs and to the 1/n power = L less than 1, then series sigma n=1 to infinity of an is absolutely convergent -if limit = L less than 1 or limit = infinity, series sigma n=1 to infinity of an is divergent -if limit = 1, inconclusive -if series is absolutely convergent series with sum s, then any rearrangement of series has same sum s

3.8.1 theorem (rolle's theorem)

- let f be cont on closed interval [a,b] and diff on open interval (a,b) - if f(a) = 0 and f(b) = 0: there is at least one point c in interval (a,b) such that f'(c) = 0

3.8.2 theorem (mean-value theorem)

- let f be cont on closed interval [a,b] and diff on open interval (a,b) - is at least one point c in (a,b) such that f'(c) = f(b) - f(a)/ b-a

Theorem 1.4.1 limit def/two-sided limit def

- let f(x) be defined for all x in some open interval containing a, with possible exception that f(x) need not be defined at a - lim of f(x) as x approaches a = L - if given any number epsilon > 0: can find number b > 0 such that the absolute value of f(x)-L < epsilon if 0 < the absolute value of x-a < b

1.6.1 theorem if c is any number in natural domain of stated trig function

- lim sin x = sin c, lim cos x = cos c, lim tan x = tan c, and still applies to inverse trig functions

theorem 1.2.2

- limit of a sum is equal to sum of the limits - limit of a difference is equal to the difference of the limits - limit of a product is equal to the product of the limits - limit of a quotient is equal to the quotient of the limits provided that the denom doesn't = 0 - limit of a root is equal to the root of the limit provided that the limit is greater than zero if a is even

Properties of differentiability

- not differentiable at any point x_0 where sec line from P(x_0, f(x_0)) to points Q(x, f(x)) distinct from P do not approach unique non vertical limiting pos as x approaches x_0 - corners/one-sided limit does not exist - points of vertical tan

Curves Defined by Parametric Equations

- sometimes better to express functions parametrically when y = f(x) because C fails the Vertical Line Test - x and y coordinates are functions of time so we can write x= f(t) and g= f(t) - parametric equations: x = f(t) and y = g(t) - as t varies: point (x,y) = (f(t), g(t)) varies and traces out a curve C (called a parametric curve) - when restrict t to lie in finite interval: x = f(t) and g(t) in int a less than or equal to t which is less than or equal to b - initial f(a), f(b) and terminal (f(b), g(b)) - parametric equations for circle with center (h,k) and rad r: x = rcost and y = rsint - Bezler curves: used extensively in manufacturing and are special parametric curves - cycloid: curved traced by point P on circumference of circle as circle rolls along straight line - brachistochrone problem: find area under one arch of cycloid - particle will take least time sliding from A to B if curve is part of inverted arch of cycloid (proposed by Swiss mathematician John Bernoulli by contrasting it with all other possible curves that join a to b) -tautochrone problem (proposed by Dutch physicist Huygens): no matter where particle P is placed on inverted cycloid, it takes same time to slide to bottom - conchoids of Nicomedes (named after ancient Greek scholar): x= a + cost and y = a tan t + sin t and when a less than -1 branches are smooth but when a reaches -1, right branch acquires sharp point called cusp , a between 0 and 1, left branch has loop which shrinks to become cusp when a =1, and for a > 1, branches become smooth again and as a increases further, they become less curved - a positive: reflections about y-axis of corresponding curves with a negative

Calculus with Parametric Curves

- suppose f and g diff functions and find tan line at point on parametric curve x = f(t) and y = g(t), where y is also diff function of x: dy/dt = dy/dx * dx/dt and dy/dx = dy/dt/dx/dt if dx/dt not equal to 0 ( think of canceling the dt's) - horizontal tangent when dy/dt = 0 and vertical targent when dx/dt = 0 - second derivative: d2y/dx2 = d/dx( dy/dx) = d/dt (dy/dx)/ dx/dt - using sub rule for integrals, can find area for parametric curve: A = integral from a to b of y dx = integral from sigma to beta g(t) * f'(t) dt ( y= g(t) and x = f(t) so that dx/dt = f'(t) and dx = f'(x)dt - theorem for finding arc length: if curve C described by parametric equations x=f(t) and y =g(t), where f prime and g prime are cont on [sigma, beta] and C is traversed exactly once as t increases from sigma to beta, then length of C is L = integral from sigma to beta of [(dx/dt)^2 + (dy/dt)^2]^1/2 dt (consistent with general formula (dx)^2 = (dx)^2+(dy)^2

3.2.2 theorem

- suppose f defined on open interval containing the point x_0 - if f has relative extremum at x = x_0: then x = x_0 is a critical point of f that is, either f'(x_0) = 0 or f is not diff at x_0

3.4.4 theorem

- suppose f is cont and has exactly one relative extremum on an interval at x_0: - if f has relative min at x_0 : f(x_0) is absolute min of f on the interval - if f has relative max at x_0: f(x_0) is absolute max of f on the interval

3.2.3 theorem (first derivative test)

- suppose f is cont at a critical point x_0 - if f'(x_0) > 0 on open interval extending left from x_0 and f'(x_0) < 0 on an open interval extending right from x_0: relative max at x_0 - if f'(x) < 0 on an open interval extending left from x_0 and f'(x) > 0 on an open interval extending right from x_0: f has relative relative min at x_0 - if f'(x) has same sign on open interval extending left from x_0 as it does on an open interval extending right from x_0: f does not have a relative extremum at x_0

3.2.5 geometric implications of multiplicity

- suppose p(x) is a polynomial with root of multiplicity m at x=r - m is even: graph of y = p(x) is tan to x-axis at x=r, does not cross x-axis there, and does not have inflection point there - m is odd and greater than 1: graph is tan to x-axis at x=r, crosses x-axis there, and also has inflection point there - if m = 1: graph is not tan to x-axis at x=r, crosses x-axis there, and may or may not have inflection point there

3.2.4 theorem (second derivative test)

- suppose that f is twice diff at point x_0 - if f'(x_0) = 0 and f"(x_0) > 0: f has relative minimum at x_0 - if f'(x_0) = 0 and f"(x_0) < 0: f has relative max at x_0 - if f'(x_0) = 0 and f"(x_0) = 0: test is inconclusive and f may have relative max, relative min, or neither at x_0

2.1.1 def

- suppose x_0 is in domain of function f. Tan line to curve y = f(x) at point P(x_0, f(x_0)): - y - f(x_0) = mtan (x-x_0) - mtan = lim of f(x)-f(x_0)/x-x_0 as x approaches x_0 - called tan line to y = f(x) at x_0

line that approxs. graph of f in vicinity of P(x_0, f(x_0))

- tan line to graph of f at x - y = f(x_0) + f'(x_0)(x-x_0) - f(x_0 + change in x): about f(x_0) + f'(x_0)(change in x)

Improper Integrals

- vertical asymptotes within interval of integration: not integrable and are called infinite discontinuities - integrals with infinite intervals or infinite discontinuities within interval of integration: improper integrals - some types: infinite intervals, infinite discontinuities, and infinite discontinuities and infinite intervals - 7.8.1 def: improper integral of F over interval [a, positive infinity] defined to be integral from a to positive infinity of f(x) dx = limit as b approaches positive finite of the integral from a to b of f(x) dx - converge: when limit exists and is defined to be value of integral - diverge: when limit does not exist and is not assigned a value - evaluating convergent improper integrals: change top limit of integration to b such that it is the limit as b approaches infinity - 7.8.2 theorem: integral from 1 to positive infinity of dx/x^p = 1/p-1 if p > 1 or diverges if p less than or equal to 1 - 7.8.3 definition: improper integral of f over int (-infinity, b] is defined: integral from - infinity to b of f(x) dx = limit as a approaches - infinity of the integral from a to b of f(x) dx - converge: limit exists; diverge: limit doesn't exist - improper integral over int (-infinity, + infinity) is defined: integral from - infinity to positive infinity of f(x) dx = integral from - infinity to c f(x) dx + integral from c to + infinity f(x) dx where c is any real number - said to converge if both terms converge; said to diverge if either term diverges - 7.8.4 def: if f is cont on int [a,b] except for infinite discont at b then improper integral of f over int [a,b]: integral from a to b of f(x) dx = limit as k approaches b- of the integral from a to k of f(x) dx ( for disconts at right-hand endpoints) - converge: limit exists and is defined to be value of integral; diverge: limit does not exist and is not assigned a value 7.8.5 def (for integrals that have disconts at left-hand endpoints or in the interval of integration): if f is cont on int [a,b], except for discont at a, then improper integral of f over int [a,b]: integral from a to b of f(x) dx = limit as k approaches a + of the integral from k to b of f(x) dx - converge: limit exists; diverge: limit does not exist - discont inside int of integration: integral from a to b of f(x) dx = integral from a to c of f(x) dx + integral of c to b of f(x) dx where the two integrals on the right side are improper - converge: both terms converge; diverge: either term on right side diverges - can also use improper integrals to evaluate arc length and surface area as most functions are not smooth (lead to convergent improper integrals)

radicals and fractional exponents

- vertical tangent lines and cusps - function is not diff at x_0 - may be because there's an inflection point at x_0 - cusp: when f'(x) approaches +infinity form one side of x_0 and -infinity from the other side

Functions as power series

- want to express known function as sum of infinitely many terms for integrating functions that don't have elementary antiderivatives, solving diff equations, and approximating functions by polynomials - 1/(1-x) = 1+x+x^2+x^3+...= sigma n=0 to infinity x^n abs x less than 1 -express function 1/(1-x) as sum of power series -term by term differentiation and integration: used to integrate/differentiate functions that are sums of power series and can do so by differentiating/integrating each individual term in series -Theorem: if power series sigma cn (x-a)^n has radius of convergence R greater than 0, then function f defined by F(x)= c0 + c1 (x-a) + c2 (x-a)^2 +...= sigma n=0 to infinity of cn(x-a)^n is diff and cont on int (a-R,a+R) And 1. F prime of x = c1 + 2c2(x-a) + 2c3 (x-a)^2+...= sigma n=1 to infinity n cn (x-a)^n-1 2. Integral of f(x) dx = C + c0 (x-a) + c1 (x-a)^2/2 + c2 (x-a)^3 /3 +...= C+ sigma n=0 to infinity cn (x-a)^n+1/n+1 -radii of convergence of power series in both equations: both R - derivative of a sum is equivalent to sum of derivative -integral of a sum is equivalent to the sum of the integral -interval of convergence may not necessarily remain the same (may converge at endpoint but differentiated form may diverge at endpoint

Taylor and Maclaurin series

- which functions have power series representations? How to find such representations? -Theorem: if f has power series representation (expansion) at a -if f(x) = sigma n=0 and infinity cn (x-a)^nc abs x-a less than R - then coefficients are given by formula cn = f^(n)(a)/ n! - f(x) = sigma from n = 0 to infinity of f^(n) (a)/n! (x-a)^n = f(a) + f'(a)/1! (X-a) + f''(a)/2! (x-a)^2 + f'''(a)/3! (X-a)^3 -called Taylor series of function f at a (about a or centered at a) -maclaurin series: a= 0 and is f(0) + f'(0)/1! + f''(0)/2! -if f can be represented as power series about a: then f is equal to sum of its Taylor series -exist functions that are not equal to sum of their Taylor series -x^n/n!: nth -degree Taylor polynomial of f at a -f(x) is sum of Taylor series if f(x) = limit n approaches infinity Tn (x) -Rn(x) = f(x) -Tn (x) do that f(x) = Tn (x) + Rn(x) then Rn(x) is called remInder of series -Theorem: if f(x) = Tn(x) + Rn (x) where Tn is nth degree Taylor polynomial of f at a and lim as n approaches infinity Rn(x)= 0 for abs x-a less than R, then f is equal to sum of Taylor series on int abs x-a less than R -Taylor's inequality: if abs f^(n+1) (x) less than or equal to M for abs x-a less than or equal to D, then remainder Rn(x) of Taylor series satisfies inequality and Rn(x) less than or equal to M/(n+1)! abs x-a ^n+1 for abs x-a less than or equal to d -limit n approaches infinity of x^n / n! = 0 for every number x -e^x = sigma n = 0 and infinity x^n /n! for all x - e = sigma n= 0 infinity of 1/n! =1+ 1 +1/2! +1/3! -sin x = x -x^3/3! + x^5/5! = sigma n=0 infinity (-1)^n (x^2n+1)/(2n+1)! for all x - cos x = 1- x^2/2!+ x^4/4! -x^6/6+...= sigma n=0 and infinity (-1)^n x^2n/(2n)! for all x -binomial series: k is any number and abs x less than 1 then (1+x)^k = sigma n=0 and infinity (k n) x^n = 1+ kx (k(k-1)/2! X^2

Infinite series

-931 def: infinite series is expression written in form sigma from k=1 to infinity of uk = u1+u2+u3+u4...uk+... (called terms) -let sn be sequence of partial sums of series u1+u2+u3+...uk+... -if sequence an converges to limit S, then series is said to converge to S (sim of series) -sequence of partial sums diverges: series said to be divergent -9.3.3: geometric series sigma from k=0 to infinity of ar^k = a+ar+ar^2... -converges if |r| less than 1 and diverges if abs of r greater than or equal to 1 then sum is a/1-r -for divergent geometric series: a(1-r^n+1) /1-r -can use partial fractions to determine convergence and sum or divergence -harmonic series: 1/k from n=1 to infinity (divergent)

Squeezing Theorem

-Let an bn and can be sequences such that an less than or equal to bn which is less than or equal to cn for all vals of n beyond index N. -if sequences an and cn have common limit L as n approaches infinity then bn also has limit L as n approaches infinity -9.1.6: if limit n approaches infinity of abs an = 0, then limit as n approaches pos infinity of an = 0

Euler's Method

-method of approximation using differentiation -essentially proceed short distance along tan line and then make midcourse correction by changing directions as indicated by direction field -says to start at point given by initial value and proceed in direction indicated by direction field -method: approximate values of solution of the initial-value problem y'= F(x,y), y(x0)=y0, with step size h, at xn=xn-1+h: yn=yn-1+hF(xn-1,yn-1) with n in set of all counting numbers excluding zero -as step gets smaller: estimates get closer to actual value

Bessel Functions

-named after Friedrich Bessel (1774-1846) -functions first arose when Bessel solved Kepler's equation for describing planetary motion -sum of power series - applied in physical situations like temp distribution in circular plate and shape of vibrating drumhead

Method of Disks

-V= integral from a to b of pi[f(x)]^2dx -problem: f be continuous and nonnegative on [a,b], and let R be region that is bounded above by y=f(x), below by x-axis, and on sides by lines x=a and x=b -find volume of solid of revolution generated by region R about x-axis

Strategy for Testing Series

-classify series based on form -if series is of form 1/n^p, it is a p series (convergent if p greater than 1 and divergent if less than 1) -if series has form sigma ar^n-1 or ar^n , it is geometric. Converges if abs r less than 1 and diverges if r greater than or equal to 1 -if series has form that is similar to geometric or o series, should use comparison (comparison tests only valid for positive terms but if sigma an has negative terms, can apply comparison test for sigma abs an -if limit b approaches infinity of an doesn't equal 0, then test for divergence should be used -if series is of form sigma (-1)^n-1 bn or sigma (-1)^n bn, then alternating series is a choice -series with factorials of exponents can be tested with ratio test (limit approaches 1 for all p series and therefore all rational or algebraic functions of n -if an is of form (bn)^n, then ratio test could be useful -an =f(n) where integral from n to infinity of f(x) dx is easily evaluated, then integral test is effective

Integration by Parts

-corresponds to product rule (can also use to derive it) -rule: integral of u dv = uv-integral of v du -can also be used in definite intervals by combining it with second fund theorem of calc -usually assign variables to expressions that, once integrated/differentiated, would give the simplest expression

Monotone sequences

-def 921: sequence (an) from n+1 to pos infinity is -strictly increasing if all terms are -increasing if they are greater than or equal to each other -strictly decreasing: all decreasing -decreasing if they are less than or equal to each other -monotone vs strictly monotone: increase and decrease and strictly increase and strictly decrease -test for monotonicity: differences of successive terms and ratios of successive terms (assumed that they are all pos) -922 def: if taking out finitely many terms from beginning of sequence produced sequence with a certain property, then original sequence is said to have property eventually -convergence and divergence: depend not on initial terms but on how sequence behaves eventually -923 theorem: if sequence an is eventually increasing, there are two possibilities 1. Constant M called upper bound such that an less than or equal to M for all n such that sequence converges to limit L satisfying L less than or equal to M 2. No upper bound/ limit as n approaches infinity of an equals pos infinity -special note: limit as n approaches pos infinity of x^n/n! = 0 -9.2.5 axiom: if nonempty set S of real numbers has upper bound, then it has a smallest upper bound/least upper bound. If nonempty set S of real numbers has lower bound, then it had largest lower bound/greatest lower bound -proof: at least

Absolute Convergence and Ratio and Root Tests

-def: a series sigma an is called absolutely convergent if series of abs values sigma abs an is convergent -def: series sigma an called conditionally convergent if it is convergent but not absolutely convergent - if series sigma an is absolutely convergent, then it is convergent

Calculus with polar functions

-dy/dx = dy/d theta / dx/d theta = r cos theta + sin theta dr/d theta/ -r sin theta + cos theta dr/d theta since x = f(theta)cos theta and y= f(theta sin theta -10.3.1 theorem: if polar curve r = f(theta) passes through origin at theta = theta 0, and if dr/d theta doesn't equal 0 at theta = theta 0, then line theta = theta 0 is tangent to curve at origin -arc length formula: integral from a to b of (r^2 + (dr/d theta)^2)^0.5 d theta if no segment of polar curve is traced more than once -area in polar coordinates: integral from a to b of 0.5 r^2 d theta enclosed by lines theta = a and theta =b -determining limits of integration: sketch r region, draw arbitrary radial line from pole to curve, over what int does theta vary in order to sweep out the area (these are the lower and upper bounds)

Surface Area

-suppose f is smooth, nonnegative function on [a,b] and that surface of revolution is generated by revolving portion of curve y=f(x) between x=a and x=b about x-axis. Define what is meant by area of surface, and find formula -def: if f is smooth, nonnegative function on [a,b] then surface area of surface of revolution that is generated y revolving portion of curve y=f(x) between x=a and x=b about x-axis is defined S = integral from a to b of 2pi f(x) {1+[f'(x)]^2}^0.5 -essentially multiplying the circumference by the arc length

absolute error/error in approx

epsilon = |x-x_0|

Simpson's Rule (numerical integration)

-errors in midpoint and trapezoidal approximations: Em= integral from a to b of f(x)dx- Mn and ET= integral from a to b of f(x)dx-Tn -define absolute values of Em and ET to be absolute errors -midpoint approx: sometimes called tangent line approx because (for each subint) and is sometimes better than trapezoidal approximation -theorem 7.7.1: let f be cont on [a,b], and let abs Em and abs Et be absolute errors that result from midpt approx and trap approx of definite integral of f using n subints a- if graph of f is either concave up or concave down on (a,b), then abs Em< abs Et(abs error from midpt approx is less than that from trap approx b- if graph is concave down on (a,b): Tn less than definite integral of f which is less than Mn C- if graph of f is concave up on (a,b): Mn less than definite integral which is less than Tn -weighted avg of midpt and trap approxs can yield even better approxs -since ET about -2EM: 2Em + Et is about 0 -3 definite intervals = 2 def int + def int= 2(Mk+Em) + (Tk +ET)= (2Mk+Tk) + 2Em+Et= 2Mk +Tk -def int: 1/3(2Mk+Tk) -for trap approx: need to use 2k+1 values, so one would need 2k equal subints -S2k= 1/3(2Mk+Tk) -for mid pt (2Mn/2): need to divide int into 2k subints so that length of each subint is 2(b-a)/n (subscripts are all odd while subscripts for trap are all even or Tn/2) -adding them together: Sn = 1/3(b-a/n)[y0+4y1+2y2+4y3+...2yn-2+4yn-1+yn] (called Simpson's rule) -two main sources of error when doing approxs: intrinsic/truncation error and roundoff error -increasing n: reduces truncation error but increases roundoff because of more calculations -midpt and trap error bounds: if f double prime is cont on [a,b] and if K2 is max value of |f double prime (x)| on (a,b) -a |Em| = | integral from a to b of f(x)ex -Mn| less than or equal to (b-a)^3K2/24n^2 -b abs ET = abs integral from a to b of f(x)dx-Tn| less than or equal to (b-a)^3K2/12n^2 -Simpson error bound: if f ^4 is cont on [a,b] and if K4 is max value of |f^(4) (x)| on [a,b]: abs Es = abs of integral from a to b of f(x)dx-Sn which is less than or equal to (b-a)^5K4/180n^4 -shows that there's less error in Simpson rule approx when reducing the width of the int

Derivatives and Integrals of Inverse Trig Functions

-identities for inverse trig functions: -sin inverse x + cosines inverse x = pi/2 -sin of cos inverse x =(1-x^2)^0.5 -cos of sin inverse x= (1-x^2)^0.5 -tan of sin inverse of x = x/(1-x^2)^0.5 -sec(tan inverse x) = (1+x^2)^0.5 -sin(sec inverse of x)= (x^2-1)^0.5/x so that x is greater than of equal to one -proved using Pythagorean by setting one side of tri as x and one side -derivative of cotan inverse: -1/1+u^2 du/dx -derivative of sec inverse: 1/|u|(u^2-1)^0.5 du/dx -derivative of csc inverse: -1/|u|(u^2-1)^0.5 du/dx -integral of du/(a^2+u^2) =1/a (tan inverse of u/a) +C -integral of du /(a^2-u^2)^0.5 = son inverse of u/a +C -integral of du/u(u^2-a^2)^0.5 =1/a sec inverse |u/a| +C

Comparison Tests

-if have series whose terms are smaller than those of a known convergent series, then series will also be convergent -if have series whose terms are larger than those of a known divergent series, then it will also be divergent -proof: let sn= sigma from I=1 to n, tn = sigma from I=1 to n of bi, & t = sigma I=1 to infinity of bn -sn and tn must also be increasing -tn approaches t for all n so tn less than or equal to t -since ai less than or equal to bi, sn less than or equal to tn; therefore, an less than or equal to t for all n -sn must then be increasing and bounded above and converges by monotonic sequence theorem; therefore, sigma an converges -if sigma bn is divergent, then t approaches infinity, but an greater than or equal to bn so sn greater than or equal to tn; therefore, sn approaches infinity (sigma an diverges) -need only to verify for n greater than or equal to N where N is some fixed integer since convergence is not affected by finite number of terms -if an greater than bn for convergence or if an less than bn for divergence, comparison test cannot be used -limit comparison test: suppose an and bn are series with positive terms -if limit as n approaches infinity of an/bn =c where c is finite and c greater than 0, then either both series converge or diverge -proof: let m and M be pos numbers such that m less than c and c less than M -since an/bn is close to c for large n, there is integer N such that m less than quotient and quotient less than M -if sigma bn converges, so does sigma M bn (sigma an converges) -if sigma bn diverges, so does sigma m bn (so does sigma an)

Ratio Test

-if limit of abs an+1/an = L less than 1, then series sigma an from n-1 to infinity is absolutely convergent -if limit as n approaches infinity of abs an+1/an = L less than 1 or limit as n approaches infinity of abs an+1/an = infinity, then series sigma n-1 to infinity of an is divergent -if limit of abs an+1/an as n approaches infinity =1, ratio test is inconclusive -no conclusion can be drawn about convergence or divergence of sigma an -usually conclusive if n term of series contains exponential or factorial

Estimating sums

-if used comparison test to show that series sigma an converges, by comparison with series sigma bn, then will estimate sum sigma an by comparing remainders -R = s-sn = an+1.... - Tn= t-tn= Bn+1... -since an less than or equal to bn for all n, then Rn less than or equal to Tn -if sigma bn is p series, can estimate remainder Tn -geometric series, can sum exactly

5.2.2 Volume Formula

-let S be solid that extends along x axis and is bounded on left and right, respectively, by planes that are perp to x axis at x=a and x=b. Find volume of solid assuming that cross sectional area A is known at each x in int [a,b] -let S be solid bounded by two parallel planes perp to the x acid at x=a and x=b -if (for each x in [a,b]) the cross-sectional area of S perp to the x-axis is A(x), then volume of solid V= integral from a to b of A(x)dx -same for cross sections perp to y-axis -volume of solid: obtained by integrating cross-sectional area from one end of solid to other

Method of washers

-let f and g be cont and nonnegative on [a,b] and suppose f(x)>/=g(x) for all x in int [a,b].Let R be region bounded above by y=f(x), below by y=g(x), and on sides by lines x=a and x=b. Find volume of solid of revolution generated by revolving region R about x-axis -since f is above g, have to take the difference of the areas to find the cross sectional area between f and g -V = integral from a to b of pi([f(x)]^2-[g(x)]^2)dx -same for when the cross section is perp to the y axis except have to integrate with respect to y and use y coordinate limits of integration

instantaneous velocity

-limit as h approaches 0 of its avg veto or v_avg over time intervals between t = t_0 & t = t_0 + h - v avt = lim of f(x_0+h)-f(x_0)/h

Polar Coordinates

-polar coordinate system: fixed point and ray from pole called polar axis -can associate each pair of polar coordinates (r,theta) -r: radial coordinate: theta: angular coordinate -relationship between polar and rect: x= r cos theta and y = r sin theta,r^2 =x^2+y^2, and tan theta y/x -given equation in r and theta: graph polar coordinates to consist of all points with at least one pair of coordinates (r,theta) -symmetry tests: a) curve in polar is symmetric about x-axis if replacing theta by negative theta in equation produces equivalent equation b)symmetric about y if replacing theta by pi -theta in equation produces equivalent equation c) symmetric about origin if replacing theta by theta + pi or r by -r produces equivalent equation -if theta 0 is fixed angle, then for all values of r point (r, theta0) lies on line that makes an angle of theta = theta 0 with polar axis -- r=a, r=2a cos theta (circle with radius a and tangent to the y axis at the origin), r= 2a sin theta(circle with radius a and is tangent to x-axis at the origin) and can think of them as triangles with hypotenuse as either r or 2a -rose curves: r=a cos n theta and r = a sin n theta -consist of n equally spaced petals of radius a if n is odd -2n petals of radius a if j is even -cardioid and limacons: r = a +/- b sin theta and r = a +/- b cos theta -limacon: a and b both greater than 0 -Cardioid: a=b -a/b less than 1: limacon with inner loop -a/b=1: cardioid -a/b greater than 1 and less than 2: dimples limacon -a/b greater than or equal to 2: convex limacon -spiral: curve that could around central point and have left-hand and right-hand versions that coil in opposite directions -Archimedean spiral: a theta; parabolic spiral:a(theta)^0.5; logarithmic: ae^b theta; Lituus spiral: r =a /(theta)^0.5, and hyperbolic spiral: r =a/theta

Cylindrical shells

-problem: let f be cont anc nonnegative on [a,b] (0</= a</=b), and let R be region that is bounded above y= f(x), below by x-axis, and on sides by x=a and x=b. Find volume of solid of revolution that is generated by revolving region R about y axis -cylindrical shell: solid enclosed by two concentric right circular cylinders -V=[area of cross section]*[height] => 2pi * avg rad*height*thickness

Arc Length

-problem: suppose y=f (x) is smooth curve on int [a,b]/is cont. Define and find formula for arc length L of curve y=f(x) over int (a,b) -if y=f(x) is smooth curve on int [a,b], then arc length L of curve over [a,b] is L = integral from a to b of {1+[f'(x)]^2}^0.5 dx -same for curve in form x=g(y) except have to integrate with respect to y

Recursion

-recursion formulas: specify how to generate each term in sequence from terms that precede it

Limit of sequence

-sequence an approaches limit L if terms in sequence eventually become arbitrarily close to L -9.1.2: sequence said to converge to limit L if given epsilon > 0, there is pos integer N such that |an-L| less than epsilon for n greater than or equal to N. limit as n approaches pos infinity of an = L -diverge: not converge to some finite limit -9.1.3: suppose sequences an and bn converge to limits L1 and L2 respectively and c is constant -limit n approaches infinity of c =c -limit n approaches infinity of can = n limit as n approaches infinity of an = L -limit as n approaches infinity of an + bn = L1 + L2 -limit as n approaches infinity of an-bn= L1-L2 -limit as n approaches infinity of anbn = L1L2 -limit as n approaches infinity of an/bn= L1/L2 provided that L2 doesn't equal zero -if f(x) approaches L as x approaches infinity, then f(n) approaches L as n approaches infinity -9.1.4: sequence converges to limit L if and only if sequence of even-numbered terms and odd-numbered terms both converge to L

Power series

-series of form sigma from n=0 to infinity of cn x^n = c0+c1x +c2x^2+c3x^3 where c is variable and cn's are constants called coefficients of series (series of constants that can be tested for converge or diverge) -resembles polynomial -sigma from n=0 to infinity of x^n (when c=1) resembles geometric and is convergent for x greater than -1 and less than 1 and divergent for abs x greater than or equal to 1 -power series in (x-a), power series centered at a, or power series centered at a: sigma from n=0 to infinity of cn(x-a)^n = c0 + c1(x-a) + c2(x-a)^2+... -power series always converges when x=a -may use ratio or root tests for these series -set of values of x for which series is convergent and turned out to be an int [finite int for geometric] -Theorem: for given power series sigma n=0 to infinity of cn(x-a)^n, are only three possibilities -series converges only when x=a -series converges for all x -positive number R such that series converges if abs x-a less than R and diverges if abs x-a greater than R -R: radius of convergence (0 for 1 and infinity for 2) -not of converge: all values x for which series converges - (a-R, a+R) , (a-R, a+R], [a-R, a+R), [a-R, a+R]: possibilities for three since abs x-a less than R -when x is endpoint, x=a +/- R and series may converge at one or both endpoints or diverge at both endpoints

Alternating Series

-series whose terms alternate in sign(+ and -) -alternating series test: if alternating series sigma from n =1 to infinity of (-1^n-1 bn = b1-b2+b3... satisfies two conditions 1. bn+1 less than or equal to bn for all n 2. Limit as n approaches infinity of bn =0 -series is then convergent -proof: even terms (n= 2,4,6...) is increasing but each time the increments are smaller; odd terms are decreasing and increments are decreasing (both series are converging)

Derivatives of Inverse Functions

-since f inverse is reflection over line y=x which gives f: f'(x) at (x,y) is equal to f inverse (x) = 1/f'(x) at (y,x) -general: f^-1 (x) = 1/f'(f^-1 (x)) -confirmed using implicit: y=f^-1(x) is same as x =f(y):1=d/dx x = d/dx [f(y)] = f'(y) dy/dx -dy/dx = 1/f prime of y = 1/f prime of f inverse of x -dy/dx = 1/dx/dy -6.3.1 theorem: suppose domain of function is open interval on which f'(x) greater than 0 or on which f prime of x less than 0. Then f is one-to-one, f prime of x is diff at all values of x in range, and derivative of f inverse prime of is formula above

Integral Test

-suppose f is cont, pos, decreasing function on [1, infinity) and let an = f(n) -series sigma from n=1 to infinity of an is convergent if and only if improper integral from 1 to infinity of an is an - if improper integral is convergent, then sim is convergent -if improper integral is divergent, sum is divergent -p series: sigma from n=1 to infinity of 1/n^p is converge if p greater than 1 and divergent if p less than or equal to 1 -remainder estimate for integral test: suppose f(k) = ak, where f is cont, pos, decreasing function for x greater than or equal to n and sigma of an is converge -if Rn=s-sn, then integral from n+1 to infinity of f(x)dx less than or equal to Rn and is less than or equal to integral from n to infinity of f(x)dx -can also add sn to both sides

Integration using Partial Fractions

-two ways of obtaining partial fractions: 5x-10=A(x+1)+B(x-4) (trial and error) or distribute and obtain (A+B)x+(A-4b) =5x-10 so that A+B= 5 and A-4b =-10 (systems of equations) -every proper rational function can be expressed as sum P(x)/Q(x)=F1(x)+F2(x)+...+Fn(x) in which the functions are rational in the form A/(ax+b)^k or Ax+B/(ax^2+bx+c)^k -sum: partial fraction decomp -should first factor Q(x) into as many irreducible linear and quadratic factors as possible -linear factor rule: for each factor of form (ax+b)^m, partial fraction decomp contains sum of m partial fractions: A1/(ax+b) + A2/(ax+b)^2 +...+Am/(ax+b)^m where the A's are constants tbd; in case where m=1, only first term in sum appears -quadratic factor rule: for each factor in form (ax^2+bx+c)^m, partial fraction decomp contains sum of m partial fractions A1x+B1/(ax^2+bx+c) + A2x+B2/(ax^2+bx+c)^2... where A's and B's are constants tbd. In case where m=1, only first term appears -integrating improper: first apply polynomial division so that there is a proper rational function

strategy

1. assign letters to quantities that vary with time and any others that seem relevant to problem and give definition of each letter 2. identify rates of change that are known & rate of change is unknown and interpret as derivative 3. find equation that relates vars whose rates of change are identified in step 2 and will often need to draw labeled figure 4. diff both sides with respect to time to produce relationship between known and unknown rates of change 5. substitute all known values for rates of change & vars and solve for unknown rate of change

Procedure for solving applied max and min problems

1. draw appropriate figure and label quantities relevant to problem 2. find formula for quantity to be maximized or minimized 3. use conditions stated in the problem to eliminate variables and express quality to be maximized or minimized as a function of one variable 4. find interval of possible values for this variable from physical restrictions in the problem 5. iuse techniques to obtain max or min

finding eq for tan line to y = f(x) at x = x_0

1. evaluate f(x_0): point of tangency is (x_0, f(x_0)) 2. find f'(x) and eval f'(x_0) which is slope m of line 3. substitute value of slope m and point (x_0, f(x_0)) into point-slope form of line: - y - f(x_0) = f'(x_0)(x-x_0) - y = f(x_0) +f'()

1.5.3 def: if functions f & g are cont at c

1. f +g are cont at c 2. f-g is cont at c 3. fg is cont at c 4. f/g cont at c if g(c) doesn't equal zero and has discount. at c if g(c) = 0

1.5.2 def: function said to be continuous on closed interval [a,b] if conditions are satisfied

1. f is cont on (a,b) 2. f is cont from right at a 3. f is cont from left at b

1.5.1 def: function f said to be continuous at x=c provided following conditions are satisfied

1. f(c): defined 2. limit of f(x) as x approaches c exists 3. limit of f(x) as x approaches c = f(c)

Procedure for finding absolute extrema of cont function f on finite closed interval [a,b]

1. find critical points of f in [a,b] 2. eval f at all critical points and at the endpoints a and b 3. largest of values in step 2 is the absolute max value of f on [a,b] and the smallest value is absolute min

1.5.6 theorem

1. function g is cont at c and function f is cont at g(c) then comp f of g is cont at c 2. function of g is cont everywhere and function f is cont everywhere comp f of g is cont everywhere

1.6.3 theorem

1. limit of sin x/x = 1 as x approaches zero & limit of 1-cos x/x = 0 as x approaches 0

1.5.4 theorem

1. poly cont everywhere 2. rational function cont at every point where denom is nonzero and has discont at points where denom is zero

Graphing rational function f(x) = P(x)/Q(x) if P(x) and Q(x) have no common factors

1. symmetries: symmetry about y-axis or origin 2. x- and y-intercepts: 3. vertical asymptotes: find values for x for which Q(x) = 0 4. sign of f(x) 5. end behavior: compute limits as x approaches +/- infinity and may have hor asymptote if L is finite 6. conclusions and graph: analyze changes of f'(x) and f"(x) to determine where function is increasing, decreasing, concave up, and concave down; determine locations of all stationary points, relative extrema, and inflection points

Volume

Area of cross section * height

9.1.1 def of sequence

Function whose domain is a set of integers

Estimating Sums

If s - sigma (-1)^n-1bn where bn greater than 0, is sum of alternating series that satisfies 1. bn-1 less than or equal to bn 2. Limit as n approaches infinity of bn =0 -then size of error smaller than bn+1 if satisfy conditions for alternating series test which is the abs value of first neglected term -abs Rn = abs s-sn less than or equal to bn+1

Volume by cylindrical shells about y-axis

Let f be font and nonnegative on int [a,b] (0</=a</=b), and let R be region that is bounded above by y=f(x), below by x-axis, and on sides by lines x=a and x=b.then volume of solid of revolution is generated by revolving region R about y-axis: V = integral from a to b of 2pixf(x)dx -also: volume of solid of revolution generated by revolving region R about axis can be obtained by integrating area of surface generated by arbitrary cross section of R taken parallel to axis of revolution

2.3.1 theorem

deriv of oonstant function: 0

limits of rational functions as x approaches +/- infinity

divide each term in num and denom by highest power of x that occurs in denom

multiplying x^n by pos or neg number

doesn't change sign & reverses sign respectively

multiplicity m

if (x-r)^m divides p(x) but (x-r)^m+1 does not

2.4.2 theorem (quotient rule)

if f & g are diff at x and if g(x) is not equal to zero:f/g is diff at x and the f'(x)/g'(x) = g(x)f'(x) - f(x)g'(x)/g(x)^2

2.3.5 theorem (sum and diff rules)

if f & g are diff at x: f+g and f-g are also diff

2.4.1 theorem (prod rule)

if f &g diff at x: so is f*g and the derivative of the product of f and g is equal to f(x)(derivative of g(x)) + g(x)(derivative of f(x))

3.1.5 def

if f changes direction of concavity at point (x_0, f(x_0)): then say that f has inflection point at x_0

3.4.3 theorem

if f has absolute extremum on open interval (a,b): must occur at critical point of f

absolute extremum

if f has either absolute max or min at that point/greatest and smallest points

relative extremum

if f has relate max or relative minimum at x_0

2.2.3 theorem

if func f differentiable at x_0: f is cont at x_0 since f'(x) exists and, therefore, the limit exists as h approaches 0 & functions not differentiable at points of discont

3.4.2 theorem (extreme-value theorem)

if function f is cont on finite closed interval [a.b]: f has both absolute max and absolute min on [a,b]

theorem 1.2.4

let f(x) = p(x)/q(x) be a rational function, and let a be any real number a) if q(a) doesn't equal zero, then lim of f(x) as x approaches a = f(x) b) if q(a) = 0, but p(a) doesn't equal zero, then the limit of f(x) doesn't exist

limit laws for limits at infinity

limit laws with finite numbers carry over to limits at +/- infinity

if values of f(x) increase or decrease without bound as x approaches =/- infinity

limit of f(x) as x approaches +/- infinity = + infinity or -infinity

if values of f(x) get close to #L as x increases or decreases without

limit of f(x) as x approaches +/- infinity = L and if either limit holds: call line y=2 a horizontal asymptote

basic theorem

limit of k as x approaches a = k, limit of x as x approaches a = a, limit of 1/x as x approaches 0- = -infinity, limit of 1/x as x approaches 0+ = +infinity

alternate way of expressing 2.1.1

m tan = lim of f(x_0 + h) - f(x_0)/ h as h approaches 0 and h = x - x_0

limits involving radicals

manipulate function so that powers of x are transformed to powers of 1/x and can be achieved by dividing num by denom y the absolute value of x and using fact that the square root of x^2 = the absolute value of x

end behavior of polynomial

matches end behavior of highest degree term

if p(x)/q(x) is rational function for which p(a) = 0 and q(a) = 0

num and denom must have one or more common factors of x-a, and limit is found by canceling all common factors of x-a

in determinant form of type 0/0

quotient of f(x)/g(x) in which num and denom both have limit of zero as x approaches a

two-sided limit

requires values of f(x) to get closer & closer to L as values of x are taken from either side of x=a

simple root

root of multiplicity 1

function diff at x_0

said to be local linear at x_0 an takes appearance of straight line when magnified

derivatives of trig functions

sin x: cos x & cos x = sin x tan x = sec^2 x & sec x = sec x (tan x) cotan x = - csc^2 x & csc x = - csc x (cotan x)

higher order derivatives

take derivative of function more than one

if variable x increases/decreases without bound

x approaches +/- infinity