lim f(x) does not exist, such that reducing a common factor between p(x) and q(x) results in the agreeable function k(x), then


 Marylou Webb
 2 years ago
 Views:
Transcription
1 AP Clculus AB/BC Formul nd Concept Chet Sheet Limit of Continuous Function If f(x) is continuous function for ll rel numers, then lim f(x) = f(c) Limits of Rtionl Functions A. If f(x) is rtionl function given y f(x) = p(x),such tht p(x) nd q(x) hve no common fctors, nd c is rel q(x) numer such tht q(c) = 0, then I. lim f(x) does not exist II. lim f(x) = ± x = c is verticl symptote B. If f(x) is rtionl function given y f(x) = p(x), such tht reducing common fctor etween p(x) nd q(x) results q(x) in the greele function k(x), then lim f(x) = lim p(x) q(x) = lim k(x) = k(c) Hole t the point (c, k(c)) Limits of Function s x Approches Infinity If f(x) is rtionl function given y (x) = p(x), such tht p(x) nd q(x) re oth polynomil functions, then q(x) A. If the degree of p(x) > q(x), lim x f(x) = B. If the degree of p(x) < q(x), lim x f(x) = 0 y = 0 is horizontl symptote C. If the degree of p(x) = q(x), lim x f(x) = c, where c is the rtio of the leding coefficients. y = c is horizontl symptote Specil Trig Limits sin x A. lim x 0 x = 1 B. lim x x 0 sin x 1 cos x = 1 C. lim = 0 x 0 x L Hospitl s Rule If results lim f(x) or lim f(x) results in n indeterminte form ( 0, x 0 f(x) = p(x), then q(x),, 0, 00, 1, 0 ), nd lim f(x) = lim = lim q(x) p(x) p (x) q (x) nd p(x) lim f(x) = lim = lim x x q(x) x p (x) q (x)
2 The Definition of Continuity A function f(x) is continuous t c if I. lim f(x) exists II. f(c) exists III. lim f(x) = f(c) Types of Discontinuities Removle Discontinuities (Holes) I. lim f(x) = L (the limit exists) II. f(c) is undefined NonRemovle Discontinuities (Jumps nd Asymptotes) A. Jumps lim f(x) = DNE ecuse lim f(x) lim f(x) + B. Asymptotes (Infinite Discontinuities) lim f(x) = ±
3 Intermedite Vlue Theorem If f is continuous function on the closed intervl [, ] nd k is ny numer etween f() nd f(), then there exists t lest one vlue of c on [, ] such tht f(c) = k. In other words, on continuous function, if f()< f(), ny y vlue greter thn f() nd less thn f() is gurnteed to exists on the function f. Averge Rte of Chnge The verge rte of chnge, m, of function f on the intervl [, ] is given y the slope of the secnt line. m = f() f() Definition of the Derivtive The derivtive of the function f, or instntneous rte of chnge, is given y converting the slope of the secnt line to the slope of the tngent line y mking the chnge is x, Δx or h, pproch zero. f f(x+h) f(x) (x) = lim h 0 h Alternte Definition f f(x) f(c) (c) = lim x c
4 Differentiility nd Continuity Properties A. If f(x) is differentile t x = c, then f(x) is continuous t x = c. B. If f(x) is not continuous t x = c, then f(x) is not differentile t x = c. C. The grph of f is continuous, ut not differentile t x = c if: I. The grph hs cusp or shrp point t x = c II. The grph hs verticl tngent line t x = c III. The grph hs n endpoint t x = c Bsic Derivtive Rules Given c is constnt, Derivtives of Trig Functions Derivtives of Inverse Trig Functions
5 Derivtives of Exponentil nd Logrithmic Functions Explicit nd Implicit Differentition A. Explicit Functions: Function y is written only in terms of the vrile x (y = f(x)). Apply derivtives rules normlly. B. Implicit Differentition: An expression representing the grph of curve in terms of oth vriles x nd y. I. Differentite oth sides of the eqution with respect to x. (terms with x differentite normlly, terms with y re multiplied y dy per the chin rule) dx II. Group ll terms with dy on one side of the eqution nd ll other terms on dx the other side of the eqution. III. Fctor dy dy nd express in terms of x nd y. dx dx Tngent Lines nd Norml Lines A. The eqution of the tngent line t point (, f()): y f() = f ()(x ) B. The eqution of the norml line t point (, f()): y f() = 1 (x ) f () Men Vlue Theorem for Derivtives If the function f is continuous on the close intervl [, ] nd differentile on the open intervl (, ), then there exists t lest one numer c etween nd such tht f (c) = f() f() The slope of the tngent line is equl to the slope of the secnt line.
6 Rolle s Theorem (Specil Cse of Men Vlue Theorem) If the function f is continuous on the close intervl [, ] nd differentile on the open intervl (, ), nd f() = f(), then there exists t lest one numer c etween nd such tht f (c) = f() f() = 0 Prticle Motion A velocity function is found y tking the derivtive of position. An ccelertion function is found y tking the derivtive of velocity function. x(t) Position x (t) = v(t) Velocity * v(t) = speed x (t) = v (t) = (t) Acclertion Rules: A. If velocity is positive, the prticle is moving right or up. If velocity is negtive, the prticle is moving left or down. B. If velocity nd ccelertion hve the sme sign, the prticle speed is incresing. If velocity nd ccelertion hve opposite signs, speed is decresing. C. If velocity is zero nd the sign of velocity chnges, the prticle chnges direction. Relted Rtes A. Identify the known vriles, including their rtes of chnge nd the rte of chnge tht is to e found. Construct n eqution relting the quntities whose rtes of chnge re known nd the rte of chnge to e found. B. Implicitly differentite oth sides of the eqution with respect to time. (Rememer: DO NOT sustitute the vlue of vrile tht chnges throughout the sitution efore you differentite. If the vlue is constnt, you cn sustitute it into the eqution to simplify the derivtive clcultion). C. Sustitute the known rtes of chnge nd the known vlues of the vriles into the eqution. Then solve for the required rte of chnge. *Keep in mind, the vriles present cn e relted in different wys which often involves the use of similr geometric shpes, Pythgoren Theorem, etc.
7 Extrem of Function A. Asolute Extrem: An solute mximum is the highest y vlue of function on given intervl or cross the entire domin. An solute minimum is the lowest y vlue of function on given intervl or cross the entire domin. B. Reltive Extrem I. Reltive Mximum: The yvlue of function where the grph of the function chnges from incresing to decresing. Another wy to define reltive mximum is the yvlue where derivtive of function chnges from positive to negtive. II. Reltive Minimum: The yvlue of function where the grph of the function chnges from decresing to incresing. Another wy to define reltive mximum is the yvlue where derivtive of function chnges from negtive to positive. Criticl Vlue When f(c) is defined, if f (c) = 0 or f is undefined t x = c, the vlues of the x coordinte t those points re clled criticl vlues. *If f(x) hs reltive extrem t x = c, then c is criticl vlue of f. Extreme Vlue Theorem If the function f continuous on the closed intervl [, ], then the solute extrem of the function f on the closed intervl will occur t the endpoints or criticl vlues of f. *After identifying criticl vlues, crete tle with endpoints nd criticl vlues. Clculte the y vlue t ech of these x vlues to identify the extrem.
8 Incresing nd Decresing Functions For differentile function f A. If f (x) > 0 in (, ), then f is incresing on (, ) Tngent line hs positive slope B. If f (x) < 0 in (, ), then f is decresing on (, ) Tngent line hs negtive slope C. If f (x) = 0 in (, ), then f is constnt on (, ) Tngent line hs zero slope (horizontl) First Derivtive Test After clculting ny discontinuities of function f nd clculting the criticl vlues of function f, crete sign chrt for f, reflecting the domin, discontinuities, nd criticl vlues of function f. A. If f (x) chnges sign from negtive to positive t x = c, then f(c) is reltive minimum of f. B. If f (x) chnges sign from positive to negtive t x = c, then f(c) is reltive mximum of f. *If there is no sign chnge of f (x), there exists shelf point Concvity For differentile function f(x), A. If f (x) > 0, the grph of f(x) is concve up This mens f (x) is incresing B. If f (x) < 0, the grph of f(x) is concve down This mens f (x) is decresing Second Derivtive Test For function f(x) tht is continuous t x = c A. If f (c) = 0 nd f (c) > 0, then f(c) is reltive minimum. B. If f (c) = 0 nd f (c) < 0, then f(c) is reltive mximum. * If f (c) = 0 nd f (c) = 0, you must use the first derivtive test to determine extrem
9 Point of Inflection Let f e functions whose second derivtive exists on ny intervl. If f is continuous t x = c, f (c) = 0 or f (c) is undefined, nd f (x) chnges sign t x = c, then the point (c, f(c)) is point of inflection. Optimiztion Finding the lrgest or smllest vlue of function suject to some kind of constrints. A. Define the primry eqution for the quntity to e mximized or minimized. Define fesile domin for the vriles present in the eqution. B. If necessry, define secondry eqution tht reltes the vriles present in the primry eqution. Solve this eqution for one of the vriles nd sustitute into the primry eqution. C. Once the primry eqution is represented in single vrile, tke the derivtive of the primry eqution. D. Find the criticl vlues using the derivtive clculted. E. The optiml solution will more thn likely e found t criticl vlue from D. Keep in mind, if the criticl vlues do not represent minimum or mximum, the optiml solution my e found t n endpoint of the fesile domin. Derivtive of n Inverse If f nd its inverse g re differentile, nd the point (c, f(c)) exists on the function f mening the point (f(c), c) exists on the function g then d dx [g(x)] = 1 f (f 1 (x)) = 1 f (f(c)) BC Only: Derivtives of Prmetric Functions If f nd g re continuous functions of t on n intervl, then the equtions x = f(t) nd y = g(t) re clled prmetric equtions, providing the position in the coordinte plne, nd t is clled the prmeter. A. The slope of the curve t the point (x, y) is dy dx = dy/dt, provided dx/dt 0 dx/dt B. The second derivtive t the point (x, y) is d 2 y dx 2 = d dt (dy dx ) dx dt
10 Antiderivtives If F (x) = f(x) for ll x, F(x) is n ntiderivtive of f. f(x) = F(x) + C * The ntiderivtive is lso clled the Indefinite Integrl Bsic Integrtion Rules Let k e constnt. Definite Integrls (The Fundmentl Theorem of Clculus) A definite integrl is n integrl with upper nd lower limits, nd, respectively, tht define specific intervl on the grph. A definite integrl is used to find the re ounded y the curve nd n xis on the specified intervl (, ). If F(x) is the ntiderivtive of continuous function f(x), the evlution of the definite integrl to clculte the re on the specified intervl (, ) is the First Fundmentl Theorem of Clculus: f(x)dx = F() F() Integrtion Rules for Definite Integrls *This mens tht c is vlue of x, lying etween nd
11 Riemnn Sum (Approximtions) A Riemnn Sum is the use of geometric shpes (rectngles nd trpezoids) to pproximte the re under curve, therefore pproximting the vlue of definite integrl. If the intervl [, ] is prtitioned into n suintervls, then ech suintervl, Δx, hs width: x = n. Therefore, you find the sum of the geometric shpes, which pproximtes the re y the following formuls: A. Right Riemnn Sum Are x [f(x 0 ) + f(x 1 ) + f(x 2 ) + + f(x n 1 )] B. Left Riemnn Sum Are x [f(x 1 ) + f(x 2 ) + f(x 3 ) + + f(x n )] C. Midpoint Riemnn Sum Are x [f(x 1/2 ) + f(x 3/2 ) + f(x 5/2 ) + + f(x (2n 1)/2 )] D. Trpezoidl Sum Are 1 2 x [f(x 0) + 2 f(x 1 ) + 2 f(x 2 ) + + 2f(x n 1 ) + f(x n )] Properties of Riemnn Sums A. The re under the curve is under pproximted when I.A Left Riemnn sum is used on n incresing function. II. A Right Riemnn sum is used on decresing function. III. A Trpezoidl sum is used on concve down function. B. The re under the curve is over pproximted when I.A Left Riemnn sum is used on decresing function. II. A Right Riemnn sum is used on n incresing function. III. A Trpezoidl sum is used on concve up function.
12 Riemnn Sum (Limit Definition of Are) Let f e continuous function on the intervl [, ]. The re of the region ounded y the grph of the function f nd the x xis (i.e. the vlue of the definite integrl) cn e found using f(x)dx n = lim f(c i ) x n Where c i is either the left endpoint (c i = + (i 1) x) or right endpoint (c i = + i x) nd x = ( )/n. i=1 Averge Vlue of Function If function f is continuous on the intervl [, ], the verge vlue of tht function f is given y 1 f(x)dx Second Fundmentl Theorem of Clculus If function f is continuous on the intervl [, ], let u represent function of x, then A. B. C. d dx d dx x [ f(t)dt] = f(x) [ f(t)dt] = f(x) x u(x) d dx [ f(t)dt ] = f(u(x)) u (x) Integrtion of Exponentil nd Logrithmic Formuls
13 Integrtion of Trig nd Inverse Trig BC Only: Integrtion y Prts If u nd v re differentile functions of x, then u dv = uv v du Tips: For your choice of the function u, mke the selection following: A. LIPET: Logrithmic, Inverse Trig, Polynomil, Exponentil, Trig B. LIATE: Logrithmic, Inverse Trig, Algeric, Trig, Exponentil Comes from Integrtion y Prts. MEMORIZE ln x dx = x ln x x + C
14 BC Only: Prtil Frctions Let R(x) represent rtionl function of the form R(x) = N(x). If D(x) is fctorle polynomil, Prtil Frctions cn D(x) e used to rewrite R(x) s the sum or difference of simpler rtionl functions. Then, integrtion using nturl log. A. Constnt Numertor B. Polynomil Numertor
15 BC Only: Improper Integrls An improper integrl is chrcterized y hving limits of integrtion tht is infinite or the function f hving n infinite discontinuity (symptote) on the intervl [, ]. A. Infinite Upper Limit (continuous function) B. Infinite Lower Limit (continuous function) C. Both Infinite Limits (continuous function) f(x)dx = lim f(x)dx f(x)dx = lim f(x)dx c f(x)dx = lim f(x)dx + lim f(x)dx, where c is n x vlue nywhere on f. D. Infinite Discontinuity (Let x = k represent n infinite discontinuity on [, ]) f(x)dx c k = lim f(x)dx + lim f(x)dx x k x k + k BC Only: Arc Length (Length of Curve) A. If the function y = f(x)is differentile function, then the length of the rc on [, ] is 1 + [f (x)] 2 dx B. If the function x = f(y)is differentile function, then the length of the rc on [, ] is 1 + [f (y)] 2 dy C. Prmetric Arc Length: If smooth curve is given y x(t) nd y(t), then the rc length over the intervl t is ( dx 2 dt ) + ( dy 2 dt ) dt
16 Exponentil Growth nd Decy When the rte of chnge of vrile y is directly proportionl to the vlue of y, the function y = f(x) is sid to grow/decy exponentilly. A. Differentil Eqution for rte of chnge: dy dt = ky B. Generl Solution: y = Ce kt I. If k > 0, then exponentil growth occurs. II. If k < 0, then exponentil decy occurs. BC Only: Logistic Growth A popultion, P, tht experiences limit fctor in the growth of the popultion sed upon the ville resources to support the popultion is sid to experience logistic growth. A. Differentil Eqution: dp dt = kp (1 P L ) B. Generl Solution: P(t) = L 1+e kt P = popultion k = constnt growth fctor L = crrying cpcity t = time, = constnt (found with intitl condition) Grph Chrcteristics of Logistics I. The popultion is growing the fstest where P = L 2 II. The point where P = L represents point of inflection 2 III. lim t P(t) = L
17 Are Between Two Curves A. Let y = f(x) nd y = g(x)represent two functions such tht f(x) g(x)(mening the function f is lwys ove the function g on the grph) for every x on the intervl [, ]. Are Between Curves = [f(x) g(x)] dx B. Let x = f(y) nd x = g(y)represent two functions such tht f(y) g(y)(mening the function f is lwys to the right of the function g on the grph) for every y on the intervl [, ]. Are Between Curves = [f(y) g(y)] dy Volumes of Solid of Revolution: Disk Method If defined region, ounded y differentile function f, on grph is rotted out line, the resulting solid is clled solid of revolution nd the line is clled the xis of revolution. The disk method is used when the defined region orders the xis of revolution over the entire intervl [, ] A. Revolving round the x xis Volume = π (f(x)) 2 dx B. Revolving round the y xis Volume = π (f(y)) 2 dy C. Revolving round horizontl line y = k Volume = π (f(x) k) 2 dx D. Revolving round verticl line x = m Volume = π (f(y) m) 2 dy
18 Volumes of Solid of Revolution: Wsher Method If defined region, ounded y differentile function f, on grph is rotted out line, the resulting solid is clled solid of revolution nd the line is clled the xis of revolution. The wsher method is used when the defined region hs spce etween the xis of revolution on the intervl [, ] A. Revolving round the x xis, where f(x) g(x)(mening the function f is lwys ove the function g on the grph) for every x on the intervl [, ]. Volume = π ([f(x)] 2 [g(x)] 2 )dx B. Revolving round the y xis, where f(y) g(y)(mening the function f is lwys to the right of the function g on the grph) Volume = π ([f(y)] 2 [g(y)] 2 )dy C. Revolving round horizontl line y = k, where f(x) g(x)(mening the function f is lwys ove the function g on the grph) for every x on the intervl [, ]. Volume = π ([f(x) k] 2 [g(x) k] 2 )dx D. Revolving round verticl line x = m, where f(y) g(y)(mening the function f is lwys to the right of the function g on the grph) Volume = π ([f(y) m] 2 [g(y) m] 2 )dy
19 Volumes of Known Cross Sections If defined region, ounded y differentile function f, is used t the se of solid, then the volume of the solid cn e found y integrted using known re formuls. For the cross sections perpendiculr to the x xis nd region ounded y function f, on the intervl [, ], nd the xis. I. Cross sections re squres II. Cross sections re equilterl tringles Volume = [f(x)] 2 dx Volume = 3 4 [f(x)]2 dx III. Cross sections re isosceles right tringles with leg in the se Volume = 1 2 [f(x)]2 dx IV. Cross sections re isosceles right tringles with the hypotenuse in the se Volume = 1 4 [f(x)]2 dx V. Cross sections re semicircles (with dimeter in se) Volume = π 8 [f(x)]2 dx VI. Cross sections re semicircles (with rdius in se) Volume = π 2 [f(x)]2 dx
20 Differentil Equtions A differentil eqution is n eqution involving n unknown function nd one or more of its derivtives dy dx = f(x, y) Usully expressed s derivtive equl to n expression in terms of x nd/or y. To solve differentil equtions, use the technique of seprtion of vriles. Given the differentil eqution dy dx = xy (x 2 +1) Step 1: Seprte the vriles, putting ll y s on one side, with dy in the numertor, nd ll x s on the other side, with dx in the numertor. Step 2: Integrte oth sides of the eqution. Step 3: Solve the eqution for y. 1 y dy = x (x 2 + 1) dx ln y = 1 2 ln x C y = C x Given the differentil eqution dy dx = 2x2 with the initil condition y(3) = 10. A. The generl solution to differentil eqution is left with the constnt of integrtion, C, undefined. dy = 2x 2 dx dy = 2x 2 dx y = 2 3 x3 + C B. The prticulr solution uses the given initil condition to clculte the vlue of C. 10 = 2 3 (3)3 + C C = 8 y = 2 3 x3 8 BC Only: Euler s Method for Approximting the Solution of Differentil Eqution Euler s method uses liner pproximtion with increments (steps), h, for pproximting the solution to given differentil eqution, dy = F(x, y), with given initil vlue. dx Process: Initil vlue (x 0, y 0 ) x 1 = x 0 + h y 1 = y 0 + h F(x 0, y 0 ) x 2 = x 1 + h y 2 = y 1 + h F(x 1, y 1 ) x 3 = x 2 + h y 3 = y 2 + h F(x 2, y 2 ) * This process repets until the desired y vlue is given.
21 Slope Field The derivtive of function gives the vlue of the slope of the function t ech point (x, y). A slope field is grphicl representtion of ll of the possile solutions to given differentil eqution. The slope field is generted y plugging in the coordintes of every point (x, y) into the differentil eqution nd drwing smll segment of the tngent line t ech point. Given the differentil eqution dy dx = x y dy dx = 0 (0,0) 0 undefined dy dx = 0 (0,±1) dy dx (1,2) = 1 2 *These re only three exmple points. You would do this for every point in the given region of the grph. BC Only: Testing for Convergence/Divergence of Series Given the series n = Sequence of Prtil Sums The sequence of prtil sums for the series is S 1 = 1 S 2 = S 3 = S n = n If lim n S n = S, then n converges to S. If the terms of sequence do not converge to 0, then the series must diverge. Nth Term I. If lim n n 0, then n diverges. II. If lim n n = 0, then the test is inconclusive. The form of p series is 1 n p P Series I. If p > 1, then the series converges. II. If p < 1, then the series diverges.
22 Geometric Series A geometric series is ny series of the form r n n=0 I. If r < 1, then the series converges to 1 r *Series must e indexed t n = 0 II. If r > 1, then the series diverges. A telescoping series is ny series of the form n n+1 Telescoping Series *Convergence nd divergence is found using sequence of prtil sums *Prtil decomposition my e used to rek single rtionl series into the difference of two series tht form the telescoping series. Integrl If f is positive, continuous, nd decresing for x 1, then n nd f(x)dx n=1 1 either oth converge or oth diverge. A series, contining oth positive terms, negtive terms, nd n > 0, of the form ( 1) n n n=1 or ( 1) n+1 n n=1 Alternting Series The series converge if oth of the following conditions re met I. n+1 n for ll n II. lim n n = 0 When compring two series, if n n for ll n, Direct Comprison I. If n diverges, then n diverges. II. If n converges, then n converges. *The convergence or divergence of the series chosen for comprison should e known
23 Limit Comprison If n > 0 nd n > 0 nd lim n = L, where L is finite nd positive, then the series n n n nd n either oth converge or oth diverge. *The convergence or divergence of the series chosen for comprison should e known *When choosing series to compre to, disregrd ll ut the highest powers (growth fctor) in the numertor nd denomintor Given series n Root n I. If lim n < 1, then n converges. n n II. If lim n > 1, then n diverges. n n III. If lim n = 1, then the root test is inconclusive. n *This is test for solute convergence Given series n Rtio I. If lim n+1 < 1, then n converges. n n II. If lim n+1 > 1, then n diverges. n n III. If lim n+1 = 1, then the rtio test is inconclusive. n n *This is test for solute convergence BC Only: Asolute vs Conditionl Convergence For series, n, with oth positive nd negtive terms A. If n converges, then n lso converges. n is sid to e solutely convergent. n=1 n=1 n=1 B. If n diverges, ut n converges, n is sid to e conditionlly convergent. n=1 n=1 n=1 BC Only: Alternting Series Reminder Theorem Given n is convergent lternting series, the error ssocited with pproximting the sum of the series y the first n terms is less thn or equl to the first omitted term. ( 1) n+1 n = S S n = ( 1) n+1 n n=1 Error = S S n n+1
24 BC Only: Power Series A. Power Series Structure nd Chrcteristics n x n = x + 2 x n x n + power series centered t x = 0 n=0 n (x c) n = (x c) + 2 (x c) n (x c) n + power series centered t x = c n=0 A function f cn e represented y power series, where the power series converges to the function in one of three wys: I. The power series only converges t the center x = c. II. The power series converges for ll rel vlues of x. III. The power series converges for some intervl of vlues such tht x c < R, where R is the rdius of convergence of the power series. B. Intervl of Convergence: Find this y pplying the Rtio to the given series. I. If R = 0, then the series converges only t x = c. II. If R =, then the series converges for ll rel vlues of x. III. If the Rtio Test results in n expression of the form x c < R, then the intervl of convergence is of the form c R < x < c + R. *The convergence t the endpoints of the intervl of convergence should e tested seprtely. BC Only: Tylor nd Mclurin Series (specific power series) If function of f hs derivtives of ll orders t x = c, then the series is clled Tylor Series for f centered t c. A Tylor series centered t 0 is lso known s Mclurin Series. A. Mclurin Series f(x) = f(0) + f (0)x + f (0) x2 2! + f (0) x3 3! + + f(n) (0) xn n! + = x n f(n) (0) n! B. Tylor Series f(x) = f(c) + f (c)(x c) + f (x c)2 (c) + f (x c)3 (c) + + f (n) (x c)n (c) 2! 3! n! n=0 + = f (n) (x c) n (c) n! n=0
25 BC Only: Common Series to MEMORIZE Series A. B. 1 1 x = 1 + x + x2 + + x n + = x n n=0 e x = 1 + x + x2 2! + + xn n! + = xn n! n=0 C. cos x = 1 x2 2! + x4 4! + x2n (2n)! + = ( 1)n n=0 x2n (2n)! D. sin x = x x3 3! + x5 5! + x2n+1 (2n + 1)! + = ( 1)n E. ln(1 + x) = x x2 2 + x ( 1)n x n n F. rctn x = x x3 3 + x ( 1)n x 2n+1 2n + 1 n=0 xn n = ( 1) n n=0 x 2n+1 (2n + 1)! x2n+1 n = ( 1) 2n + 1 n=0 Intervl of Convergence 1 < x < 1 < x < < x < < x < 1 < x 1 < x < BC Only: Lgrnge Reminder of Tylor Polynomil When pproximting function f(x) using n nth degree Tylor polynomil, P n (x), the ssocited error, R n (x), is ounded y (x c)n+1 R n (x) = f(x) P n (x) mx f (n+1) (z) where c z x (n + 1)! BC Only: Polr Coordintes A. The polr coordintes (r, θ)of point re relted to the rectngulr coordintes (x, y) s follows x = r cos θ y = r sin θ r 2 = x 2 + y 2 tn θ = x y B. If f is differentile function of θ (smooth curve), then the slope of the line tngent to the grph of r = f(θ) t the point (r, θ) is dy dx = dy/dθ dx/dθ = r sin θ + r cos θ r cos θ r sin θ = f (θ) sin θ + f(θ) cos θ f (θ) cos θ f(θ) sin θ C. If r = f(θ) is smooth curve on the intervl [α, β], where α nd β re rdil lines, then the re enclosed y the grph is Are = 1 β 2 r2 dθ α = 1 2 [f(θ)]2 dθ β α
( ) Same as above but m = f x = f x  symmetric to yaxis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.
AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find
More information( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB/ Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 Find the intersection of f ( x) nd g( x). A3 Show tht f ( x) is even. A4 Show tht
More informationTopics Covered AP Calculus AB
Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.
More information( ) as a fraction. Determine location of the highest
AB Clculus Exm Review Sheet  Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if
More information( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
More informationAB Calculus Review Sheet
AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is
More informationMATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
More informationMath 100 Review Sheet
Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s
More informationDisclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know.
Disclimer: This is ment to help you strt studying. It is not necessrily complete list of everything you need to know. The MTH 33 finl exm minly consists of stndrd response questions where students must
More informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More informationcritical number where f '(x) = 0 or f '(x) is undef (where denom. of f '(x) = 0)
Decoding AB Clculus Voculry solute mx/min x f(x) (sometimes do sign digrm line lso) Edpts C.N. ccelertion rte of chnge in velocity or x''(t) = v'(t) = (t) AROC Slope of secnt line, f () f () verge vlue
More informationA. Limits  L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.
A. Limits  L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c
More informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
More information2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).
AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following
More informationA. Limits  L Hopital s Rule. x c. x c. f x. g x. x c 0 6 = 1 6. D. 1 E. nonexistent. ln ( x 1 ) 1 x 2 1. ( x 2 1) 2. 2x x 1.
A. Limits  L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( ) lim where lim f or lim f limg. c g = c limg( ) = c = c = c How to find it: Try nd find limits by
More informationf(a+h) f(a) x a h 0. This is the rate at which
M408S Concept Inventory smple nswers These questions re openended, nd re intended to cover the min topics tht we lerned in M408S. These re not crnkoutnnswer problems! (There re plenty of those in the
More information. Doubleangle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) =
Review of some needed Trig. Identities for Integrtion. Your nswers should be n ngle in RADIANS. rccos( 1 ) = π rccos(  1 ) = 2π 2 3 2 3 rcsin( 1 ) = π rcsin(  1 ) = π 2 6 2 6 Cn you do similr problems?
More informationMA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations
LESSON 0 Chpter 7.2 Trigonometric Integrls. Bsic trig integrls you should know. sin = cos + C cos = sin + C sec 2 = tn + C sec tn = sec + C csc 2 = cot + C csc cot = csc + C MA 6200 Em 2 Study Guide, Fll
More informationChapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...
Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting
More informationAntiderivatives/Indefinite Integrals of Basic Functions
Antiderivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second
More information0.1 Chapters 1: Limits and continuity
1 REVIEW SHEET FOR CALCULUS 140 Some of the topics hve smple problems from previous finls indicted next to the hedings. 0.1 Chpters 1: Limits nd continuity Theorem 0.1.1 Sndwich Theorem(F 96 # 20, F 97
More informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 23, Sections 4.14.8, nd Sections 5.15.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
More informationCalculus AB. For a function f(x), the derivative would be f '(
lculus AB Derivtive Formuls Derivtive Nottion: For function f(), the derivtive would e f '( ) Leiniz's Nottion: For the derivtive of y in terms of, we write d For the second derivtive using Leiniz's Nottion:
More informationMain topics for the Second Midterm
Min topics for the Second Midterm The Midterm will cover Sections 5.45.9, Sections 6.16.3, nd Sections 7.17.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the
More informationCalculus AB Bible. (2nd most important book in the world) (Written and compiled by Doug Graham)
PG. Clculus AB Bile (nd most importnt ook in the world) (Written nd compiled y Doug Grhm) Topic Limits Continuity 6 Derivtive y Definition 7 8 Derivtive Formuls Relted Rtes Properties of Derivtives Applictions
More information( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
More information. Doubleangle formulas. Your answer should involve trig functions of θ, and not of 2θ. cos(2θ) = sin(2θ) =.
Review of some needed Trig Identities for Integrtion Your nswers should be n ngle in RADIANS rccos( 1 2 ) = rccos(  1 2 ) = rcsin( 1 2 ) = rcsin(  1 2 ) = Cn you do similr problems? Review of Bsic Concepts
More informationThomas Whitham Sixth Form
Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos
More informationFINALTERM EXAMINATION 9 (Session  ) Clculus & Anlyticl GeometryI Question No: ( Mrs: )  Plese choose one f ( x) x According to PowerRule of differentition, if d [ x n ] n x n n x n n x + ( n ) x n+
More informationMath 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8
Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite
More informationMath& 152 Section Integration by Parts
Mth& 5 Section 7.  Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
More informationFinal Review, Math 1860 Thomas Calculus Early Transcendentals, 12 ed
Finl Review, Mth 860 Thoms Clculus Erly Trnscendentls, 2 ed 6. Applictions of Integrtion: 5.6 (Review Section 5.6) Are between curves y = f(x) nd y = g(x), x b is f(x) g(x) dx nd similrly for x = f(y)
More informationChapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
More informationInstantaneous Rate of Change of at a :
AP Clculus AB Formuls & Justiictions Averge Rte o Chnge o on [, ]:.r.c. = ( ) ( ) (lger slope o Deinition o the Derivtive: y ) (slope o secnt line) ( h) ( ) ( ) ( ) '( ) lim lim h0 h 0 3 ( ) ( ) '( ) lim
More informationFirst midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009
Mth 3B Review Steve 18 Mrch 2009 About the finl Fridy Mrch 20, 3pm6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationMath 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More informationCalculus II: Integrations and Series
Clculus II: Integrtions nd Series August 7, 200 Integrls Suppose we hve generl function y = f(x) For simplicity, let f(x) > 0 nd f(x) continuous Denote F (x) = re under the grph of f in the intervl [,x]
More informationImproper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.
Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationf a L Most reasonable functions are continuous, as seen in the following theorem:
Limits Suppose f : R R. To sy lim f(x) = L x mens tht s x gets closer n closer to, then f(x) gets closer n closer to L. This suggests tht the grph of f looks like one of the following three pictures: f
More informationAP Calculus Multiple Choice: BC Edition Solutions
AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this
More informationDefinite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +
Definite Integrls 5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the lefthnd
More informationTopics for final
Topics for 161.01 finl.1 The tngent nd velocity problems. Estimting limits from tbles. Instntneous velocity is limit of verge velocity. Slope of tngent line is limit of slope of secnt lines.. The limit
More informationBig idea in Calculus: approximation
Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:
More informationHow can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?
Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those
More informationIntegration Techniques
Integrtion Techniques. Integrtion of Trigonometric Functions Exmple. Evlute cos x. Recll tht cos x = cos x. Hence, cos x Exmple. Evlute = ( + cos x) = (x + sin x) + C = x + 4 sin x + C. cos 3 x. Let u
More informationMath 0230 Calculus 2 Lectures
Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two
More information7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the righthnd side limit equls to the lefthnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More information7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?
7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge
More informationEdexcel GCE Core Mathematics (C2) Required Knowledge Information Sheet. Daniel Hammocks
Edexcel GCE Core Mthemtics (C) Required Knowledge Informtion Sheet C Formule Given in Mthemticl Formule nd Sttisticl Tles Booklet Cosine Rule o = + c c cosine (A) Binomil Series o ( + ) n = n + n 1 n 1
More informationOverview of Calculus I
Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,
More informationFinal Exam Review. Exam 1 Material
Lessons 24: Limits Limit Solving Strtegy for Finl Exm Review Exm 1 Mteril For piecewise functions, you lwys nee to look t the left n right its! If f(x) is not piecewise function, plug c into f(x), i.e.,
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More informationMathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
More informationMATH SS124 Sec 39 Concepts summary with examples
This note is mde for students in MTH124 Section 39 to review most(not ll) topics I think we covered in this semester, nd there s exmples fter these concepts, go over this note nd try to solve those exmples
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More informationx = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b
CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick
More informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
More informationThe area under the graph of f and above the xaxis between a and b is denoted by. f(x) dx. π O
1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the xxis etween nd is denoted y f(x) dx nd clled the
More informationDERIVATIVES NOTES HARRIS MATH CAMP Introduction
f DERIVATIVES NOTES HARRIS MATH CAMP 208. Introduction Reding: Section 2. The derivtive of function t point is the slope of the tngent line to the function t tht point. Wht does this men, nd how do we
More informationIf u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du
Integrtion by Substitution: The Fundmentl Theorem of Clculus demonstrted the importnce of being ble to find ntiderivtives. We now introduce some methods for finding ntiderivtives: If u = g(x) is differentible
More informationMath 113 Exam 1Review
Mth 113 Exm 1Review September 26, 2016 Exm 1 covers 6.17.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between
More informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rellife exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More informationFunctions of Several Variables
Functions of Severl Vribles Sketching Level Curves Sections Prtil Derivtives on every open set on which f nd the prtils, 2 f y = 2 f y re continuous. Norml Vector x, y, 2 f y, 2 f y n = ± (x 0,y 0) (x
More informationTest 3 Review. Jiwen He. I will replace your lowest test score with the percentage grade from the final exam (provided it is higher).
Test 3 Review Jiwen He Test 3 Test 3: Dec. 46 in CASA Mteril  Through 6.3. No Homework (Thnksgiving) No homework this week! Hve GREAT Thnksgiving! Finl Exm Finl Exm: Dec. 1417 in CASA You Might Be Interested
More informationSection 7.1 Integration by Substitution
Section 7. Integrtion by Substitution Evlute ech of the following integrls. Keep in mind tht using substitution my not work on some problems. For one of the definite integrls, it is not possible to find
More informationWe divide the interval [a, b] into subintervals of equal length x = b a n
Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:
More informationUnit Six AP Calculus Unit 6 Review Definite Integrals. Name Period Date NONCALCULATOR SECTION
Unit Six AP Clculus Unit 6 Review Definite Integrls Nme Period Dte NONCALCULATOR SECTION Voculry: Directions Define ech word nd give n exmple. 1. Definite Integrl. Men Vlue Theorem (for definite integrls)
More informationAP CALCULUS AB & BC. Course Description
AP CALCULUS AB & BC Course Description Clculus is n dvnced mthemtics course tht uses meningful problems nd pproprite technology to develop concepts nd pplictions relted to continuity nd discontinuity of
More informationMath 3B Final Review
Mth 3B Finl Review Written by Victori Kl vtkl@mth.ucsb.edu SH 6432u Office Hours: R 9:4510:45m SH 1607 Mth Lb Hours: TR 12pm Lst updted: 12/06/14 This is continution of the midterm review. Prctice problems
More information38 Riemann sums and existence of the definite integral.
38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the xxis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
More informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion  re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
More informationCalculus III Review Sheet
Clculus III Review Sheet 1 Definitions 1.1 Functions A function is f is incresing on n intervl if x y implies f(x) f(y), nd decresing if x y implies f(x) f(y). It is clled monotonic if it is either incresing
More information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl new nme for ntiderivtive. Differentiting integrls. Tody we provide the connection
More informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion  re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
More informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More informationMath 107H Topics for the first exam. csc 2 x dx = cot x + C csc x cotx dx = csc x + C tan x dx = ln secx + C cot x dx = ln sinx + C e x dx = e x + C
Integrtion Mth 07H Topics for the first exm Bsic list: x n dx = xn+ + C (provided n ) n + sin(kx) dx = cos(kx) + C k sec x dx = tnx + C sec x tnx dx = sec x + C /x dx = ln x + C cos(kx) dx = sin(kx) +
More information5: The Definite Integral
5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce
More informationAB Calculus Path to a Five Problems
AB Clculus Pth to Five Problems # Topic Completed Definition of Limit OneSided Limits 3 Horizontl Asymptotes & Limits t Infinity 4 Verticl Asymptotes & Infinite Limits 5 The Weird Limits 6 Continuity
More informationMath 118: Honours Calculus II Winter, 2005 List of Theorems. L(P, f) U(Q, f). f exists for each ǫ > 0 there exists a partition P of [a, b] such that
Mth 118: Honours Clculus II Winter, 2005 List of Theorems Lemm 5.1 (Prtition Refinement): If P nd Q re prtitions of [, b] such tht Q P, then L(P, f) L(Q, f) U(Q, f) U(P, f). Lemm 5.2 (Upper Sums Bound
More informationSection 6: Area, Volume, and Average Value
Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find
More informationOverview of Calculus
Overview of Clculus June 6, 2016 1 Limits Clculus begins with the notion of limit. In symbols, lim f(x) = L x c In wors, however close you emn tht the function f evlute t x, f(x), to be to the limit L
More informationn=0 ( 1)n /(n + 1) converges, but not n=100 1/n2, is at most 1/100.
Mth 07H Topics since the second exm Note: The finl exm will cover everything from the first two topics sheets, s well. Absolute convergence nd lternting series A series n converges bsolutely if n converges.
More informationSample Problems for the Final of Math 121, Fall, 2005
Smple Problems for the Finl of Mth, Fll, 5 The following is collection of vrious types of smple problems covering sections.8,.,.5, nd.8 6.5 of the text which constitute only prt of the common Mth Finl.
More informationSummary Information and Formulae MTH109 College Algebra
Generl Formuls Summry Informtion nd Formule MTH109 College Algebr Temperture: F = 9 5 C + 32 nd C = 5 ( 9 F 32 ) F = degrees Fhrenheit C = degrees Celsius Simple Interest: I = Pr t I = Interest erned (chrged)
More informationMA 124 January 18, Derivatives are. Integrals are.
MA 124 Jnury 18, 2018 Prof PB s oneminute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,
More informationMat 210 Updated on April 28, 2013
Mt Brief Clculus Mt Updted on April 8, Alger: m n / / m n m n / mn n m n m n n ( ) ( )( ) n terms n n n n n n ( )( ) Common denomintor: ( ) ( )( ) ( )( ) ( )( ) ( )( ) Prctice prolems: Simplify using common
More informationFinal Exam  Review MATH Spring 2017
Finl Exm  Review MATH 5  Spring 7 Chpter, 3, nd Sections 5.5.5, 5.7 Finl Exm: Tuesdy 5/9, :37:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or.
More informationStuff You Need to Know From Calculus
Stuff You Need to Know From Clculus For the first time in the semester, the stuff we re doing is finlly going to look like clculus (with vector slnt, of course). This mens tht in order to succeed, you
More informationx 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx
. Compute the following indefinite integrls: ) sin(5 + )d b) c) d e d d) + d Solutions: ) After substituting u 5 +, we get: sin(5 + )d sin(u)du cos(u) + C cos(5 + ) + C b) We hve: d d ln() + + C c) Substitute
More informationSpace Curves. Recall the parametric equations of a curve in xyplane and compare them with parametric equations of a curve in space.
Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xyplne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)
More informationKeys to Success. 1. MC Calculator Usually only 5 out of 17 questions actually require calculators.
Keys to Success Aout the Test:. MC Clcultor Usully only 5 out of 7 questions ctully require clcultors.. FreeResponse Tips. You get ooklets write ll work in the nswer ooklet (it is white on the insie)
More informationdifferent methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).
Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different
More informationCHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx
CHAPTER 0 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS 0.. PARAMETRIC FUNCTIONS A) Recll tht for prmetric equtions,. B) If the equtions x f(t), nd y g(t) define y s twicedifferentile function of x, then t
More informationFINALTERM EXAMINATION 2011 Calculus &. Analytical GeometryI
FINALTERM EXAMINATION 011 Clculus &. Anlyticl GeometryI Question No: 1 { Mrks: 1 )  Plese choose one If f is twice differentible function t sttionry point x 0 x 0 nd f ''(x 0 ) > 0 then f hs reltive...
More informationIntegrals  Motivation
Integrls  Motivtion When we looked t function s rte of chnge If f(x) is liner, the nswer is esy slope If f(x) is nonliner, we hd to work hrd limits derivtive A relted question is the re under f(x) (but
More information