B Veitch. Calculus I Study Guide

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1 Clculus I Stuy Guie This stuy guie is in no wy exhustive. As stte in clss, ny type of question from clss, quizzes, exms, n homeworks re fir gme. There s no informtion here bout the wor problems. 1. Some Algebr Review () Fctoring n Solving i. Qurtic Formul: x 2 + bx + c = 0 ii. Difference of Two Squres x = b ± b 2 4c 2 u 2 2 = (u )(u + ) Creful! Sometimes you hve to rewrite the eqution, ex. x = 4 x 2 = (2 x)(2 + x) iii. Fctor Trinomils: x 2 + bx + c. Some exmples (b) Exponentil Properties x x + 16, 6x 2 + x 12, 2x 2 6x 10 i. n m = n+m ii. ( n ) m = nm iii. (b) n = n b n n iv. m = n m = 1 m n v. n = 1 n vi. 1 = n n vii. m/n = ( m ) 1/n = ( 1/n ) m (c) Properties of Ricls i. n = 1/n ii. n m = m/n 2. Limits () Nottion i. Generl Limit Nottion: lim f(x) = L ii. Left Hn Limit: lim f(x) = L iii. Right Hn Limit: lim f(x) = L + iv. If lim f(x) = lim f(x) = L then lim f(x) = L + 1

2 Clculus I Stuy Guie (b) Limits t ±. Assume ll polynomils re in escening orer. i. lim x x r = 0, r > 0 ii. If n > m, then lim x x m +... bx n +... = 0 iii. x n +... lim x bx n +... = b iv. If m > n, then lim x x m +... bx n +... = ± (c) Evlution Techniques i. If f(x) is continuous, then lim f(x) = f() ii. Fctor n Cncel lim x2 + 4 = = 13 x 3 If you evlute rtionl function n get 0, then try to fctor. 0 x 2 9 lim x 3 x 3 = lim (x 3)(x + 3) (x + 3) = lim = 6 x 3 x 3 x 3 1 iii. Rtionlizing Numertors / Denomintors Try this technique if you hve ricls. Multiply top n bottom by the conjugte. x 4 x 4 x + 4 lim x 4 x 16 = lim x 4 x 16 x 16 = lim x + 4 x 4 (x 16)( x + 4) = lim 1 x 4 x + 4 iv. Combine by Using Common Denomintors Try this when you nee to combine frctions within frctions = lim lim 2 x x = lim 2 x x 2 2x (x )(x + 4)( + 4) = lim (x + 4)( + 4) (x + 4)( + 4) = lim 2( + 4) 2(x + 4) (x )(x + 4)( + 4) 2(x ) (x )(x + 4)( + 4) = lim 2 (x + 4)( + 4) = 2 ( + 4)( + 4) () Piecewise Functions - Know how to grph n evlute Piecewise Functions. These re goo ones to test your unerstning of left-hn, right-hn, n generl limits. They re lso use to test your unerstning of continuity. (e) Definition of Continuity A function f is continuous t x = if the following three conitions re stisfie: i. f() must exist ii. lim f(x) must exist iii. lim f(x) = f() (f) Asymptotes 2

3 Clculus I Stuy Guie i. Verticl Asymptotes: The line x = is verticl symptote if ny of the following occur: lim f(x) = ± or lim From lgebr, you cn fin verticl symptotes by f(x) = ± + A. Reucing your rtionl function (no common fctors in numertor n enomintor). B. Fining the x vlues tht mke the enomintor 0. ii. Horizontl Asymptotes The line y = b is horizontl symptote if either or both occur: lim f(x) = b or lim x f(x) = b x Refer to 2(b) of this guie for shortcuts on evluting these limits. iii. Slnt Asymptotes: Given the function f(x) = P (x), slnt symptote occurs when the egree of the numertor Q(x) is 1 greter thn the egree of the enomintor. Ex. f(x) = x2 4 x 1 You fin the slnt symptote by oing long ivision. 3

4 Clculus I Stuy Guie 3. Derivtives () Limit Definition of Derivtive (b) Tngent Lines f (x) = lim h 0 f(x + h) f(x) h or f () = lim f(x) f() x When ske to fin the eqution of tngent line on f t x =, you nee two things: A point n slope. i. You re usully given the x vlue. If they on t tell you the y vlue, you must plug the x vlue into f(x) to get the y vlue. Now you hve point (, f()) ii. To fin the slope m, you fin m = f (). iii. The eqution of the tngent line is y f() = f ()(x ). You my nee to write this in slope-intercept form. (c) Derivtive Formuls i. x (c) = 0 ii. x (f ± g) = f ± g iii. x (x) = 1 iv. x (kx) = k v. Power Rule: x (xn ) = nx n 1 vi. Chin Rule: x f(g(x)) = f (g(x)) g (x) vii. Prouct Rule: (f g) = f g + fg ( ) f viii. Quotient Rule: = f g fg g g 2 ix. sin(x) = cos(x) x x. cos(x) = sin(x) x xi. x tn(x) = sec2 (x) xii. sec(x) = sec(x) tn(x) x xiii. csc(x) = csc(x) cot(x) x xiv. x cot(x) = csc2 (x) () Criticl Points x = c is vlue of f(x) if f (c) = 0 or f (c) oes not exist. (e) Incresing / Decresing i. If f (x) > 0 on n intervl I, then f(x) is incresing. ii. If f (x) < 0 on n intervl I, then f(x) is ecresing. (f) Concve Up / Concve Down i. If f (x) > 0 on n intervl I, then f(x) is concve up. ii. If f (x) < 0 on n intervl I, then f(x) is concve own. (g) Inflection Points x = c is n inflection point of f if 4

5 Clculus I Stuy Guie i. The point t x = c must exist. ii. Concvity chnges t x = c (h) First Derivtive Test to fin Locl (Reltive) Extrem i. Fin ll criticl vlue of f(x) where f (c) = 0 ii. Locl Mx t x = c if f (x) chnges from (+) to ( ) t x = c. iii. Locl Min t x = c if f (x) chnges from ( ) to (+) t x = c. iv. If f (x) oes not chnge signs, it s still n importnt point to plot. It my be plce where the slope is 0, corner, n symptote, verticl tngent line, etc. v. Mke sure you write your reltive mx n mins s points (c, f(c)) (i) Secon Derivtive Test to fin Locl (Reltive) Extrem i. Fin ll criticl vlue of f(x) where f (c) = 0 ii. Locl Mx t x = c if f (c) < 0 iii. Locl Min t x = c if f (x) > 0 iv. Mke sure you write your reltive mx n mins s points (c, f(c)) (j) Absolute Extrem i. (c, f(c)) is n bsolute mximum of f(x) if f(c) f(x) for ll x in the omin. ii. (c, f(c)) is n bsolute minimum of f(x) if f(c) f(x) for ll x in the omin. (k) Fining Absolute Extrem of continuous f(x) over [, b] i. Fin ll criticl vlues of f(x) on [, b]. ii. Evlute f(x) t ll criticl vlues. iii. Evlute f(x) t the enpoints, f() n f(b). iv. The bsolute mx is the lrgest function vlue n the bsolute min is the smllest function vlue. (l) Differentils / Lineriztion i. Lineriztion of f(x) t x = L(x) = f() + f ()(x ) ii. Differentils A. x - is the true chnge in x B. ifferentil x - is our inepenent vrible tht represents the chnge in x. We let x = x. C. y - is the true chnge in y D. ifferentil y - is the estimte chnge in y E. Formul: y x = f (x) or y = f (x) x 5

6 Clculus I Stuy Guie 4. Summry of Curve Sketching Alwys strt by noting the omin of f(x) () x n y intercepts i. x-intercepts occur when f(x) = 0 ii. y-intercept occurs when x = 0 (b) Fin ny verticl, horizontl symptotes, or slnt symptotes. i. Verticl Asymptote: Fin ll x-vlues where lim f(x) = ±. Usully when the enomintor is 0 n the numertor is not 0. Rtionl function MUST be reuce. ii. Horizontl Aymptotes: Fin lim f(x) n lim f(x). There re shortcuts bse on the x x egree of the numertor n enomintor. (c) Fin f (x) iii. Slnt Asymptotes: Occurs when the egree of the numertor is one lrger thn the enomintor. You must o long ivision to etermine the symptotes. i. Fin the criticl vlues, ll x-vlues where f (x) = 0 or when f (x) oes not exist. ii. Plot the criticl vlues on number line. iii. Fin incresing / ecresing intervls using number line iv. Use The First Derivtive Test to fin locl mximums / minimums (if ny exist). () Fin f (x) Remember to write them s points. A. Locl Mx t x = c if f (x) chnges from (+) to ( ) t x = c. B. Locl Min t x = c if f (x) chnges from ( ) to (+) t x = c. C. Note: If f (x) oes not chnge signs, it s still n importnt point. It my be plce where the slope is 0, corner, n symptote, verticl tngent line, etc. i. Fin ll x-vlues where f (x) = 0 or when f (x) oes not exist. ii. Plot these x-vlues on number line. iii. Fin intervls of concvity using the number line iv. Fin points of inflection (e) Sketch A. Must be plce where concvity chnges B. The point must exist (i.e, cn t be n symptote, iscontinuity) i. Drw every symptote ii. Plot ll intercepts iii. Plot ll criticl points (even if they re not reltive extrem). They were criticl points for reson. iv. Plot ll inflection points. v. Connect points on the grph by using informtion bout incresing/ecresing n its concvity. 6

7 Clculus I Stuy Guie 5. Integrls () Definitions i. Antieritive: An ntierivtive of f(x) is function F (x), where F (x) = f(x). ii. Generl Antierivtive: The generl ntierivtive of f(x) is F (x) + C, where F (x) = f(x). Also known s the Inefinite Integrl f(x) x = F (x) + C iii. Definite Integrl: (b) Common Integrls b f(x) x = F (b) F () i. ii. iii. iv. k x = kx + C x n x = xn+1 + C, n 1 n + 1 sin(x) x = cos(x) + C cos(x) x = sin(x) + C v. vi. vii. viii. sec 2 (x) x = tn(x) + C sec(x) tn(x) x = sec(x) + C csc 2 (x) x = cot(x) + C csc(x) cot(x) = csc(x) + C (c) u Substitution Given b i. Let u = g(x) f(g(x))g (x) x, ii. Then u = g (x) x iii. If there re bouns, you must chnge them using u = g(b) n u = g() b f(g(x))g (x) x = g(b) g() f(u) u () Every integrl must be written into the proper form in orer to use the formuls. For exmple, (e) Fining Are uner or between Curves x x = x 1/2 x 3 3 5x 4 x = 5 x 4 x 8 5 x = 8x 9/5 x x 9 7

8 Clculus I Stuy Guie 6. Approximtion Integrtion Techniques Given the integrl x i = + i x, then b f(x) x n n (for Simpson s Rule n must be even), with x = b n n Left-hn b Right-hn Mipoint b b f(x) x = x [f(x 0 ) + f(x 1 ) + f(x 2 ) + f(x 3 ) f(x n 1 )] f(x) x = x [f(x 1 ) + f(x 2 ) + f(x 3 ) + f(x 4 ) f(x n )] f(x) x = x [f(x 1) + f(x 2) + f(x 3) + f(x 4) f(x n)], where x i is mipoint in [x i 1, x i ] 8

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