Trigonometric, Differentiation and Integration Cheat Sheet, Cheat Sheet of Calculus

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Trigonometric formulas sin? @+cos* @=1 1+ tan? d= sect @ Lt cot? @ = cect a sunt -@) = —sin 2 cos(-8) =cos 2 tan{—@) =— tan @ sin(.4+ 8) =sn Acos 2 +sin Bcos A sin(.A- 8) =sin Acos#—sin 8cos A cos(.4 + B) = cos Acos #- sin Asin B cos(.4-8)=cosAcos# + sin Asin BF sin 26 = 2sin cos? cos 28 = cos’ @- sin? 9 = 2c087G-1=1-2sin 79 sn@ 1 _ cose 1 1 tan A= = cote =— sec @ = cos@ coté sin tan @ cosd 1 ca = — co 5-4] ane sin a sin 8 2 2 Differentiation formulas fat) = ns “tig = fetes a(t). fore ax dx axl g zg = Fela) = #(elaye'a) dx d.. —(sin x) = cos x am ) —(cos x) = sin x a } da a — (tan x) = sec’ x a d 2 —(cobx) = —cse" x a } a —(sec x) = secatanx ax d EP —(cse x) = -esc xcotx —(e" =e ae ) 7 } =2 - ayy ra : ao xy Fo © tan“) = 1 ax lex? Integration formulas sin ( )y D A B x C= + − A is amplitude B is the affect on the period (stretch or shrink) C is vertical shift (left/right) and D is horizontal shift (up/down) Limits: 0 0 sin sin 1 cos lim 1 lim 0 lim 0 x x x x x x x x x−> −>∞ −> − = = = 13. Properties of y = ex a. The exponential function y = ex is the inverse function of y = ln x. b. The domain is the set of all real numbers, −∞ < x < ∞. c. The range is the set of all positive numbers, y > 0. d. e. 14. Properties of y = ln x a. The domain of y = ln x is the set of all positive numbers, x > 0. b. The range of y = ln x is the set of all real numbers, −∞ < y < ∞. c. y = ln x is continuous and increasing everywhere on its domain. d. ln(ab) = ln a + ln b. e. ln(a / b) = ln a − ln b. f. ln ar = r ln a. 15. Fundamental theorem of calculus , where F'(x) = f(x), or . 16. Volumes of solids of revolution a. Let f be nonnegative and continuous on [a,b], and let R be the region bounded above by y = f(x), below by the x-axis, and the sides by the lines x = a and x = b. b. When this region R is revolved about the x-axis, it generates a solid (having circular cross sections) whose volume . c. When R is revolved about the y-axis, it generates a solid whose volume . 17. Particles moving along a line a. If a particle moving along a straight line has a positive function x(t), then its instantaneous velocity v(t) = x'(t) and its acceleration a(t) = v'(t). b. v(t) = ∫ a(t)dt and x(t) = ∫ v(t)dt. 18. Average y-value The average value of f(x) on [a,b] is . Summary of Convergence Tests for Series Test Series Convergence or Divergence Comments n'? term test (or the zero test) Da, Diverges if lim a, £0 Tnconclusive if lim a, = 0. Geometric series n=0 nal a j Converges to 7 only if [a] <1 Diverges if Jar] > 1 Useful for comparison th tests if the n' term a, of a series is similar to ax”. p-series Converges if p > 1 Diverges if p <1 Useful for comparison tests if the n!* term a, of . 1 a sories is similar to — Ww Integral Lew (20) a,, = f(n) for all n Converges if [ f(x) dx converges Diverges it [ f(a) der diverges The finction f obtained from ay = f(r) must be continuous, positive, decreasing and readily integrable for x > c. Comparison SY ay and Soy with 0 <a, < by for all n Yi by converges => JT a, converges Y- an diverges => Sb, diverges The comparison series YT by is often a geometric series or a p-series, Comparison* Ya and So. with dn,ba > 0 for all n and lim = =L>0 noc On S7b, converges > S* a, converges Y da diverges = Yay, diverges The comparison series bn is often a geometric series or a p-series. To find b, consider only the terms of a, that have the greatest effect on the magnitude. Converges (absolutely) if L <1 Inconclusive if L = Qn. ., “ . Ratio San with lim {ental Useful if a,, involves mvco a Diverges if L > 1 or if L is infinite factorials or nt" powers Converges (absolutely) if L <1 Test is inconclusive if L = 1 Root* Yan with lim Yan] = L Useful if a,, involves mee Diverges if L > 1 or if L is infinite ni* powers. ‘Absolute Value Useful for series containing laa Yan SY |an| converges + > an converges both positive and negative terms, Alternating series yen" a, net (a, > 0) Converges if 0 <any1 < an for all n 0 and lim a, Applicable only to serie with alternating terms. iy Sequence and Series Summary Formulas Ifa sequence {ay} has a limit L. that is, lim ay, = L, then the sequence is said to Noo converge to L. If there is no limit, the series diverges. If the sequence {ay} converges, then its limit is unique. Keep in nund that i | Ft Inn . } . x lim —=0; Jim x’ st lim Yn=1 lim ——=0. These limits io Noo noo noo 7! are useful and arise frequently. = = a The harmonic series > — diverges: the geometric series > a! converges to n=" n=0 7 if |r| <1 and diverges if Ir| =landa#0 oo 1 op . The p-series > —p converges if p> anddivergesif pS. n=l] 7 oo 00 Limit Comparison Test; Let dy and Son ‘be a series of nonnegative terms, with n=l n=l = On Gy, # 0 for all sufficiently large and suppose that lim ——=c > 0. Then the two H—-300 ay series either both converge or both diverge oo Alternating Series: Let > Gy, be a series such that n=l i) the series is altemating ii) | any dp | for all 1, and iii) lim ay, =0 ni-eo Then the series converges. A series Qj, is absolutely convergent if the series ty | converges. If a, n A n converges, but > la n | does not converge, then the series is conditionally convergent. Keep oa 00 im mind that if 7 |ayj| converges, then ST ay converges n=l n=l Indeterminate Form: 5 “ = Apply L’Hopital Directly Cc . a 8 © 0-0 => Rewrite as either — or — O ie) Then apply L’Hopital 1”, 0°,00° => 1. Consider the limit of the In of the function. 2. Use laws of logs to rewrite in the form 0-2. 3. Rewrite as either 9 or -. ce 4. Apply L’Hopital. 5. Exponentiate your answer. co-c => Try to rewrite so that you can use one of the previous forms. To convert polar coordinates into rectangular coordinates, we use the basic relations x =prcos 0, y=rsin 6 Converting in the opposite direction we use re =x? + y*, tan 0 = y/x if x0 What does the graph look like? r=a = Circle r=0 => Line a+bsinO OR r=a+tbcos 0 b => Dimpled Limacon b => Limacon with an inner loop b => Cardiod 22a 3 a vi iil r=acosnd OR r=asin nd n even (n > 2) => Rose with 2n petals. n odd (n > 3) = Rose with n petals.