AP Calculus Cheat Sheet for 2020 from, Lecture notes of Calculus

Download AP Calculus Cheat Sheet for 2020 from and more Lecture notes Calculus in PDF only on Docsity! AP Calculus Cheat Sheet for 2020 from https://fiveable.me/ap-calculus-ab-bc/ap- calc-study-guide-2020/ and Unit Circle and trig reference sheet The link about will take you to more practice and review resources, but just be careful about sharing any personal information with them, since they haven’t been vetted by NPCSD. This is a good comprehensive list of formulas and theorems (plus a couple extra things not in the AP curriculum!). Remember: You CAN AND SHOULD use notes while taking the AP exams. The exams are designed with that in mind. So have your unit circle, your derivative rules, your trig identities, and anything else you need handy. But also remember that time will be an issue. You don’t want to waste time scanning notes when you don’t need to! And also remember: The best way to prepare for the exam is to do as many free response problems as possible! Do all the problems in my lessons!!! Limit definition of a derivative: /"(x) = lim te if f(x) fs ho continuous ‘ oe ee _— ae faetia) Numerical definition of a derivative: f(x) = lim — Differentiabili les: © fixjis continuous ata lim f(x) = lim f(x) ea a Properties of derivatives: © Power rule: gx = nyt! © Product rule: + fixjg(x) = fg(x) + fidg"(X) d (2) = = Lig) — fare") six) [ea Remember: If x is a function (ex. x’, 2x + 5, 3x), use the Chain Rule when taking the derivative. ° tien: function rivatives: o £cosx = —sinx ° ie. SINX = COSX ° 4 fanx = sec?x ° 4 secx = secx* tanx ° 4 esex = —csex + cotx 0 Leon = —ese?x ° £ a’ =a"(ina) ° = eae ° & inx = i ° £ log,x = aS  5: Analytical ® Optimization: Use these handy tests to find the extrema of a function! = First derivative test: "Ow = i i i i is Differentiation . If f'() = 0 or is undefined and f is continuous at x. © and f’(x) changes from + to - > relative maximum at x © and f’(x) changes from + to - > relative minimum at x © and and /‘(x) has no change > neither at x *® Finding concavity: o If f(s) > 0 > concave up at x © If (x) < 0 > concave down at x e Finding inflection points: ° If £"(@) = 0 or is undefined and /” (x) changes signs > inflection point at x ° ni ivati o If f’() — 0 and f(x) > 0 + relative maximum at x o Sx) = 0 and f" (x) < 0 > relative minimum at x Mean Value Theorem (MVT): © If / is continuous on closed interval [a, b], and is differentiable on (a, b), there is a point c such that f’(c) = fuia bora © Rolle’s Theorem: Same as MVT, but just the specific case when f'(e) =0= fib) ~ fia b-a@ « Extreme Value Theorem: © If £ is continuous on closed interval [a, b], then / has both an absolute maximum and an absolute minimum  6: Integration and Accumulation of Change ® Reimann sums: These estimate the area under a curve by dividing it into rectangles (subdivisions) and adding their areas Right Reimann sum: For each subdivision, the height is the right point’s y-coordinate m If the curve increases, It overestimates = \f the curve decreases, it underestimates © Left Reimann sum: For each subdivision, the height is the left point’s y-coordinate m If the curve increases, it underestimates = |f the curve decreases, it overestimates ° Midpoint Reimman sum: For each subdivision, the height is the y-coordinate of the midpoint of the two x-coordinates © Trapezoidal Reimann sum: Most accurate of the types. Use the h(8,+ 85) i: ) a h=distance between x values a 5b,=y-coordinate of right point a b,=y-coordinate of left point e Fundamental Theorem of Calculus (FTC): b area of a trapezoid (Ad = | fix) = F(b) — F(a), where F is the antiderivative of a o Partt J = If the limits are functions, use Chain Rule: ko I fix) = FRx) k(x) — F(a) hi'(xy idx) o pata: 4] fin) dt = fix) = Corollary: If the limits are functions, use Chain Rule: A(x) £ oS fix) de = fUex)] - k) — fT] + hx) dix) © Beverse power rule: |x” dy = aT +C  ¢ Indefinite integrals: | f(x) dv =F(x) + C , where F is the antiderivative of / (do not forget + C !) e Integral properties and techniques: b © U-Substitution: If you have | f(g(x))- g'(x) dy, let = g(x) and a b du = g(x) dv. Substitute and solve: | f(w) du. Don’t forget to a replace any w's with g(x)! b b b 0 Sum: I[f(x) + g(x)| dv = I fixydy + | g(x) de b b b ° Difference: |[/(x) — g(x)] dv = | fix) de — | g(x) de b b 0 Multiplying a constant: J a- f(x) dv = aJ f(x) dr a