AP Calc BC Midterm Review Creative Commons License

Alex Beutel AP Calc BC Midterm Review Log Rules y=logax is same as ay=x logaxy = logax + logay logaxy = ylogax logax = lnx/lna log10 = 1 lne = 1 ln1 = 0 you not have ln of a nonpositive number ln of a number between 0 and 1 is negative Trig Angle 30 45 60 90 (x2) = |x| sin2 sin2 cos2 cos2 cos2 + cos2 = 1 = 2sin cos = 2cos2 – 1 = 1-2sin2 = cos2 – sin2 Sin 1/2 2/2 3/2 1 Cos 3/2 2/2 ½ 0 Tan 1/ 3 1 3 Horizontal Asymptote at number as x goes to Definition = A limit will exist iff lim(x  x ­ )f(x) = lim(x  x +­ )f(x) Top degree = bottom degree then just coefficients  lim(x  ) (2x2)/(3x2) = 2/3 If top degree is greater than bottom degree then goes to infiniti If bottom degree is greater than top degree then goes to 0 (horizontal asymptote) Vertical asymptote – in a simplified problem the roots of the bottom or zeros one of the horizontal asymptotes ?????????? Definition of Continuity: a point C is continuous iff (not ): f(c) exists and lim(x  c ­ )f(c) = lim(x  c +­ )f(c) INTERMEDIATE VALUE THEOREM: A function y=f(x) that is continuous on a closed interval [a,b] takes on every value between f(a) and f(b). In other words, if yo is between f(a) and f(b) then yo=f(c) for some c in [a,b] y = x(a/b) do b first then a A point c is considered differentialbe iff point c has a finite slope (not ) Differentiability implies continuity 4 Cases of Non-Differentiable: discontinuity corner (multirule) vertical tangent (inflection point) cusp f’(x) = lim(h 0) f(x+h) - f(x) /h Average Slope (Average Rate of Change) = f(b) – f(a)/b-a Power Rule: Definition y= cxn Product Rule: y=fg y’=fg’ + gf’ Quotient Rule y=f/g y’= (gf’ – fg’)/g2 y’=cnxn-1 d(sinx)/dx = cosx d(cosx)/dx = -sinx d(tanx)/dx = sec2x d(cscx)/dx = -cscx · cotx d(secx)/dx = secx · tanx d(cotx)dx = -csc2x s’’(t) = v’(t) = a(t) speed = |v(t)| If velocity is negative then going in reverse v(t) = 0 standing stll If velocity and acceleration have same sign then particle is speeding up. If velocity and acceleration have different signs then particle is slowing down. Implicit Differentiate and then isolate dy/dx d(au)/dx = au · ln|a| · du/dx deu/dx = eu · du/dx d(lnu)/dx = 1/u · du/dx Critical Points: x-value in the interior domain where f’(x)=0 or DNE NEVER ENDPOINTS Not always max/min Mean Value Theorem: requirement: must be closed and differentiable everywhere f(b) – f(a) /(b-a) = f’(c) Extreme Value Theorem: If a function is closed and continuous on [a,b] there must be an absolute max or an absolute min. *an extrema if it exists must occur at end points or critical points * max value means the absolute max y-value y=f(x) is concave up if y’ is increasing (y’’(x) = (+)) y=f(x) is concave down if y’ decreaseing (y’’(x) = (-)) y’’(x) = 0 visa versa INTEGRALS possibly inflection point ONLY if switches from positive to negative or Area is NEVER negative DERIVATIVE OF ARC[TRIG] y=sin-1u y’=1/( (1-u2)) du/dx y=cos-1u y’= -1/( (1-u2)) du/dx y=tan-1u y’= 1/(1+u2) du/dx Average Value: of f(x) on [a,b] = 1/(b-a) b a f(x) dx If f is continuous on [a,b] there will always be a C such that: f(c) = 1/(b-a) ab f(x) dx Trapezoidal Rule A= h/2 (f(x1) + 2f(x2) + 2f(x3) + f(x4)) h is dt INTEGRATION BY PARTS u dv = uv - vdu EXAMPLE xcosx dx u=x du=dx =xsinx - sinx dx =xsinx + cosx + c dv= cosx v= sinx FOR U: LIPET L = ln I = inv. trig P = poly E = exponential T = trig L’Hopital’s Rules *** 0/0 or / are called indeterminant IF f(a) = g(a)=0 or Then lim (x a) f(x)/g(x) = lim(x a) f’(x)/g’(x) Euler’s Method Step size = h f(x1) = f(x0) + hf’(x0) dy /dt = ky y=y0ekt y0 = initial value Population Growth dy/dt = ky(1- y/L) L is carrying capacity lim(t ) y(t) = L Genearly unaffected by initial population Population is growing fastest when it is half its carrying capacity 1 /(b-a) a b f(x) dx  x ­ )f(x) = lim(x  x ­ )  x ­ )f(x) x +­ )f(x)  lim(x  lim(x  lim(x  · ±


About this Document

This is a review from Mr. Lee's AP Calc BC class. This covers half of Calc BC and therefore about all of AP Calc AB. This goes from basic idea of slope, continuity, non-differentiable, through how to differentiate, integrals, l'hopital's rule, euler's method, etc.


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