Calculus BC

• lim as x aprroaches a of f(x)= 0/0 or ∞/∞, then


  limit as x⟶a f(x)/g(x) = limit as x⟶a f'(x)/g'(x)

  (l'Hopital's Rule)
• dy/dx of a parametric equation is


  (dy/dt)/(dx/dt)
• Integration by Parts


  ∫udv= uv-∫vdu
• When using Euler's method, to find the new y


  old y + dy/dx • ∆x
• Area of a polar curve


  ∫(1/2)• r^2 dΘ
• slope of a polar curve


  derivative of (r(Θ)•sinΘ)/(r(Θ)•cosΘ)
• sin^2(x) + cos^2(x)=


  1
• Maclaurin series for e^x

  1 + x + (x^2)/2! + (x^3)/3! + ...
• Maclaurin series for cosx


  1 - (x^2)/2! + (x^4)/4! -...
• Maclaurin series for sinx


  x - (x^3)/3! + (x^5)/5! - ...
• Maclaurin series for 1/(1-x)


  1 + x + x^2 + x^3 + ...
• What do you do when using Ratio Test?

  replace n with n+1 in the series, then multiply that by 1/original series. Simplify. To find the radius of convergence : final answer<1

  -If you make final answer(L) >1 this tells you where the series diverges

  -If you make the final answer(L)=1 you can gather nothing from this

  -always check endpoints to figure out if it is < or ≤
• Taylor series format


  f(x)= f(a) + f'(a)(x-a) + (f"(a)(x-a)^2)/2!+...+(f^n(a)(x-a)^n)/n!
• How to find the Alternating Series Error Bound


  The error bound is the next number in the series. For example if the second degree of the series is 1-x^2/2!, the error would be the following number, which is x^4/4!
• Lagrange Error Bound


  |f(x)-Pn(x)|≤ M/(n+1!) • |x-c|^n+1
• What conditions must be met to have a max or min?


  dy/dx =0 at that point and also dy/dx must change sign at that point. Remember to also check endpoints!
• What conditions must be met to have a point of inflection?


  d2y/dx2=0 at that point and d2y/dx2 must change sign at that point.
• When a question asks for the maximum value of a function what do they want?


  The y value, not the x value!
• derivative of x^n


  nx^n-1
• derivative of sinx


  cosx
• derivative of cosx


  -sinx
• derivative of tanx


  (secx)^2
• derivative of cotx


  -(cscx)^2
• derivative of sec x


  secxtanx
• derivative of cscx


  -cscxcotx
• derivative of lnu


  (1/u) • du/dx
• derivative of e^u


  (e^u) • du/dx
• derivative of inverse sin u


  (1/√(1-x^2)) • derivative of u
• derivative of inverse cos u


  (-1/√(1-x^2)) • derivative of u
• derivative if inverse tan u


  (1/(1+x^2)) • derivative of u
• derivative of inverse cot u


  (-1/(1+x^2)) • derivative of u
• derivative of inverse sec u


  (1/|x|•√(x^2-1)) • derivative of u
• derivative of inverse csc u


  (-1/|x|•√(x^2-1)) • derivative of u
• derivative if a^x


  a^x • lna
• derivative of loga x


  (1/x)lna
• Formula for radius of convergence for a geometric series


  a/(1-r) <1

  -remember you never need to check endpoints for geometric series
• lnx^x can become


  xlnx
• lnx + ln3 can become

  ln3x
• ln3-lnx can become


  ln3/x
• derivative of f(u)


  f'(u) • du/dx

  ALWAYS REMEMBER THE CHAIN PIECE