Semester One Final BC

Derivative Rules:

d/dx[cu] = cu’

d/dx[uv] = uv’ + vu’

d/dx[c] = 0

d/dx[x] = 1

d/dx[u +— v] = u’ +— v’

d/dx[u/v] = vu’— uv’/v^2

d/dx[u^n] = nu^(n—1) u’

d’dx[|u|] = (u/|u|)(u’)

d/dx[ln u] = u’/u

d/dx[log_a u] = u’/(ln a)u

d/dx[sinu] = (cosu)u’

d/dx[tanu] = (sec^2u)u’

d/dx[secu] = (secu)(tanu)u’

d/dx[e^u] = (e^u)u’

d/dx[a^u] = a^u(ln a)u’

d/dx[cosu] = (—sinu)u’

d/dx[cotu] = —(csc^2u)u’

d/dx[cscu] = —(cscu)(cotu)u’

d/dx[arcsin u] = u’/squareroot(1 — u^2)

d/dx[arccos u] = —u’/squareroot(1–u^2)

d/dx[arctan u] = u’/(1 + u^2)

d/dx[arccot u] = —u’/(1 + u^2)

d/dx[arcsec u] = u’/|u| squareroot(u^2 — 1)

d/dx[arccsc u] = —u’/|u| squareroot(u^2 — 1)

product rule: d/dx[f(x)g(x)] = f’(x)g(x) + f(x)g’(x)

quotient rule: d/dx[f(x)/g(x)] = (f’(x)g(x) — f(x)g’(x))/(g(x))^2

chain rule: d/dx[f(g(x))] = f’(g(x)) * (g’(x))

Integration Rules:

integral(k*f(u)du) = k * integral(f(u)du)

integral(du) = u + c

integral(e^(u)du) = e^u + c

integral(cosu du) = sinu + c

integral(cotu du) = ln|sinu| + c

integral(cscu du) = -ln|cscu + cotu| + c

integral(csc²u du) = -cotu + c

integral(cscucotu du) = -cscu + c

integral(du/(a² + u²)) = 1/a * arctan(u/a) + c

integral([f(u) +- g(u)]) = integral(f(u) du) +- integral(g(u) du)

integral(a^u du) = (1/lna)*a^u + c

integral(sinu du) = -cosu + c

integral(tanu du) = -ln|cosu| + c

integral(secu du) = ln|secu + tanu| + c

integral(sec²u du) = tanu + c

integral(secutanu du) = secu + c

integral(du/sqrt(a² - u²)) = arcsin(u/a) + c

integral(du/u*sqrt(u² - a²)) = 1/a*arcsec(|u|/a) + c

Inverse formula: f^(-1)(x) = 1/f’(g(x))

Maclaurin Series:

e^x = 1 + x + x²/2! + x³/3! + … + x^n/n! + …

cosx = 1 - x²/2! + x^4/4! - x^6/6! + … + ((-1)^n * x^(2n))/(2n)! + …

sinx = x - x³/3! + x^5/5! - x^7/7! + … + ((-1)^n * x^(2n+1))/(2n+1)! + …

1/1-x = 1 + x + x² + x³ + … + x^n + …

1/1+x = 1/1-(-x) = 1 - x + x² - x³ + … + (-x)^n + …

ln(1+x) = x - x²/2 + x³/3 - x^4/4 + … + ((-1)^(n-1) * x^n)/n + …

arctanx = x - x³/3 + x^5/5 - x^7/7 + … + ((-1)^n * x^(2n+1))/(2n+1) + …

Alternating series error bound: error <= |next term|

Lagrange error bound: Lagrange error bound <= |(M*(a-c)^(n+1))/(n+1)!| ; where M = maximum value of either f^(n+1)(C) or f^(n+1)(a); c is value at which taylor polynomial is centered; a is the value you are approximating;

Mean Value Theorem: you set the fcn derivative equal to the slope of the endpoints given and solve (for point(s) guaranteed by MVT)