Terms : Hide Images
| 772336699 | (f*g)' | f'g+g'f | 0 | |
| 772336700 | (f/g)' | (f'g-g'f)/g² | 1 | |
| 772336701 | (f(g))' | f'(g)*g' | 2 | |
| 772336702 | (sin(x))' | cos(x) | 3 | |
| 772336703 | (cos(x))' | -sin(x) | 4 | |
| 772336704 | (tan(x))' | sec(x)² | 5 | |
| 772336705 | (cot(x))' | -csc(x)² | 6 | |
| 772336706 | (sec(x))' | sec(x)tan(x) | 7 | |
| 772336707 | (csc(x))' | -csc(x)cot(x) | 8 | |
| 772336708 | (e^x)' | e^x | 9 | |
| 772336709 | (a^x)' | a^x*ln(a) | 10 | |
| 772336710 | (ln(x))' | 1/x | 11 | |
| 772336711 | (arcsin(x) or arccos(x))' | ±dx/√(1-x²) | 12 | |
| 772336712 | (arctan(x) or arccot(x))' | ±dx/(1+x²) | 13 | |
| 772336713 | (arcsec(x) or arccsc(x))' | ±dx/(x²-1) | 14 | |
| 772336716 | ∫x⁻¹dx | ln(x) | 15 | |
| 772336717 | ∫e^xdx | e^x+C | 16 | |
| 772336718 | ∫a^xdx | a^x/ln(a)+C | 17 | |
| 772336719 | ∫ln(x)dx | xln(x)-x+C | 18 | |
| 772336720 | ∫sin(x)dx | -cos(x)+C | 19 | |
| 772336721 | ∫cos(x)dx | sin(x)+C | 20 | |
| 772336722 | ∫tan(x)dx | ln|sec(x)|+C | 21 | |
| 772336723 | ∫cot(x)dx | ln|sin(x)|+C | 22 | |
| 772336724 | ∫sec(x)dx | ln|sec(x)+tan(x)|+C | 23 | |
| 772336725 | ∫csc(x)dx | ln|csc(x)-cot(x)|+C | 24 | |
| 772336726 | ∫sec²(x)dx | tan(x)+C | 25 | |
| 772336727 | ∫csc²(x)+C | -cot(x)+C | 26 | |
| 772336728 | ∫sec(x)tan(x)dx | sec(x)+C | 27 | |
| 772336729 | ∫csc(x)cot(x) | -csc(x)+C | 28 | |
| 772376073 | ∫dx/(a²+x²) | a⁻¹arctan(x/a)+C | 29 | |
| 772376074 | ∫dx/√(a²-x²) | arcsin(x/a)+C | 30 | |
| 772376075 | ∫dx/√(x²-a²) | a⁻¹arcsec(x/a)+C | 31 | |
| 772376076 | a function f(x) is continuous at x=a if | 1. f(a) exists 2. lim x→a f(x) exists 3. lim x→a f(x) = f(a) | 32 | |
| 772376077 | intermediate value theorem | a function f(x) that is continuous on a closed interval [a,b] takes on every value between f(a) and f(b) | 33 | |
| 772397866 | average rate of change | if (x₀, y₀) and (x₁, y₁) are points on the graph of y=f(x), then the average rate of change of y with respect to x over the interval [x₀, x₁] is (f(x₁)-f(x₀))/(x₁-x₀) | 34 | |
| 772397867 | definition of the derivative | f'(x) = lim h→0 (f(x+h)-f(x))/h f'(a) = lim x→a (f(x)-f(a))/(x−a) | 35 | |
| 772397868 | lim x→+∞ (1+x/n)^n | e^x | 36 | |
| 772397869 | lim x→0 (1+n)^(1/n) | e | 37 | |
| 772421578 | rolle's theorem | if f(x) is continuous on [a,b] and differential on (a,b) and f(a)=f(b), then there is at least one number c in the open interval (a,b) such that f'(c)=0 | 38 | |
| 772421579 | mean value theorem | if f(x) is continuous on [a,b] and differential on (a,b), then there is at least one number c in (a,b) such that f'(c)=(f(b)-f(a))/(b-a) | 39 | |
| 772421580 | extreme value theorem | if f is continuous on a closed interval [a,b] , then f(x) has both an absolute maximum and an absolute minimum on (a,b) | 40 | |
| 772513372 | 1st fundamental theorem of calculus | from a to b ∫f(x)dx = F(b)-F(a), where F'(x)=f(x) | 41 | |
| 772552563 | 2nd fundamental theorem of calculus | if F(x)=from x to 0 ∫f(t)dt, then F'(x)=f(x) if F(x)=from u(x) to 0 ∫f(t)dt, then F'(x)=f(u(x))*u'(x) | 42 | |
| 772552564 | dP/dt (logistics) | kP(1-P/L) L is carrying capacity; k is constant of proportionality | 43 | |
| 772552565 | P (logistics) | L/(1+A^(-kt)) A=(L-P₀)/P₀ | 44 | |
| 772589901 | washer method | from a to b π∫(r₁²-r₂²)dx | 45 | |
| 772589902 | disc method | from a to b π∫r²dx around x axis from c to d π∫r²dy around y axis | 46 | |
| 772589903 | shell method | from a to b 2π∫p(y)h(y)dy around x axis from c to d 2π∫p(x)h(x)dx around y axis | 47 | |
| 772589904 | maclaurin polynomial of e^x | 1+x+x²/2!+x³/3!+x⁴/4!+...x^n/n! | 48 | |
| 772589905 | maclaurin polynomial of sinx | x−x³/3!+x⁵/5!+...(-1)^n*x^(2n+1)/(2n+1)! | 49 | |
| 772589906 | maclaurin polynomial of cosx | 1-x²/2!+x⁴/4!+...(-1)^n*x^(2n)/(2n)! | 50 | |
| 772589907 | maclaurin polynomial of (1-x)⁻¹ | 1+x+x²+x³+x⁴+...x^n | 51 | |
| 772589908 | maclaurin polynomial of lnx | (x-1)−(x-1)²/2+(x-1)³/3−(x-1)⁴/4+...(x-1)^n/n | 52 | |
| 772595110 | nth term test | lim x→∞f(x)≠0, the series diverges | 53 | |
| 772614773 | lim n→∞ ln(n)/n | 0 | 54 | |
| 772614774 | lim n→∞ x^(1/n) | 1 | 55 | |
| 772614775 | lim n→∞ n^(1/n) | 1 | 56 | |
| 772614776 | lim n→∞ x^n; |x|<0 | 0 | 57 | |
| 772614777 | lim n→∞ x^n/n! | 0 | 58 | |
| 772622316 | sin2x | 2sinxcosx | 59 | |
| 772622317 | cos2x | cos²x-sin²x=2cos²x-1=1-2sin²x | 60 | |
| 772622318 | tan2x | 2tanx/(1-tan²x) | 61 | |
| 773461205 | if g(x) is the inverse of f(x), then | g'(x)=1/(f'(g(x))) | 62 | |
| 773461206 | average value of f(x) on [a,b] | 1/(b-a)∫ (from a to b) f(x) dx | 63 | |
| 773461207 | integration by parts formula | ∫udv=uv - ∫vdu | 64 | |
| 773461208 | LIATE | in order: logarithmic function, inverse trig function, algebraic function, trig function, exponential function | 65 | |
| 773461209 | euler's method | y₁ = y₀ + h*f'(x₀, y₀) informally: ynew= yold + step size*derivative at the previous point | 66 | |
| 773461210 | arc length | s = ∫ (from a to b) √(1+[f'(x)]²) dx | 67 | |
| 773461211 | arc length in parametric form | s = ∫ (from a to b) √[(dx/dt)² + (dy/dt)²] dt | 68 | |
| 773482019 | sin²x | (1-cosx)/2 | 69 | |
| 773482020 | cos²x | (1+cosx)/2 | 70 | |
| 773590388 | newton's law of cooling | dy/dt=k(y−y₀) | 71 | |
| 773590389 | work | ∫ (from a to b) F*d*dx | 72 |
