# Calculus Formulas Flashcards

Terms Definitions
 ∫cscx -ln|cscx+cotx| ∫cotx ln|sinx|+C position x(t) 1-tanh^2(x) sech^2(x) d/dx tanx sec^2x d/dx cscx -cscxcotx Marginal Cost C'(q) d/dx(tanh x) sech^2(x) d/dx(csc^-1 x) -1/[x*√(x^2-1)] d/dx(csch x) -csch(x)*coth(x) d/dx(sech x) -sech(x)tanh(x) volume of sphere 4/3πr^3 lim (x->0) cosx-1/x 0 lim (x->0) sinx/x 1 d/dx(cosh x) sinh x sinh^-1(x) ln[x+√(x^2 + 1)] surface area of sphere 4πr^3 d/dx logau 1/u lna du/dx d/dx a^u a^u (lna) du/dx optimization minimum or maximum problem d/dx(cosh^-1 x) 1/√(x^2 - 1) d/dx(csch^-1 x) 1/[abs(x)*√(x^2 + 1)] d/dx(coth^-1 x) 1/(1 - x^2) d/dx sinh x cosh x ∫udv aka integration by parts uv-∫vdu ∫[1/(x^2 + a^2)dx] (1/a)*tan^-1(x/a) + C critical point local minimum or maximum Volume of shell 2π (radius) (altitude) (thickness) sinh x e^x - e^-x / 2 sinh x e^x - e^-x / 2 derivative lim (h->0) f(x1+h) - f(x1) / h sigma notation definite integralnumber at the top tells the number of intervalsnumber at the bottom tells where to begin Profit Revenue - cost usually written as pi continuity terms 1. f(c) exists 2. lim (x->c) f(x) exists 3. lim (x->c) f(x) = f(c) L'Hopital's Rule if the derivative is in intederminate form then use which is 0/0 and infinity/infinitylim x->a f(x)/g(x) = lim x->a f'(x)/g'(x) L'Hopital's Rule if the derivative is in intederminate form then use which is 0/0 and infinity/infinity lim x->a f(x)/g(x) = lim x->a f'(x)/g'(x) point discontinuity curve has a "hole" (limit as x approaches a ≠ f at a) simpson's rule b-a/3n [f(xo) + 4f(x1) + 2f(x2) + 4f(x3) .... f(xn)] global max and min overall maximum and minimum highest and lowest y values substitute the cp into the f(x) set equal to zero to find y location on the graph to see which is the highest Mean Value theorem f'(c) = f(b) - f(a) / b-a The distance between two points of a line can be found using the ? Distance formula The distance between two points of a line can be found using the ? d = square root of ( X2 - X1 )2 + ( Y2 -Y1)2 average value 1/b-a integral from b to a f(x) dx x is what on the graph x is right or left horizontal hyperbolic functions to know cosh(0) = 1 sinh(0) = 0 cosh(-x) = cosh x sinh (-x) = -sinh x cosh^2 x - sinh^2 x = 1 definite integral F(b) - F(a) = integral from b to a F'(t) dt definite integral F(b) - F(a) = integral from b to a F'(t) dt y is what on the graph y is vertical or up and down slope of a line is positive if line is increasing from left to right acceleration x''(t) ∫du/u ln|u|+C velocity x'(t) d/dx(a^x) a^x*ln(a) d/dx sinx cosx Marginal Revenue R'(q) d/dx(sin^-1 x) 1/√(1-x^2) d/dx(sec^-1 x) 1/[x*√(x^2-1)] cosh(-x) cosh x Marginal Revenue R'(q) d/dx e^u e^u du/dx Volume of cylinder πr^2h Revenue price x quantity sinh(x+y) sinh(x)*cosh(y) + cosh(x)*sinh(y) ∫(a^x)dx a^x/ln(a) + C surface area of cyclinder 2πrh d/dx cosh x sinh x d/dx(log a X) 1/[x ln(a)] d/dx(sech^-1 x) 1/[x*√(1 - x^2)] d/dx(sinh^-1 x) 1/√(x^2 + 1) d/dx cosh x sinh x semicircle y = root(r^2 - x^2) antiderivative working backwards from the derivative antiderivative working backwards from the derivative Volume of disk π (radius) ^2 (thickness) cosh x e^x + e^-x / 2 removable discontinuity rational expression w/ common factors that cancel out maximum profit Marginal Cost = Marginal revenueR'(q) = c'(q)on graph where space between is widest maximum profit Marginal Cost = Marginal revenue R'(q) = c'(q) on graph where space between is widest Modeling Optimization problems need two equationsthe quantity needed to be optimized is the thing you want the derivative of sketchesfind the cp and ep to evaluate global max and min Modeling Optimization problems need two equations the quantity needed to be optimized is the thing you want the derivative of sketches find the cp and ep to evaluate global max and min Second fundamental theorem dF/dx=d/dx ∫[a to x] f(t)dt = f(x) delta t b-a/n n is the number of intervals and b and a is the interval global max and min overall maximum and minimum highest and lowest y values substitute the cp into the f(x) set equal to zero to find y location on the graph to see which is the highest A function is a relation in which each element of the domain (x value - independent variable ) is paired with only one element of the range ( y value - dependent variable ) A function is A relation can be tested to see if it is a function by the vertical line test. Draw a vertical line through any graph, and if it hits an x-value more than once, it is not a function. Mean value theorem for integrals f(c) = 1/b-a ∫[a to b] f(x)dx Extreme Value theorem if f is continuous then must have a global max and global min on the closed interval left and right increments in a chart begin with the first value but don't take the last for the right hand value begin with the second value and end with the last value for the left hand sum average the two to find the most acurate measurement of the integral left and right increments in a chart begin with the first value but don't take the last for the right hand value begin with the second value and end with the last value for the left hand sum average the two to find the most acurate measurement of the integral Linear functions take the form of f(x) = mx + b or y = mx + b Linear functions take the form of Where m = the slope b = the y-intercept so f(x) = 4x - 1 the slope is 4/1 (rise over run) and the y-intercept is -1
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