Calculus Cheat Sheet All Reduced

Calculus Cheat Sheet Limits Definitions Precise Definition: We say lim x -> a f(x) = L for every epsilon > 0 there is a delta > 0 such that whenever 0 < |x - a| < delta| then |f(x) - L| < epsilon_{x} "Working" Definition: We say lim x -> a f(x) = Lif we can make (2) as close to as we want by taking sufficiently close to (on either side of) without letting Right hand limit: lim t -> a f(x) = L This has the same definition as requires Limit at Infinity: We if we can make as close to L. as we want by taking large enough and positive. lim x -> ∞ f(x) = L There is a similar definition for lim x -> - ∞ f(x) = L except we require large and Infinite Limit: We say lim x -> ∞ f(x) = ∞ * f(x) we can arbitrarily by taking a sufficiently close to (on either side of a) without letting. Left hand limit: lim x -> a f(x) = L_{x} This has the same definition as the requires. There is a similar definition for lim x -> ∞ f(x) = - ∞ except we make (2) arbitrarily . Relationship between the limit and one-sided limits Continuous Functions Calculus Cheat Sheet Evaluation Techniques If f(x) overline v continuous at then lim x -> a f(x) = f(a) Continuous Functions and Composition continuous at & and lim x -> a g(x) = b then lim x -> 0 f(g(x)) = f(lim x -> ∞ g(x)) = f(b) Factor and Cancel Rationalize Numerator/Denominator L'Hospital's/L'Hôpital's Rule If or or lim x -> a (f(x))/(g(x)) = (plus/minus ∞)/(plus/minus ∞) , lim x a f(x) g(x) =lim a a (f' * (x))/(g' * (x)) , is a number, or-x lim x -> ∞ (f(x))/(g(x)) = 0/0 Polynomials at Infinity and are polynomials. To compute lim a -> 0 (p(x))/(q(x)) factor largest power of ing out of both and then compute limit. lim) Does Not Exist lim x -> ∞ f(x) = lim x -> ∞ f(x) =L Rightarrow lim x infty f(x)=L lim x a f(x)=L Rightarrow lim x -> a f(x) = lim x -> a f(x) = L lim x -> a f(x) ne lim x a^ - f(x) Properties Assume lim x -> ∞ f(x) and lim x -> w g(x) both exist and is any number then, 1. lim x -> a [cf(x)] = c * lim x -> a f(x) 4. provided lim x infty g(x) ne0 lim x -> a [(f(x))/(g(x))] = (lim x -> a f(x))/(lim x -> a g(x)) 2. lim epsilon -> a [f(x) plus/minus g * (x)] = lim x -> a f(x) plus/minus lim x -> a g(x) 6. lim x -> n [f(x)] ^ n = [lim x -> n f(x)] ^ n 3. lim x -> a [f(x) * g(x)] = lim x -> a f(x) * lim x -> a g(x) 6. lim x -> ∞ [root(f(x), 3)] = root(lim x -> ∞ f(x), 3) 1. lim x -> ∞ \mathfrak{e} ^ x = ∞ & lim x infty bullet^ x = 0 Basic Limit Evaluations at tx 5. even 2. lim x -> ∞ ln(x) = ∞ & lim x -> 0 ln(x) = - ∞ 3. Ifthen lim x -> ∞ h/(x ^ sigma) = 0 4. If and is real for negative then lim epsilon - infty b/(x ^ x) = 0 6. nodd: lim x -> ∞ x ^ n = ∞ & lim x -> - ∞ x ^ n = - ∞ 7. even lim e pm infty a.x^ n +***+bx+c= mathfrak sgn(a) * ∞ 8. odd lim c infty ax^ n +***+ bx + c = sgn(a) * ∞ 9.odd: lim x - infty ax^ n +***+cx+d=- mathfrak sgn(a) * ∞ Note: s mathfrak gn * (a) = 1 and sgn()=-1 if a<0. Combine Rational Expressions. Piecewise Function lim x -> - 2 g(x) where Compute two one sided limits, One sided limits are different so doesn't exist. If the two one sided limits equal then lim t -> - 1 g(x) ) would have existed and had the same lim x -> - 3 g(x) Some Continuous Functions Partial list of continuous functions and the values of for which they are continuous. 1. Polynomials for all. 2. Rational function, except for's that give division by zero. root(x, 3) n odd) for all. 4. n even) for all 5. e for all Suppose that continuous a number such that and and let M be any number between (4) and Then there exists 5. for 7. and for all . 8. and provided x ne..., - (3pi)/2 ,- pi 2 , pi 2 , 3 pi 2 , 9. and csc) provided x ne***, - 2pi ,- pi,0, pi,2 pi, Intermediate Value Theorem = lim h -> 0 1/h * ((- h)/(x(x + h))) = lim h -> 0 - 1/(x(x + h)) = - 1/(x ^ 2) = lim x -> 9 (9 - x)/((x ^ 2 - 81)(3 + sqrt(x))) = lim x -> 9 - 1/((x + 9)(3 + sqrt(x))) = - 1/((18)(6)) = - 1/108 lim x 2 g(x)=lim x -2^ - x ^ 2 + 5 = 9 lim x -2^ + g(x)=lim x -2^ + 1 - 3x = 7 lim 0 x ^ (∞) = ∞ =lim 8 2 (x + 6)/x = 8 2 =4 lim x -> ∞ (3x ^ 2 - 4)/(5x - 2x ^ 2) = lim x -> - ∞ (x ^ 2 * (3 - 4/(x ^ 2)))/(x ^ 2 * (1/x - 2)) = lim x -> - ∞ (3 - 4/(x ^ 2))/(1/x - 2) = - 3/2 lim h -> 0 1/h * (1/(x + h) - 1/x) = lim h -> 0 1/h * ((x - (x + h))/(x(x + h))) lim x -> 2 (x ^ 2 + 4x - 12)/(x ^ 2 - 2x) = lim x -> 2 ((x - 2)(x + 6))/(x(x - 2)) lim x -> 0 (3 - sqrt(x))/(x ^ 2 - 81) = lim x -> 0 (3 - sqrt(x))/(x ^ 2 - 81) * (3 + sqrt(x))/(3 + sqrt(x)) g(x) = x ^ 2 + 5 ffx < - 2; 1 - 3x fx >= - 2 Calculus Cheat Sheet Derivatives Definition and Notation If y = f(x) then the derivative is defined to be f' * (x) = lim h -> 0 (f(x + h) - f(x))/h If y = f(x) all of the following are equivalent notations for derivative evaluated at If y = f(x) then all of the following are equivalent notations for the derivative. if y = f(x) then, Interpretation of the Derivative 1. m = f' * (a) the slope of the tangent line y = f(x) at and the equation of the tangent line at given by y = f(a) + f' * (a)(x - a) f' * (a) is the instantaneous rate of change of f(x) 3. If f(t) is the position of an object at time then f' * (a) the velocity of the object at Basic Properties and Formulas If f(x) and g(x) are differentiable functions (the derivative exists), and are any real numbers, 4. (f(x) plus/minus g * (x))' = f' * (x) plus/minus g' * (x) 2. (ef(x))' = e * f' * (x) 5. (f(x) * g(x))' = f' * (x) * g(x) + f(x) * g' * (x) - Product Rule 3. d dx (x^ n )= n * x ^ (n - 1) - -Power Rule 6. Quotient Rule ((f(x))/(g(x))) ^ r = (f' * (x) * g(x) - f(x) * g' * (x))/((g(x)) ^ 2) 7. d/dx (f(g(x))) = f' * (g(x)) * g' * (x) Chain Rule Common Derivatives d d x ( mathfrak csc(x) )=- mathfrak csc(x) mathfrak cot (x) d/dx (a ^ x) = a ^ x * ln(a) d/dx (ln(x)) = 1/x, x > 0 d dx ( mathfrak soc(x))= mathfrak sec(x) * tan(x) (10g.())- Calculus Cheat Sheet Chain Rule Variants The chain rule applied to some specific functions. 1. d/dx ([f(x)] ^ n) = n * [f(x)] ^ (n - 1) * f' * (x) d/dx (\mathfrak{sigma} ^ (f(x))) = f' * (x) * \mathfrak{sigma} ^ (f(x)) 2 d/dx (ln[f(x)]) = (f' * (x))/(f(x)) 4. d dx ( mathfrak sin[f(x)])= f' * (x) mathfrak cos[f(x)] 5. d dx ( mathfrak cos[f(x)])= - f' * (x) * sin[f(x)] 6. d/dx (tan[f(x)]) =f^ prime (x) mathfrak sec ^ 2 [f(x)] 7. d dx ( mathfrak sec [f(x)])=f^ prime (x) mathfrak soc * [f(x)] * tan[f(x)] 8. d/dx (arctan[f(x)]) = (f' * (x))/(1 + [f(x)] ^ 2) Higher Order Derivatives The 2 ^ (nd) Derivative is denoted as The Derivative is denoted as and is defined as f ^ (n) * (x) = (f ^ (n - 1) * (x))' ie the derivative of the (n - 1) ^ (at) f ^ (n) * (x) = (d ^ n * f)/(d * x ^ n) * i f ^ (n - 1) * (x) t^ prime prime (x) = f ^ (2) * (x) = (d ^ 2 * f)/(d * x ^ 2) . and is defined as f^ prime prime (x) = (f' * (x))' , le the derivative of the first f' * (x) Implicit Differentiation y' bullet^ 2x - 8y + x ^ 3 * y ^ 2 = mathfrak sin(y)+11x. y = y(x) here, so products/quotients of and will use the product/quotient rule and derivatives of will use the chain rule. The "trick" is to differentiate as normal and every time you differentiate a you tack on a (from the chain rule). Then solve for Critical Points Increasing/Decreasing - Concave Up/Concave Down a critical point of provided either f(x) 1. f' * (c) = 0 or 2. f' * (c) doesn't exist. Increasing/Decreasing 1. If f' * (x) > 0' for all in an interval / then f(x) increasing on the interval I. 2. Iffor all in an interval / then f(x) decreasing on the interval. f(x) < 1 3. If f' * (x) = 0 for all in an interval / then f(x) constant on the interval 1. Concave Up/Concave Down 1. If f^ prime prime (x) > 0 for all in an interval / then is concave up on the interval J. I(x) 2. If f^ prime prime (x) < 0 for all in an interval / then f(x) is concave down on the interval 1. Inflection Points is a inflection point of f(x) if the concavity changes x = e_{i} d/dx (cot(x)) = - c * s * e ^ 2 * (x) d/dx (sigma ^ z) = sigma ^ z d d x ( mathfrak cos(x) )=*** mathfrak sin(x) d/dx (c) = 0 d d x ( mathfrak sin(x) )=cos(x) d/dx (tan(x)) = sec^2 (x) d/dx (arccos(x)) = - 1/(sqrt(1 - x ^ 2)) d/dx (arcsin(x)) = 1/(sqrt(1 - x ^ 2)) d/dx (ln|x|) = 1/x ,x ne0 f' * (x) = y' = d/dx (f) = d/dx (y) = d/dx (f(x)) = Df(x) d/dx (x) = 1 d/dx (arctan(x)) = 1/(1 + x ^ 2) 8 ^ (2x - 9y) * (2 - 9 * y') + 3x ^ 2 * y ^ 2 + 2x ^ 3 * y * y' = cos(y) * y' + 11 2e ^ (2x - 8y) - 9 * y' * e ^ (2x - 8y) + 3x ^ 2 * y ^ 2 + 2x ^ 3 * y * y' = mathfrak cos(y)y^ prime +11 (2x^ 3 y-9e^ 2x-0y - mathfrak cos(y)) y' = 11 - 2e ^ (2x + 0y) - 3x ^ 2 * y ^ 2 y' = (11 - 2theta ^ (2x - 8y) - 3x ^ 2 * y ^ 2)/(2x ^ 3 * y - 9a ^ (2x - 3y) - cos(y)) f^ prime (a)=y^ prime | x=a = df dx | x=a = dy dx | x = a = Df(a) Calculus Cheat Sheet Calculus Cheat Sheet Absolute Extrema 1.2 is an absolute maximum of if f(c) >= f(x) for all in the domain. 2. an absolute minimum of f(c) <= f(x) for all in the domain. Fermat's Theorem If has a relative (or local) extrema at then is a critical point of f(x) Extreme Value Theorem Extrema is continuous on the closed intervala. then there exist numbers and if so that, 2. f(c) is the absolute maximum in a.. 3. the absolute minimum in [a, b] Finding Absolute Extrema To find the absolute extrema of the continuous function on the interval [a, b] process use the following 1. Find all critical points of [a, b] 2. Evaluate at all points found in Step 1. 3. Evaluate f(a) and f(b) 4. Identify the absolute maximum (largest function value) and the absolute minimum (smallest function value) from the evaluations in Steps 2 & 3. Relative (local) Extrema 1. is a relative (or local) maximum of f(c) >= f(x) for all near a relative (or local) minimum of f(x) <= f(x) for all near c 1 ^ (st) Derivative Test If a critical point of x =c 1 1. a relative maximum of left of f' * (x) < 0! the right of to the f' * (x) > 0' 2. a relative minimum of if f' * (x) < 0 the left of f' * (x) > 0! the right of 3. not a relative extrema of if f' * (x) is the same sign on both sides of 2 ^ (nd) | Derivative Test x =c 1 a critical point of such that f' * (c) = 0 then 1. is a relative maximum of if f^ prime prime (c) < 0 . 2. is a relative minimum of f^ prime prime (c) > 0 . 3. may be a relative maximum, relative minimum, or neither if f^ prime prime (c) = 0 , Finding Relative Extrema and/or Classify Critical Points 1. Find all critical points of 2. Use the 1 derivative test or the 2 ^ (nd) derivative test on each critical point. Related Rates Sketch picture and identify known/unknown quantities. Write down equation relating quantities and differ-entiate with respect to using implicit differentiation (le. add on a derivative every time you differentiate a function of t). Plug in known quantities and solve for the unknown quantity. Example A 15 foot ladder is resting against a wall. The bottom is initially 10 ft away and is being pushed towards the wall at 1/4 /sec. How fast is the top moving after 12 sec? y 15ft is negative because is decreasing. Using Pythagorean Theorem and differentiating. x' Example Two people are 50 ft apart when one starts walking north. The angle changes at 0.01 rad/min. At what rate is the distance between them changing when theta = 0.5 rad? Moving Person 50 ft Stationary Person We have vartheta' = 0.01 rad/min. and want to find x' We can use various trig functions but easiest is, After 12 sec we have x = 10 - 12(1/4) = 7 and so y = sqrt(15 ^ 2 - 7 ^ 2) = sqrt(176) Plug in and solve for 7(- 1/4) + sqrt(176) * y' =0 Rightarrow y^ prime = 7 4 sqrt 176 fisec y ^ prime prime = theta = 0.5 theta ^ * sec( theta)= x 50 Rightarrow sec(theta) * tan(theta) * theta' = (x')/50 sec(0.5) tan(0.5) (0.01) Remember to have calculator in radians! x' = 0.3112f / m * in Optimization Sketch picture if needed, write down equation to be optimized and constraint. Solve constraint for one of the two variables and plug into first equation. Find critical points of equation in range of variables and verify that they are min/max as needed. Example We're enclosing a rectangular field with 500 ft of fence material and one side of the field is a building. Determine dimensions that will maximize the enclosed area Building Example Determine point(s) on y = x ^ 2 + 1 that are closest to (0,2).. () () and differentiable on the open interval (a,b) then there is a If is continuous on the closed interval [a, b] Mean Value Theorem number such that f' * (c) = (f(b) - f(a))/(b - a) a < c < b Newton's Method If is the n ^ (tk) guess for the root/solution of then (n + 1) ^ prime prime guess is f' * (x_{n}) exists. f(x) = 0 provided x n + 1 =x n - f(x n ) f^ prime (x n ) ; Maximize A = xy subject to constraintx x + 2y = 500 Solve constraint for and plug into area. Differentiate and find critical point(s). Minimize and the constraint is y = x ^ 2 + 1 Solve constraint for x ^ 2 : and r = d ^ 2 = (x + 0) ^ 2 + (y - 2) ^ 2 ) plug into the function. = y - 1 + (y - 2) ^ 2 = y ^ 2 - 3y + 3 r^ 2 =y-1 Rightarrow f = x ^ 2 + (y - 2) ^ 2 Differentiate and find critical point(s) f^ prime =2y-3 Rightarrow y = 1/2 By the 2 ^ (md) derivative test this is a relative minimum and so all we need to do is find value(s). By 2 derivative test this is a relative maximum and so is the answer we're after. Finally, find . The dimensions are then 250 * 125 The 2 points are then (1/(sqrt(2)), 3/2) and (- 1/(sqrt(2)), 1/2) x ^ 2 = 1/2 - 1 = 1 2 Rightarrow x= pm 1 sqrt 2 x = 500 - 2(125) = 250 A^ * =500-4y Rightarrow y = 125 x^ 2 +y^ 2 =15^ 2 Rightarrow 2x * x' + 2y * y' = 0 aligned x=500-2y Rightarrow A = y(500 - 2y) \\ =500y-2y^ 2 aligned a <= c, d <= b rfloor (0, 2) Calculus Cheat Sheet Integrals Definitions Definite Integral: Suppose is continuous on 6. Divide a, b into subintervals of width and choose from each interval. Then integrate f(x) dx from a to b =lim n infty sum i=1 ^ n f(x i ^ * ) Delta x. Anti-Derivative: An anti-derivative of is a function, such that Indefinite Integral: int c f(x)dx= F(x) + c where is an anti-derivative Fundamental Theorem of Calculus Part 1: If is continuous on, then g(x) = integrate f(t) dt from a to x also continuous and g' * (x) = d/dx (integrate f(t) dt from a to x) = f(x) Variants of Part I: d/dx (integrate f(t) dt from a to a(x)) = u' * (x) * f[u(x)] d/dx (integrate f(t) dt from v(x) to b) = - v' * (x) * f[v(x)] d/dx (integrate f(t) dt from v(x) to u(x)) = u' * (x) * f[u(x)] - v' * (x) * f[v(x)] Part continuous on an antiderivative -of ie. F(x)= integrate f(x) dx i then integrate f(x) dx from a to b = F(b) - F(a) Properties integrate ef(x) dx = c * integrate f(x) dx, c is a constant integrate cf(x) dx from a to b =c int a ^ b f(x)dx i e is a constant integrate c dx from a to b = c(b - a) e is a constant |integrate f(x) dx from a to b| <= integrate |f(x)| dx from a to b int f(x)=g(x)dx= integrate f(x) dx plus/minus integrate g(x) dx integrate f(x) plus/minus g * (x) dx from a to b = integrate f(x) dx from a to b plus/minus integrate g(x) dx from a to b integrate f(x) dx from a to a = 0 integrate f(x) dx from a to b = - integrate f(x) dx from b to a integrate f(x) dx from a to b = integrate f(x) dx from a to v + integrate f(x) dx from a to b any value r. If on then integrate f(x) dx from a to b >= integrate g(x) dz from a to b If on integrate f(x) dx from a to b >= 0 If on then m(b - a) <= integrate f(x) dx from a to b <= M(b - a) Calculus Cheat Sheet Common Integrals integrate \mathfrak{e} ^ u du = \mathfrak{e} ^ u + c int mathfrak cos (u)du= mathfrak sin(u) + c int mathfrak sin (u)du=- mathfrak cos(u) + c int mathfrak sec^2 (u) * du = tan(u) + c int mathfrak cs * \mathcal{C} ^ 2 * (u) du=- mathfrak cot(u)+c integrate x ^ n dx = 1/(n + 1) * x ^ (n + 1) + c ,n ne-1 integrate 1/(ax + b) dx = 1/a * ln|ax + b| + c int mathfrak sec(u) * tan(u) du= mathfrak sec (u)+c int mathfrak c sc(u) mathfrak cot(u)du=- mathfrak cos c * (u) + c int mathfrak soc * (u) du= ( mathfrak esc * (u) * du = -ln| mathfrak cso * (u) + mathfrak cot(u)|+c integrate x ^ - 1 dx = integrate 1/x dx = ln|x| + c integrate ln(u) du = u * ln(u) - u + c integrate tan(u) du =ln| mathfrak sec (u)|+c integrate tan(u) du =-ln| mathfrak cos(u)|+c integrate 1/(a ^ 2 + u ^ 2) du = 1/a * arctan(u/a) + c integrate 1/(sqrt(a ^ 2 - u ^ 2)) du = arcsin(u/a) + c Standard Integration Techniques Substitution: integrate f(g(x)) * g' * (x) dx from a to b convert the integral into the substitution where For indefinite integrals integrate f(g(x)) * g' * (x) dx from a to b = integrate f(u) dz from g(a) to g(b) using Example integrate 5x ^ 2 * cos(x ^ 3) dx from 1 to 2 u=x^ 2 Rightarrow du=3x^ 2 dx Rightarrow x ^ 2 * dx = 1/3 * du x=1 Rightarrow u=1^ 3 =1 Rightarrow x=2 Rightarrow u = 2 ^ 3 = 8 integrate 5x ^ 2 * cos(x ^ 3) dx from 1 to 2 = integrate 5/3 * cos(u) du from 1 to 8 = 5/3 * sin(u) | 1 ^ 8 = 5 3 ( mathfrak sin(8)- mathfrak sin(1)) Products and (some) Quotients of Trig Functions int mathfrak sin ^ n (x) mathfrak cos^(m(x)) (dx) we have the following: 1. and convert rest to cosines using sin^2 (x) = 1 - cos^2 (x) then use the 2. m odd. Strip 1 cosine out and convert rest to sines using mathfrak ss ^ 2 * (x) = 1 - sin^2 (x) . then use the 3. and both oddUse either 1, or 2. 4. and both even. Use double angle and/or half angle formulas to reduce the integral into a form that can be integrated. int tan^ prime prime (x) * sec^(m(x)) (dx) we have the following: 1. and 1 secant out and convert the rest to secants using tan^ 2 (x)= mathfrak sec^2 (x) - 1 , then use the substitution u= mathfrak seeee (x) . 2. even. Strip 2 secants out and convert rest to tangents using iecc^ 2 (x) = 1 + tan^2 (x) . then use the substitution 3. n odd and m even, Use either 1. or 2. 4. n even and modd. Each integral will be dealt with differently. Trig Formulas: sin(2x) =2 sin(x) mathfrak cos(x), mathfrak cos^ 2 (x)= 1 2 (1+ mathfrak cos(2x)), mathfrak sin^ 2 (x)= 1 2 (1- mathfrak cos(2x)) Example integrate tan^3 ((x) * ve) * \mathcal{C} ^ 3 * (x) dx integrate tan^5 x * sec^5 x dx = integrate tan^2 x * sec^4 x * tan x * sec(x) dx = int (sec^2 (x) - 1) * sec^4 (x) * tan(x) mathfrak sec (x)dx = int u (u ^ 2 - 1) * u ^ 4 du[u=sec(x)] =/ mathfrak sac * c ^ 7 * (x) - 1 5 mathfrak sac^ 5 (x)+c Example integrate (sin^3 (s))/(cos^2 (s)) dx int sin^ x sin^ x dx= int sin^2 x * sin x cos^ x dx= int sin^ 2 x)^ 2 sin x cos^ x dx = integrate ((1 - cos^2 (x)) ^ 2 * sin(x))/(cos^2 (x)) dx * [u = cos(x)] = - integrate ((1 - u ^ 2) ^ 2)/(u ^ 2) du = - integrate (1 - 2u ^ 4 + u ^ 4)/(u ^ 3) du = 1 3 sec^ 2 (x)+2 ln(cos(x)| - 1/3 * cos^2 (x) + c Calculus Cheat Sheet Integration by Parts: integrate u dv = uv - integrate v du and int a ^ b u dv=uv| a ^ b - integrate v du from a to b , Choose and die from integral and compute du by compute v = integrate 1 dv Example = 5 * ln(5) - 3 * ln(3) - 2 ) integrate ln(x) dx from 3 to 5 int 3 ^ 5 ln(x)dx=x ln(x)| 3 ^ 5 - integrate 1 dx from x to 5 =(x ln(x)-x)| 3 ^ 5 u = ln(x) du = 1/s * dx Trig Substitutions: If the integral contains the following root use the given substitution and formula to convert into an integral involving trig functions. integral we'll assume positive and drop absolute value bars. If we had a definite integral we'd need to computes and remove absolute value bars based on that and, sqrt(x ^ 2) = |x| sqrt(4 - 9x ^ 2) = sqrt(4 - 4sin^2 (theta)) = sqrt(4cos^2 (theta)) = 2|cos(theta)| x= 2 3 mathfrak sin( theta) Rightarrow dx= 2 3 mathfrak cos(theta) * d*theta In this case we have sqrt(4 - 9x ^ 2) =2 mathfrak coos(0). Use Right Triangle Trig back substitution we have sin(theta) = (2pi)/2 so 3 From this we see that cot( theta)= (sqrt(pi * E ^ 2))/(3pi) *\$0, Partial Fractions: If integrating a rational expression involving polynomials, integrate (P(x))/(Q(x)) dx 1 where the degree of smaller than the degree of Factor denominator as completely as and find the partial fraction decomposition of the rational expression. Integrate the partial fraction decomposition (P.F.D.). For each factor in the denominator we get term(s) in the decomposition according to the following table. Factor of Term in P.F.D (Ax + B)/(a * x ^ 2 + bx + c) A/(ax + b) Factor is Q(x) (ax + b) ^ k Term in P.F.D az+b(a+b) A A (a+b) Net Area: represents the net area between (2) and the-axis above-axis positive and area below-axis negative. integrate f(x) dx from a to b Calculus Cheat Sheet Applications of Integrals upper function lower function dir &= f() Aright function)-(let function) dy If the curves intersect then the area of each portion must be found individually. Here are some sketches of Area Between Curves: The general formulas for the two main cases for each are, a couple possible situations and formulas for a couple of possible cases. (3) y Volumes of Revolution: The two main formulas are V = integrate A(x) dx and V = integrate A(y) dy Here is some general information about each method of computing and . Rings A(outer radius)-(inner radius)) Limits of right/bot ring to /y of left/top ring Horz. Axis use and dr Vert. Axis use and dy. Ex, Axis Ex, Axis g = a <= 0 Cylinders/Shells A= 2pi () radius) (width/height) Limits of inner cyl. outer cyl. Horz. Axis use Vert. Axis and dy and dr. Ex. Axis=4>0 outer radius (2) inner radius: outer radius: |a|+ g * (x) | inner radius: |a| + f(x) radius: width: f(y) - g(y) Ex, y = a <= 0 radius: width f(y) - g(y) Example int (7x ^ 2 + 13x)/((x - 1)(x ^ 2 + 4)) d Here is = 4 * ln[x - 1] + 2/2 * ln(x ^ 2 + 4) + 8arctan(x/2) ] Set numerators equal and collect like terms. 7x ^ 2 + 13x = (A + B) * x ^ 2 + (C - B) * x + 4A - C Set coefficients equal to get a system and solve to get constants. C - B = 13 4A - C = 0 These are only a few cases for horizontal axis of rotation. If the axis of rotation is the-axis use y = a <= 0 case with For vertical axis of rotation and interchange and to get appropri ate formulas mu * x = a > 0 x = a <= 0 \ Work: If a force of moves an object in the work done is W = integrate F(x) dx from a to b a <= x <= b Average Function Value: The average value of a <= x <= b f avg = 1 b-a int a ^ b f(x)dx = integrate 12cot c ^ 2 * (theta) dtheta = - 12cot(theta) + c integrate 16/(4/5 * sin^2 (theta) * (2cos theta)) * (2/5 * cos theta) dtheta = integrate 12/(sin^2 (theta)) dtheta = integrate 4/(x - 1) + (3x)/(x ^ 2 + 4) + 16/(x ^ 2 + 4) dx integrate (7x ^ 2 + 13x)/((x - 1)(x ^ 2 + 4)) dx = integrate 4/(x - 1) + (3x + 16)/(x ^ 2 + 4) dx A = integrate f(x) - g(x) dx from a to b integrate 16/(x ^ 2 * sqrt(4 - 9x ^ 2)) dx = - (4sqrt(4 - 9x ^ 2))/x + c |x||= x ifx >= 0; - x ifx < 0 integrate x * e ^ (- x) dx x * theta ^ (- x) * dx = - x * theta ^ (- x) + integrate theta ^ (- x) dx = - x * \mathfrak{e} ^ (- x) - \mathfrak{e} ^ (- x) + c . x_{1} = x de= bullet^ -x Rightarrow du = dx v=- bullet^ - pi A = integrate f(y) - g(y) dy from a to a A = integrate f(x) - g(x) dx from a to x + integrate g(x) - f(x) dx from a to b integrate 16/(x ^ 2 * sqrt(4 - 9x ^ 2)) dx (2x ^ 3 + 12x)/((x - 1)(x ^ 2 + 4)) = A/(x - 1) + (Bx + C)/(x ^ 2 + 4) = (A(x ^ 2 + 4) + (Bx + C)(x - 1))/((x - 1)(x ^ 2 + 4)) matrix sqrt a^ 2 -b^ 2 x^ 2 & Rightarrow x= g 2 sin( theta)\\ cos^ 2 ( theta)=1-sin^ 2 ( theta)&[ matrix sqrt b^ 2 x^ 2 -a^ 2 & Rightarrow x= g 2 sec( theta)\\ tan^ 2 ( theta)=1-sin^ 2 ( theta)-1&[ matrix sqrt a^ 2 +b^ 2 x^ 2 & Rightarrow&x= underline g tan\\ tan^2 (theta) - se * \mathcal{C} ^ 2 * (theta) - 1 matrix y= bigoplus x' (x) ax^ 2 +hx+c)^ k A 1 x+B 2 ax^ 2 +bx+c +***+ (A_{k}*x + B_{k})/((a * x ^ 2 + bx + c) ^ k) a * x ^ 2 + bx + c sqrt(4 - 5x ^ 2) Calculus Cheat Sheet Arc Length & Surface Area: The three basic formulas are, L = integrate 1 ds from a to b SA = integrate 2pi*y ds from a to b rotate about-axis) SA= int a ^ b 2 pi x.ds rotate about y-axis) where ds is dependent upon the form of the function being worked with as follows. ds = sqrt(1 + (d/dx (y)) ^ 2) * dx de= 1+ if y=f(z), ab da= dy if z = f(y), a≤ysb dr 2 2 dt if f(t), y=g(t), a ≤ t ≤ b ds = sqrt(r ^ 2 + (d/dtheta (r)) ^ 2) * d*theta*fr = f(theta), a <= theta <= b With surface area you may have to substitute in for the ory depending on your choice of de to match the differential in the de. With parametric and polar you will always need to substitute. Improper Integral An improper integral is an integral with one or more infinite limits and/or discontinuous integrands. Integral is called convergent if the limit exists and has a finite value and divergent if the limit doesn't exist or has infinite value. Infinite Limit 1. integrate f(x) dx from a to ∞ = lim t -> ∞ integrate f(x) dx from x to t integrate f(x) dx from - ∞ to k = lim t -> - ∞ integrate f(x) dx from t to b 3. integrate f(x) dx from - ∞ to ∞ = integrate f(x) dx from - ∞ to epsilon + integrate f(x) dx from epsilon to ∞ provided both integrals are convergent. Discontinuous Integrand 1. Discontinuity at a / (integrate f(x) dx from a to b) = lim t -> a integrate f(x) dx from t to b 2. Discontinuity at : integrate f(x) dx from a to b = lim x -> b integrate f(x) dx from a to b 3. Discontinuity at a < c < b : integrate f(x) dx from a to b = integrate f(x) dx from a to c + integrate f(x) dx from c to b provided both are convergent. Comparison Test for Improper Integrals: If f(x) >= g(x) >= 0 on [a, ∞) then, 1. If integrate f(x) dx from a to ∞ is convergent then integrate g(x) dx from a to ∞ convergent (if larger converges so does the smaller). 2. If integrate g(x) dx from a to ∞ is divergent then Useful fact: If a > 0 then. integrate 1/(x ^ p) dx from a to ∞ integrate f(x) dx from a to ∞ divergent (if smaller diverges so does the larger). converges if p > 1 and diverges for p <= 1 Approximating Definite Integrals For given integral into subintervals [ integrate f(x) dx from a to b and (must be even for Simpson's Rule) define [x_{0}, x_{1}], [x_{1}, x_{2}] ,...,[x n-1 ,x n ]] with x_{0} = a and x_{n} = b then, Delta*v = (b - a)/n and divide [a,b] Midpoint Rule: integrate f(x) dx from a to b approx Delta x[f(x 1 ^ * )+f(x 2 ^ * )+***+f(x n ^ * )],x; is midpoint [x i - 1 ,x i ] Trapezoid Rule: 10: int a ^ b f(x)dx approx Delta x 2 [ f(x_{0}) + 2f(x_{1}) + 2f(x_{2}) +***+2f(x n-1 )+f(x n )] Simpson's Rule: int a ^ n f(x)dx approx Delta x 3 [ f(x_{0}) + 4f(x_{1}) + 2f(x_{2}) +***+2f(x n-2 )+4f(x n-1 )+f(x n )]