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Distance in R3 Acceleration to Velocity if a(t) is acceleration at time t The distance between the points P1 = (x1 , y1 , z1 ) and P2 = Z b (x2 , y2 , z2 ) is Change in velocity between times a and b = a(t)dt p a 2 2 2 d(P1 , P2 ) = (x2 − x1 ) + (y2 − y1 ) + (z2 − z1 ) Rate of Growth to Total Growth if r(t) is rate of growth at time t Equation of a Tangent Plane Z b The equation of the plane through a point (x1 , y1 , z1 ) have slope a in the x-direction and slope b in the y-direction is Total Growth between times a and b = r(t)dt a z − z1 = a(x − x1 ) + b(y − y1 ) Properties of the Definite Integral Rb Rb Rb Linear Approximation for (x, y) near (a, b) 1. a [f (x) ± g(x)]dx = a f (x)dx ± a g(x)dx Rb Rb f (x, y) ≈ f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b) 2. any constant k, a kf (x)dx = k a f (x)dx Rb Rc Rb this can be written as ∆f ≈ fx (a, b)∆x + fy (a, b)∆y. 3. if a ≤ c ≤ b, a f (x)dx = a f (x)dx + c f (x)dx 2nd Derivative Test Suppose (a, b) satisfies fx (a, b) = 0 Ra Rb and fy (a, b) = 0. 4. b f (x)dx = − a f (x)dx 2 Let D = fxx (a, b)fyy (a, b) − fxy (a, b) . Then, Ra 5. f (x)dx = 0 a • if D > 0 and fxx (a, b) > 0 then f has a local min. at (a, b) Fundamental Theorem of Calculus Suppose f (x) is a continuous function on the interval [a, b] and F (x) is an antiderivative of f (x), that is F 0 (x) = f (x) for all x in [a, b]. • if D > 0 and fxx (a, b) < 0 then f has a local max. at Then, Z b (a, b) f (x)dx = F (b) − F (a) a • if D < 0 then f has neither a local min nor local max. at (a, b) The Area Function If f (x) is continuous on [a, b], then the Rt area function A(t) = a f (x)dx with a ≤ t ≤ b is an antiderivative of f , that is A0 (t) = f (t). Total Change in Cost The change in cost from changing • if D = 0 the test is inconclusive. production from a units to b units is Method of Lagrange Multipliers To optimize f (x, y) subZ b Z b ject to the constraint g(x, y) = c, solve the system C(b) − C(a) = C 0 (x)dx = M C(x)dx a fx = λgx fy = λgy g(x, y) = c a Also, C(x) = C 0 (x)+ constant to be determined. Total Change in Revenue The change in revenue from changing production from a units to b units is Z b Z b R(b) − R(a) = R0 (x)dx = M R(x)dx R for x, y, and λ. Rules for (Basic) Indefinite Integrals R k 1 1. k 6= 1, x dx = k+1 xk+1 + c R 1 2. k = 1, x dx = ln|x| + c R kx 3. k 6= 0, e dx = k1 ekx + c R R 4. any k, kf (x)dx = k f (x)dx R R R 5. [f (x) ± g(x)]dx = f (x)dx ± g(x)dx a a Also, R(x) = R0 (x)+ constant to be determined. Total Change in Profit The change in profit from changing production from a units to b units is Z b Z b π(b) − π(a) = π 0 (x)dx = [R0 (x) − C 0 (x)]dx R a R a 0 Also, π(x) = π (x)+ constant to be determined. Area Between 2 Curves if f (x) ≥ g(x), the the area between f (x) and g(x) over [a, b] is Z b Area = [f (x) − g(x)]dx Integration by Parts Z Z u · dv = u · v − v · du a Definition of the Definite Integral Z Average Value of a Function b f (x)dx = lim [f (x1 )∆x + f (x2 )∆x + ... + f (xn )∆x] a Average value of f (x) on [a, b] = n→∞ provided the limit on the right exists. Velocity to Displacement if v(t) is velocity at time t Z Displacement between times a and b = 1 b−a Z b f (x)dx a Midpoint Rule and Error Bound if n subintervals are used, and xi is the midpoint of the ith subinterval, then Z b f (x)dx ≈ [f (x1 ) + f (x2 ) + ... + f (xn )]∆x b v(t)dt a a 1 where ∆x = EM ≤ b−a n . The error is bounded by K(b − a)3 , 24n2 Median of a Continuous Random Variable The median of a random variable X that has probability density function f (x) is a number m such that Z m 1 f (x)dx = 2 −∞ K = max of f 00 (x) on [a, b] Trapezoidal Rule and Error Bound if n subintervals are used, and xi is the left end of the ith subinterval, then Z b 1 1 f (x)dx ≈ f (x1 ) + f (x2 ) + ... + f (xn−1 ) + f (xn ) ∆x 2 2 a where ∆x = b−a n . Mean of a Continuous Random Variable For a random variable X with density function f (x), a ≤ x ≤ b, the mean µ, or expected value E(X), is given by The error is bounded by Z µ = E(x) = K(b − a)3 ET ≤ , 12n2 xf (x)dx a 00 K = max of f (x) on [a, b] Variance of a Continuous Random Variable The variance , V ar(X) or σ 2 , of a random variable X with probability density function f (x), a ≤ x ≤ b and mean µ is Consumer Surplus and Producer Surplus If qe is the equilibrium quantity and pe is the equilibrium price, then Z qe Z qe CS = D(q)dq − pe qe , and P S = pe qe − S(q)dq 0 b 2 Z V ar(X) = σ = b x2 f (x)dx − µ2 a 0 Standard Deviation of a Continous Random Variable The standard deviation, σ, of the random variable X is nt • compounded n times per year: F V = P V 1 + nr , given by √ p −nt σ = σ 2 = V ar(X) and P V = F V 1 + nr Integrals of Trigonometric Functions • continuously compounded: F V = P V ert , and P V = R −rt FV e . • sin(x)dx = −cos(x) + c R PV and FV of a Continuous Income Stream • cos(x)dx = sin(x) + c R Z T Z T • tan(x)dx = ln|sec(x)| + c FV = S(t)er(T −t) dt, PV = S(t)e−rt dt R 0 0 • cot(x)dx = −ln|csc(x)| + c R if S(t) = S is a constant • sec(x)dx = ln|tan(x) + sec(x)| + c R S S P V = (1 − e−rT ) F V = (erT − 1), • csc(x)dx = −ln|cot(x) + csc(x)| + c r r Integrals Involving Inverse Trig. Functions Useful Limits (for Improper Integrals) R √ 1 • dx = arcsin(x) + c 1−x2 • limb→∞ ebb = 0 R 1 m dx = arccos(x) + c • − √1−x 2 • limb→∞ beb = 0 for any m > 0 R 1 • 1+x2 dx = arctan(x) + c • limb→∞ ln(b) bm = 0 for any m > 0 Present and Future Value of a Single Investment • limb→∞ 1 xm • limb→∞ xm = ∞ = 0 for any m > 0 for any m > 0 Useful Improper Integral Z 1 ∞ 1 dx : xp ( diverges if p ≤ 1 converges if p > 1 Present Value of a Perpetual Income Stream Z PV = ∞ S(t)e−rt dt 0 Properties of Probability Density Functions 1. f (x) ≥ 0 for all x in (a, b) ((a, b) is the domain) Rb 2. a f (x)dx = 1, and Rd 3. for any a ≤ c < d ≤ b, P (c < X < d) = c f (x)dx. 2