Contents 1 Derivatives 3 1.1 Common derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Implicit Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Derivative of an inverse function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.5 Critical pts, extrema, increasing/decreasing functions, and more with derivatives . 4 1.5.1 Critical points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5.2 Increasing/decreasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5.3 Concavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5.4 Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.6 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2 Integrals 6 2.1 Definitions, theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.2 Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Common Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Integration techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3.1 U-substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3.2 Integration by parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3.3 Trig substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3.4 Partial fraction decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4.1 Area between curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4.2 Solids of revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4.3 Arc length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4.4 Average value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.5 Improper integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5.1 Integrals with bounds that approach ±∞ . . . . . . . . . . . . . . . . . . . 9 2.5.2 Integrals with discontinuities at bounds . . . . . . . . . . . . . . . . . . . . 9 2.6 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.6.1 Trapezoidal rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3 Differential equations 10 3.1 Solving equations given a point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Slope fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 Euler’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4 Series 12 4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2 Convergence tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2.1 nth term test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2.2 p-series test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2.3 Alternating series test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1 4.2.4 Direct comparison test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2.5 Limit comparison test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2.6 Integral test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.2.7 Ratio test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.3 Taylor polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.3.1 How to write a Taylor polynomial . . . . . . . . . . . . . . . . . . . . . . . 13 4.3.2 Common Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.4 Finding intervals of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.4.1 Radii of convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.4.2 Endpoint testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 1 Derivatives 1.1 Common derivatives d 1 (tan−1 x) = dx 1 + x2 d d 1 (sin x) = cos x (cot−1 x) = − dx dx 1 + x2 d d 1 (cos x) = − sin x (sec−1 x) = √ dx dx |x| x2 − 1 d d 1 (tan x) = sec2 x (csc−1 x) = − √ dx dx |x| x2 − 1 d d x (cot x) = − csc2 x (c ) = cx ln c dx dx d d x (sec x) = sec x tan x (e ) = ex dx dx d d 1 (csc x) = − csc x cot x (ln(x)) = , x > 0 dx dx x d 1 d 1 (sin−1 x) = √ (ln |x|) = , x 6= 0 dx 1 − x2 dx x d 1 d 1 (cos−1 x) = − √ (loga (x)) = ,x > 0 dx 1 − x2 dx x ln a 1.2 Rules d (f (x)g(x)) = f 0 (x)g(x) + f (x)g 0 (x) dx g(x)f 0 (x) − f (x)g 0 (x) d f (x) = dx g(x) g(x)2 d f (g(x)) = f 0 (g(x)) · g 0 (x) dx d n (x ) = nxn−1 dx 1.3 Implicit Differentiation Step 1: Differentiate with respect to x dy Step 2: Seperate dx terms from others dy Step 3: Factor out dx from other terms, and divide 1.4 Derivative of an inverse function 0 1 (f −1 ) (x) = f 0 (f −1 (x)) 3 1.5 Critical pts, extrema, increasing/decreasing functions, and more with derivatives 1.5.1 Critical points x = c is a critical point of f (x) provided either f 0 (c) = 0 or f 0 (c) doesn’t exist. 1.5.2 Increasing/decreasing If f 0 (x) > 0 for all x in an interval, then f (x) is increasing on that interval If f 0 (x) < 0 for all x in an interval, then f (x) is decreasing on that interval If f 0 (x) = 0 for all x in an interval, then f (x) is constant on that interval 1.5.3 Concavity If f 00 (x) > 0 for all x in an interval, then f (x) is concave up on that interval If f 00 (x) < 0 for all x in an interval, then f (x) is concave down on that interval x = c is an inflection point of f (x) if the concavity changes at x = c 1.5.4 Extrema Absolute extrema x = c is an absolute maximum of f (x) if f (c) ≥ f (x) for all x in the domain of f x = c is an absolute minimum of f (x) if f (c) ≤ f (x) for all x in the domain of f Relative extrema x = c is a relative maximum if f (c) ≥ f (x) for all x near c x = c is a relative minimum if f (c) ≤ f (x) for all x near c 1st derivative test If x = c is a critical point of f (x) then x = c is a relative maximum if f 0 (x) > 0 to the left of c and f 0 (x) < 0 to the right of c a relative minimum if f 0 (x) < 0 to the left of c and f 0 (x) > 0 to the right of c not a relative extrema if f 0 (x) is the same sign on both sides of x = c 4 2nd derivative test If x = c is a critical point of f (x) and f 0 (c) exists then x = c is a relative maximum if f 00 (c) < 0 a relative minimum if f 00 (x) > 0 unknown if f 00 (c) = 0 1.6 Mean Value Theorem If f (x) is continuous on [a, b] and differentiable on (a, b) then there is a number a < c < b such f (b) − f (a) that f 0 (c) = b−a 5 2 Integrals 2.1 Definitions, theorem 2.1.1 Definitions Definite integral The area under a curve. If f (x) is continuous on [a, b] and divided into Z b n X subintervals of width ∆x, with xi chosen from each interval, then f (x)dx = lim f (xi )∆x a n→∞ i=1 Anti-derivative An anti-derivative of f (x) is a function F (x) such that F 0 (x) = f (x). Z Indefinite integral f (x)dx = F (x) + c where F (x) is an anti-derivative of f (x) 2.1.2 Fundamental Theorem of Calculus Z x Part 1 If f (x) is continuous on [a, b] then g(x) = f (t)dt is also continuous on [a, b] and Z x a 0 d g (x) = f (t)dt = f (x). dx a Variants of Part 1 Z u(x) d f (t)dt = u0 (x)f [u(x)] dx a Z b d f (t)dt = −v 0 (x)f [v(x)] dx v(x) Z u(x) d f (t)dt = u0 (x)f [u(x)] − v 0 (x)f [v(x)] dx v(x) Z b Part 2 If f (x) is continuous on [a, b] and F (x) is an anti-derivative of f (x) then f (x)dx = a F (b) − F (a) Z 2.2 Common Integrals 1 1 dx = ln |ax + b| + c Z ax + b a Z kdx = kx + c ln(x)du = u ln(x) − x + c Z 1 Z xn dx = xn+1 + c, n 6= −1 cos xdx = sin x + c n+1 Z 1 Z dx = ln |x| + c sin xdx = − cos x + c x 6 Z Z 2 sec xdx = tan x + c sec xdx = ln | sec x + tan x| + c Z Z 1 1 −1 x sec x tan xdx = sec x + c dx = tan +c a2 + x2 a a Z Z 1 x tan xdx = ln | sec x| + c √ dx = sin−1 +c a2 − x 2 a 2.3 Integration techniques 2.3.1 U-substitution Find a u such that substituting in u and du makes life easier. ex Z Example: dx 1 + e2x Let u = ex Then, du = ex dx Z 1 We can rewrite the integral as du 1 + u2 Integrating, we get tan−1 u + c Substitute u back in to get tan−1 (ex ) 2.3.2 Integration by parts Z Z udv = uv − vdu Pick a u with a derivative that goes to 0, and a dv which is easy to integrate. Z Example: xex dx Let u = x and dv = ex dx Then, du = dx and v = ex Z Substituting in, we get xe + x ex dx Z Evaluating, we get xex dx = xex + ex 2.3.3 Trig substitution Substitute in trigonometric equations to simplify problems with square roots. p For a2 − (bx)2 , let x = ab sin θ 7 p For a2 + (bx)2 , let x = a b tan θ p For (bx)2 − a2 , let x = a b sec θ 2.3.4 Partial fraction decomposition For polynomial fractions with a greater order denominator. x−8 Steps: Example: x2 −x−2 Factor the denominator x−8 x−8 = x2 −x−2 (x + 1)(x − 2) Write out a partial fraction for each factor x−8 A B = + ) (x + 1)(x − 2) x+1 x−2 Multiply both sides by the denominator x − 8 = A(x − 2) + B(x + 1) Plug in zeros for x 2 − 8 = A(2 − 2) + B(2 + 1) 2.4 Applications 2.4.1 Area between curves Z b A= [f (x) − g(x)] dx where a and b are intersection points of f and g and f (x) ≥ g(x) a 2.4.2 Solids of revolution Z x=b Disc method V = π (r(x))2 dx where a and b are the bounds, and r(x) is the distance x=a between the curve and the axis of rotation. Z b Shell method V = 2π r(x)h(x)dx where r(x) is the distance between the curve and the axis a of rotation, and h(x) is the height of the segment being rotated around the axis. 2.4.3 Arc length s 2 Z b dy L= 1+ dx a dx 2.4.4 Average value Z b 1 The average value of a function f (x) on a ≤ x ≤ b is favg = f (x)dx b−a a 8 2.5 Improper integrals 2.5.1 Integrals with bounds that approach ±∞ Z ∞ Z n f (t)dt = lim f (t)dt a n→∞ a Z b Z b f (t)dt = lim f (t)dt −∞ n→−∞ n Z ∞ Z c Z ∞ f (t)dt = f (t)dt + f (t)dt if both integrals converge. =∞ −∞ c 2.5.2 Integrals with discontinuities at bounds Z b Z b If discontinuity is at x = a: f (x)dx = lim+ f (x)dx a n→a n Z b Z n If discontinuity is at x = b: f (x)dx = lim− f (x)dx a n→b a Z b Z c Z b If discontinuity is at x = c such that a < c < b: f (x)dx = f (x)dx + f (x)dx a a c 2.6 Estimation 2.6.1 Trapezoidal rule Z b ∆x f (x)dx ≈ Tn = [f (x0 ) + 2f (x1 ) + 2f (x2 ) + · · · + 2f (xn )] a 2 OR 1 1 f (x0 )∆x0 + f (x1 )∆x1 + f (x2 )∆x2 + · · ·f (xn−1 )∆xn−1 + f (xn )∆xn 2 2 9 3 Differential equations 3.1 Solving equations given a point Applicable when the equation can be seperated into x and y terms. dy Steps: Example: = 2xy passing through (0, 1) dx Seperate x and y terms dy = 2xdx y Z Z Integrate both sides, adding c to the x side dy = 2xdx → ln |y| = x2 + c y Isolate y 2 +c y = ±ex Substitute in c1 if applicable 2 c1 = ±ec → y = c1 ex Find c1 given the point 1 = c1 e0 =⇒ c1 = 1 Substitute in c1 2 y = ex 3.2 Slope fields Gives a visual of what a graph may look like when a differential equation isn’t solveable. Steps: 1. Draw a grid 2. Draw a line at each point with the slope defined by the differential equation 10 3.3 Euler’s method Used to find an approximate value of a point on a graph when given an unsolveable differential equation and a point it passes through. Steps: Example: dy = 2x + y, through (0, 1), at x = 1 dx Choose an increment h. The smaller, the h = 0.1 more accurate. Starting at the inital point, increase x by h, dy (0, 1) → (0.1, 1.1) and y by h dx Pick up from that point, and repeat. (0.1, 1.1) → (0.2, 1.23) Continue until you reach the x value to ap- (0.2, 1.23) → (0.3, 1.393)...... proximate. 11 4 Series 4.1 Definitions 4.2 Convergence tests 4.2.1 nth term test ∞ X If lim an 6= 0, then the series an must diverge. n→∞ n=1 4.2.2 p-series test ∞ 1 X If an = p , an converges if p > 1 and diverges if p ≤ 1 n n=1 4.2.3 Alternating series test ∞ X ∞ X n (−1) an and (−1)n+1 an converge if both: n=1 n=1 0 < an+1 ≤ an for all n > 0 (an is positive and decreasing) lim an = 0 n→∞ 4.2.4 Direct comparison test ∞ X ∞ X If bn is absolutely convergent, and 0 < bn ≤ an for all n > c, then an converges absolutely. n=1 n=1 ∞ X ∞ X If bn diverges, so does an n=1 n=1 4.2.5 Limit comparison test ∞ ∞ an X X If an , bn > 0 and lim = L > 0 then an converges if and only if bn converges. n→∞ bn n=1 n=1 4.2.6 Integral test ∞ X If an = f (n) and f (n) is continuous, positive, and decreasing, then an converges if and only if Z ∞ n=1 f (x)dx converges. 1 12 4.2.7 Ratio test ∞ X an+1 an+1 an converges if lim < 1 and diverges if lim > 1. The test is inconclusive if n=1 n→∞ an n→∞ an an+1 lim =1 n→∞ an 4.3 Taylor polynomials 4.3.1 How to write a Taylor polynomial Pn (x) is the taylor series of nth degree centered around x = c for f (x) if f 00 (c)(x − c)2 f 000 (c)(x − c)3 f n (c)(x − c)n Pn = f (c) + f 0 (c)(x − c) + + +···+ 2 6 n! ∞ X f (n) (c) P∞ = (x − c)n n=0 n! 4.3.2 Common Taylor series ∞ 1 X = xn , x ∈ (−1, 1) 1 − x n=0 ∞ x X xn e = n=0 n! ∞ X x2n cos x = (−1)n n=0 (2n)! ∞ X x2n+1 sin x = (−1)n n=0 (2n + 1)! ∞ X xn ln(1 + x) = (−1)n+1 , x ∈ (−1, 1] n=1 n 4.4 Finding intervals of convergence 4.4.1 Radii of convergence The radius of convergence of a Taylor series can be determined using the ratio test, getting |x − c| = R, where c is where the series is centered, and R is the radius of convergence. 13 4.4.2 Endpoint testing To determine the convergence of endpoints, simply plug x = c + R and x = c − R into the series, and use another convergence test to determine whether it converges or diverges. 14
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