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 n th 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 dx (sin x ) = cos x d dx (cos x ) = − sin x d dx (tan x ) = sec 2 x d dx (cot x ) = − csc 2 x d dx (sec x ) = sec x tan x d dx (csc x ) = − csc x cot x d dx (sin − 1 x ) = 1 √ 1 − x 2 d dx (cos − 1 x ) = − 1 √ 1 − x 2 d dx (tan − 1 x ) = 1 1 + x 2 d dx (cot − 1 x ) = − 1 1 + x 2 d dx (sec − 1 x ) = 1 | x |√ x 2 − 1 d dx (csc − 1 x ) = − 1 | x |√ x 2 − 1 d dx ( c x ) = c x ln c d dx ( e x ) = e x d dx (ln( x )) = 1 x , x > 0 d dx (ln | x | ) = 1 x , x 6 = 0 d dx ( log a ( x )) = 1 x ln a , x > 0 1.2 Rules d dx ( f ( x ) g ( x )) = f ′ ( x ) g ( x ) + f ( x ) g ′ ( x ) d dx ( f ( x ) g ( x ) ) = g ( x ) f ′ ( x ) − f ( x ) g ′ ( x ) g ( x ) 2 d dxf ( g ( x )) = f ′ ( g ( x )) · g ′ ( x ) d dx ( x n ) = nx n − 1 1.3 Implicit Differentiation Step 1: Differentiate with respect to x Step 2: Seperate dy dx terms from others Step 3: Factor out dy dx from other terms, and divide 1.4 Derivative of an inverse function ( f − 1 ) ′ ( x ) = 1 f ′ ( 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 ′ ( c ) = 0 or f ′ ( c ) doesn’t exist. 1.5.2 Increasing/decreasing If f ′ ( x ) > 0 for all x in an interval, then f ( x ) is increasing on that interval If f ′ ( x ) < 0 for all x in an interval, then f ( x ) is decreasing on that interval If f ′ ( x ) = 0 for all x in an interval, then f ( x ) is constant on that interval 1.5.3 Concavity If f ′′ ( x ) > 0 for all x in an interval, then f ( x ) is concave up on that interval If f ′′ ( 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 1 st derivative test If x = c is a critical point of f ( x ) then x = c is a relative maximum if f ′ ( x ) > 0 to the left of c and f ′ ( x ) < 0 to the right of c a relative minimum if f ′ ( x ) < 0 to the left of c and f ′ ( x ) > 0 to the right of c not a relative extrema if f ′ ( x ) is the same sign on both sides of x = c 4 2 nd derivative test If x = c is a critical point of f ( x ) and f ′ ( c ) exists then x = c is a relative maximum if f ′′ ( c ) < 0 a relative minimum if f ′′ ( x ) > 0 unknown if f ′′ ( 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 that f ′ ( c ) = f ( b ) − f ( a ) 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 subintervals of width ∆ x , with x i chosen from each interval, then ∫ b a f ( x ) dx = lim n →∞ n ∑ i =1 f ( x i )∆ x Anti-derivative An anti-derivative of f ( x ) is a function F ( x ) such that F ′ ( x ) = f ( x ). 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 Part 1 If f ( x ) is continuous on [ a, b ] then g ( x ) = ∫ x a f ( t ) dt is also continuous on [ a, b ] and g ′ ( x ) = d dx ∫ x a f ( t ) dt = f ( x ). Variants of Part 1 d dx ∫ u ( x ) a f ( t ) dt = u ′ ( x ) f [ u ( x )] d dx ∫ b v ( x ) f ( t ) dt = − v ′ ( x ) f [ v ( x )] d dx ∫ u ( x ) v ( x ) f ( t ) dt = u ′ ( x ) f [ u ( x )] − v ′ ( x ) f [ v ( x )] Part 2 If f ( x ) is continuous on [ a, b ] and F ( x ) is an anti-derivative of f ( x ) then ∫ b a f ( x ) dx = F ( b ) − F ( a ) 2.2 Common Integrals ∫ kdx = kx + c ∫ x n dx = 1 n + 1 x n +1 + c, n 6 = − 1 ∫ 1 x dx = ln | x | + c ∫ 1 ax + bdx = 1 a ln | ax + b | + c ∫ ln( x ) du = u ln( x ) − x + c ∫ cos xdx = sin x + c ∫ sin xdx = − cos x + c 6 ∫ sec 2 xdx = tan x + c ∫ sec x tan xdx = sec x + c ∫ tan xdx = ln | sec x | + c ∫ sec xdx = ln | sec x + tan x | + c ∫ 1 a 2 + x 2 dx = 1 a tan − 1 ( x a ) + c ∫ 1 √ a 2 − x 2 dx = sin − 1 ( x a ) + c 2.3 Integration techniques 2.3.1 U-substitution Find a u such that substituting in u and du makes life easier. Example: ∫ e x 1 + e 2 x dx Let u = e x Then, du = e x dx We can rewrite the integral as ∫ 1 1 + u 2 du Integrating, we get tan − 1 u + c Substitute u back in to get tan − 1 ( e x ) 2.3.2 Integration by parts ∫ udv = uv − ∫ vdu Pick a u with a derivative that goes to 0, and a dv which is easy to integrate. Example: ∫ xe x dx Let u = x and dv = e x dx Then, du = dx and v = e x Substituting in, we get xe x + ∫ e x dx Evaluating, we get ∫ xe x dx = xe x + e x 2.3.3 Trig substitution Substitute in trigonometric equations to simplify problems with square roots. For √ a 2 − ( bx ) 2 , let x = a b sin θ 7 For √ a 2 + ( bx ) 2 , let x = a b tan θ For √ ( bx ) 2 − a 2 , let x = a b sec θ 2.3.4 Partial fraction decomposition For polynomial fractions with a greater order denominator. Steps: Example: Factor the denominator Write out a partial fraction for each factor Multiply both sides by the denominator Plug in zeros for x x − 8 x 2 − x − 2 x − 8 x 2 − x − 2 = x − 8 ( x + 1)( x − 2) x − 8 ( x + 1)( x − 2) = A x + 1 + B x − 2) x − 8 = A ( x − 2) + B ( x + 1) 2 − 8 = A (2 − 2) + B (2 + 1) 2.4 Applications 2.4.1 Area between curves A = ∫ b a [ f ( x ) − g ( x )] dx where a and b are intersection points of f and g and f ( x ) ≥ g ( x ) 2.4.2 Solids of revolution Disc method V = π ∫ x = b x = a ( r ( x )) 2 dx where a and b are the bounds, and r ( x ) is the distance between the curve and the axis of rotation. Shell method V = 2 π ∫ b a r ( x ) h ( x ) dx where r ( x ) is the distance between the curve and the axis of rotation, and h ( x ) is the height of the segment being rotated around the axis. 2.4.3 Arc length L = ∫ b a √ 1 + ( dy dx ) 2 dx 2.4.4 Average value The average value of a function f ( x ) on a ≤ x ≤ b is f avg = 1 b − a ∫ b a f ( x ) dx 8 2.5 Improper integrals 2.5.1 Integrals with bounds that approach ±∞ ∫ ∞ a f ( t ) dt = lim n →∞ ∫ n a f ( t ) dt ∫ b −∞ f ( t ) dt = lim n →−∞ ∫ b n f ( t ) dt ∫ ∞ = ∞ f ( t ) dt = ∫ c −∞ f ( t ) dt + ∫ ∞ c f ( t ) dt if both integrals converge. 2.5.2 Integrals with discontinuities at bounds If discontinuity is at x = a : ∫ b a f ( x ) dx = lim n → a + ∫ b n f ( x ) dx If discontinuity is at x = b : ∫ b a f ( x ) dx = lim n → b − ∫ n a f ( x ) dx If discontinuity is at x = c such that a < c < b : ∫ b a f ( x ) dx = ∫ c a f ( x ) dx + ∫ b c f ( x ) dx 2.6 Estimation 2.6.1 Trapezoidal rule ∫ b a f ( x ) dx ≈ T n = ∆ x 2 [ f ( x 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) + · · · + 2 f ( x n )] OR 1 2 f ( x 0 )∆ x 0 + f ( x 1 )∆ x 1 + f ( x 2 )∆ x 2 + · · · f ( x n − 1 )∆ x n − 1 + 1 2 f ( x n )∆ x n 9 3 Differential equations 3.1 Solving equations given a point Applicable when the equation can be seperated into x and y terms. Steps: Example: Seperate x and y terms Integrate both sides, adding c to the x side Isolate y Substitute in c 1 if applicable Find c 1 given the point Substitute in c 1 dy dx = 2 xy passing through (0 , 1) dy y = 2 xdx ∫ dy y = ∫ 2 xdx → ln | y | = x 2 + c y = ± e x 2 + c c 1 = ± e c → y = c 1 e x 2 1 = c 1 e 0 = ⇒ c 1 = 1 y = e x 2 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: Choose an increment h The smaller, the more accurate. Starting at the inital point, increase x by h , and y by h dy dx Pick up from that point, and repeat. Continue until you reach the x value to ap- proximate. dy dx = 2 x + y, through (0 , 1), at x = 1 h = 0 1 (0 , 1) → (0 1 , 1 1) (0 1 , 1 1) → (0 2 , 1 23) (0 2 , 1 23) → (0 3 , 1 393)...... 11 4 Series 4.1 Definitions 4.2 Convergence tests 4.2.1 n th term test If lim n →∞ a n 6 = 0, then the series ∞ ∑ n =1 a n must diverge. 4.2.2 p-series test If a n = 1 n p , ∞ ∑ n =1 a n converges if p > 1 and diverges if p ≤ 1 4.2.3 Alternating series test ∞ ∑ n =1 ( − 1) n a n and ∞ ∑ n =1 ( − 1) n +1 a n converge if both: 0 < a n +1 ≤ a n for all n > 0 ( a n is positive and decreasing) lim n →∞ a n = 0 4.2.4 Direct comparison test If ∞ ∑ n =1 b n is absolutely convergent, and 0 < b n ≤ a n for all n > c , then ∞ ∑ n =1 a n converges absolutely. If ∞ ∑ n =1 b n diverges, so does ∞ ∑ n =1 a n 4.2.5 Limit comparison test If a n , b n > 0 and lim n →∞ a n b n = L > 0 then ∞ ∑ n =1 a n converges if and only if ∞ ∑ n =1 b n converges. 4.2.6 Integral test If a n = f ( n ) and f ( n ) is continuous, positive, and decreasing, then ∞ ∑ n =1 a n converges if and only if ∫ ∞ 1 f ( x ) dx converges. 12 4.2.7 Ratio test ∞ ∑ n =1 a n converges if lim n →∞ ∣ ∣ ∣ ∣ a n +1 a n ∣ ∣ ∣ ∣ < 1 and diverges if lim n →∞ ∣ ∣ ∣ ∣ a n +1 a n ∣ ∣ ∣ ∣ > 1. The test is inconclusive if lim n →∞ ∣ ∣ ∣ ∣ a n +1 a n ∣ ∣ ∣ ∣ = 1 4.3 Taylor polynomials 4.3.1 How to write a Taylor polynomial P n ( x ) is the taylor series of n th degree centered around x = c for f ( x ) if P n = f ( c ) + f ′ ( c )( x − c ) + f ′′ ( c )( x − c ) 2 2 + f ′′′ ( c )( x − c ) 3 6 + · · · + f n ( c )( x − c ) n n ! P ∞ = ∞ ∑ n =0 f ( n ) ( c ) n ! ( x − c ) n 4.3.2 Common Taylor series 1 1 − x = ∞ ∑ n =0 x n , x ∈ ( − 1 , 1) e x = ∞ ∑ n =0 x n n ! cos x = ∞ ∑ n =0 ( − 1) n x 2 n (2 n )! sin x = ∞ ∑ n =0 ( − 1) n x 2 n +1 (2 n + 1)! ln(1 + x ) = ∞ ∑ n =1 ( − 1) n +1 x n n , x ∈ ( − 1 , 1] 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