ES_Math137_Exam

ES_Math137_Exam - MATH 137 Exam Equation Sheet PROPERTIES...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 137 Exam Equation Sheet PROPERTIES OF REAL NUMBERS Properties of real numbers: associativity, commutativity, existence of identity, existence of inverses Subsets of real numbers: natural numbers (e.g. 1,2,3,…), integers (e.g. -2,1,0,1,2,…), rational numbers (i.e. p for p, q xx= q integers, q ≠ 0 ). PROPERTIES OF FUNCTIONS Domain: The domain is the set of values of x for which f(x) is defined Range: The range is the set of all possible values of f(x) Even Function: Even functions are functions that have the property f (− x ) = f ( x ) for all x Odd Function: Odd functions are functions that have the property f (− x ) = − f (x ) for all x Invertible Function: An invertible function is a function f is invertible if there exists a function g such that y = f x if and () only if g ( y ) = x Increasing Function: An increasing function is a function is increasing on an interval I if for all x, such that f (x1 ) < f ( x 2 ) whenever x1 < x2 on I Decreasing Function: A decreasing function is a function is decreasing on an interval I if for all x, such that f ( x1 ) > f (x2 ) whenever x1 < x2 on I MATHEMATICAL INDUCTION Induction step by step Step 1—Base case. Show that the statement holds in the simplest case (normally for n = 0 and/or for n = 1). Step 2—Induction Hypothesis. Assume the statement holds for an arbitrary k, i.e. for n = k. Step 3. Prove that it holds for n = k + 1 given the above induction hypothesis. Strong (or complete) induction: Assume that the statement holds for n < k , and then prove it true for n = k . Upper/lower bound: A has an upper bound if there is a number a such that for all x ∈ A , x ≤ a . A has a lower bound if there is a number a such that for all x ∈ A , a ≤ x . Bounded: A is bounded iff A has both an upper bound and a lower bound; iff there exists a number b such that for all x ∈ A , x ≤ b . Least Upper Bound: b is the least upper bound of A iff b is an upper bound of A and if for all other upper bounds a of A, b ≤ a . Greatest Lower Bound: b is the greatest lower bound of A iff b is a lower bound of A and if for all lower bounds a of A, a ≤ b. Completeness of real numbers: If A is a non empty set of real numbers and A has an upper bound then A has a least upper bound If A is a non empty set of real numbers and A has a lower bound then A has a greatest lower bound If you’re taking the limit as x → ∞ of a quotient, and substitution yields an invalid answer, then try first to divide both the numerator and the denominator of the limit expression by its highest power of x If the limit expression contains absolute values, then try breaking the limit up into two one-sided limits. If the limit expression is a quotient, and if factoring doesn’t work, then try l’Hopital’s Rule: f ( x) f ′( x) • • • lim x→a • • • f (a ) and f (b ) then there exists a c such that lim f ( x ) = lim h( x ) = L then lim g ( x ) exists and is x→a x→ a x→a equal to L. THE DERIVATIVE h→0 h Rules for differentiation: for functions u,v and for constants c, n • dy n n −1 du dx • log b b = 1 log b b r = r • • • • g ′( x) a < c < b and f (c ) = k Squeeze theorem: If f ( x ) ≤ g (x ) ≤ h( x ) near a and f ′( x ) = lim log b 1 = 0 • x→a Differentiation by first principles f ( x + h ) − f (x ) . log b m r = r log b m 1 log b = − log b m m • = lim Intermediate value theorem: If f ( x ) is continuous on [a, b] and if k is between LOGARITHMIC FUNCTIONS Properties of logarithms: • log b (mn) = log b m + log b n ⎛ m⎞ • log b ⎜ ⎟ = log b m − log b n ⎝ n⎠ g ( x) b logb m = m log x 10 =x e ln x = x log a m log b m = log a b • LIMITS AND CONTINUITY (cu ) = cnu dx dy (uv ) = u dv + v du dx dx dx du dv v −u dy ⎛ u ⎞ dx dx ⎜ ⎟= dx ⎝ v ⎠ v2 Chain rule: If f ( x) = (a o b )( x) , then f ′( x) = a ′(b( x )) ⋅ b′( x) Power rule: d u(x ) du c = c u ( x ) ⋅ ln c ⋅ dx dx Higher order derivatives: f ′′( x) = d ⎛ dy ⎞ . ⎜⎟ dx ⎝ dx ⎠ More free study sheet and practice tests at: INEQUALITIES AND ABSOLUTE VALUES Absolute value rules: • a+b ≤ a + b • a−b ≥ a − b • a ⋅b = a ⋅ b Formal Definition: The function f(x) has limit L, as x approaches a, denoted lim f (x ) = L if given any ε > 0 , there DERIVATIVES OF LOGARITHMIC AND EXPONENTIAL FUNCTIONS x→a exists a δ > 0 such that |f(x) - L| < ε for all x satisfying 0 < |x – a|< δ. Properties of limits: • lim[ f ( x ) ± g ( x )] = lim f (x ) ± lim g ( x ) x→a • x→ a x→a x→a • • x→a x→a lim[ f (x ) ⋅ g (x )] = lim f ( x ) ⋅ lim g (x ) lim f ( x ) ⎡ f (x )⎤ x →a = lim ⎢ x→a g (x ) ⎥ ⎦ lim g ( x ) ⎣ x→a n [ x→a dx • • • lim[ f ( x )] = lim f ( x ) x →a For u, a function of x: • d 1 n Techniques for finding limits: • First try substitution, factoring (ln x ) = x d 1 du (ln u) = ⋅ du u dx d 1 du (log b u) = ⋅ (ln b)u dx dx dx e = ex dx • du du (e ) = e u dx dx • du du (a ) = a u (ln a ) dx dx More free Study Sheets and Practice Tests at: w ww.prep101.com Need help for exams? Check out our classroom prep sessions - customized to your exact course - at www.prep101.com TECHNIQUES OF DIFFERENTIATION Implicit differentiation: • to differentiate an implicit function Step 1. Take the derivative of both sides with respect to x. Use the Chain Rule on terms involving y (and note that the derivative of y with respect to x must be left as dy dx .) Step 2. Collect all terms involving dy dx on one side of the equation. Step 3. Solve for dy dx . Logarithmic differentiation: • to differentiate a function with complicated exponent/a product of several functions, etc. • to differentiate functions with an x in both the base and the exponent Step 1. Take the ‘ln’ of both sides (to find an expression of the form ln y = ln[ f ( x )] ). Step 2. Simplify ln( f ( x )) by using the properties of logarithms. Step 3. Differentiate both sides with respect to x (thus: 1 dy d ). ⋅ = (ln f (x )) y dx dx Step 4. Solve for dy dx . Step 5. Express the answer in terms of x only (substitution f(x) for y). APPLICATIONS OF THE DERIVATIVE L’Hopital’s rule: If the limit of the quotient of differentiable functions f (x ) and g ( x ) are of types 0 or ∞ , 0 ∞ and if g ′(a ) is not 0, then lim x →a Vertical asymptote: The line x = a is a vertical asymptote for the graph of the function f(x) if and only if lim f ( x ) = ±∞ x→a + or lim f ( x) = ±∞ . Horizontal asymptote: The line x = b is a horizontal asymptote for the graph of the function f(x) if and only if lim f ( x ) = b x → +∞ or lim f ( x) = b . x → −∞ Mean value theorem: If f is continuous on [a, b] and differentiable on (a, b), there exists some x in (a, b) for which f (b ) − f ( a ) . b−a INTEGRATION • minimum at p. If f ′( p ) = 0 and f ′′( p ) < 0 , then f has a relative maximum at p. If f ′( p ) = 0 and f ′′( p ) = 0 , then the test fails. Concavity: If f ′′( p ) > 0 on an interval, then f(x) is concave up on that interval; if f ′′( p ) < 0 , then f(x) is concave down on that interval. A point of inflection occurs when f ′′( p ) changes sign (and thus concavity). • • • n ∫ x ⋅ dx = ∫ dx = ln x + C x ∫ sin xdx = − cos x + C ∫ cos xdx = sin x + C ∫ tan xdx = ln sec x + C b a ∫ kf ( x) ⋅ dx = k ∫ f ( x) ⋅ dx b b b ∫ [ f (x ) ± g (x )]⋅ dx = ∫ f (x ) ⋅ dx ± ∫ g (x ) ⋅ dx a • 0 if , then −a TECHNIQUES OF INTEGRATION Integration by substitution Step 1. Choose a substitution u such that the resulting integral, written in terms of u and du, will be easy to evaluate (easiest with practice). Step 2. Differentiate u with respect to x to find du = u ′(x )dx . Step 3. Replace dx with du u ′(x ) . Step 4. Integrate with respect to u. Step 5. Substitute back to the initial variable x. Integration by parts Step 1. Write the given integral ∫ f ( x) g ( x) ⋅ dx . Step 2. Introduce the intermediary functions u(x) and v(x): ⎧ u = f (x ) ⎨ ⎩dv = g ( x )dx Step 3. Differentiate u and integrate dv to get: ⎧ ⎪ du = f ′( x ) . ⎨v = g ( x )dx ⎪ ∫ ⎩ Step 4. Use the following formula: u (x )dv = u (x )v ( x ) − v( x )du and solve. ∫ ∫ P(x ) Q( x ) Integration by Trigonometric substitution • To be used when the integrand contains the b a • −a Step 1. If the degree (P) ≥ degree (Q), perform polynomial long-division. If it is not, proceed to step 2. Step 2. Factor the denominator Q(x) into irreducible polynomials: linear and irreducible quadratic polynomials. Step 3. Find the partial fraction decomposition. Step 4. Integrate the result of step 3. x n +1 +C n +1 Properties of integration: • w ww.prep101.com • f (− x ) = − f ( x ) f (x ) = ∫ positive at p, then f has a relative minimum at p. If f ’ changes from positive to negative at p, then f has a relative maximum at p. Second derivative test: If f ′( p ) = 0 and f ′′( p ) > 0 , then f has a relative a Partial fractions: • to integrate rational functions of the form Rules of integration: • dx = x + c • a ∫ f (x )dx = 2∫ f (x )dx ∫ f (x )dx = 0 Optimization problems Step 1. Determine what we’re trying to maximize/minimize and write an equation for this. Step 2. Write a second equation from additional information given in the problem, isolate one of the two variables, and substitute this into the first equation. Step 3. Take the derivative of this equation, set it to zero and solve for the remaining variable. Step 4. Plug the value of this variable into the original equations to solve for the remaining variables. First derivative test: If f ′( p ) = 0 and f ’ changes from negative to Odd and even functions: • if f (− x ) = f ( x ) , then a Curve sketching Step 1. Find the intercepts. Step 2. Find all the asymptotes. Step 3. Find critical points and the intervals of increase/decrease. Step 4. Find inflection points and the intervals in which the function is concave up/down. f ( x) f ′( x ) = lim g ( x ) x → a g ′( x) Critical points: The values of x ∈ domain of f(x) such that f ′( x ) = 0 or f ′( x ) is not defined. b ∫ f (x )dx = F (b) − F (a ) a x →a − f ′( x) = The fundamental theorem of Calculus: For F ′(x ) = f (x ) a a b c b a a expressions a 2 ± x 2 or x 2 ± a 2 . o For a 2 − x 2 set x = a sin(t) o for a 2 + x 2 set x = a tan(t) o for x 2 − a 2 set x = a sec(t) c ∫ f ( x) ⋅ dx = ∫ f ( x) ⋅ dx + ∫ f ( x) ⋅ dx Our Course Booklets - free at prep sessions - are the “Perfect Study Guides.” Need help for exams? Check out our classroom prep sessions - customized to your exact course - at www.prep101.com Improper Integration • If the integral exists (and equals, or can be written as, a number), then it’s convergent • If the integral doesn’t exist (is infinite), then it’s divergent The moment about the x-axis is given by b n 1 1 2 2 M x = lim ∑ ρ ⋅ [ f ( xi )] ∆x = ρ ∫ [ f ( x) ] dx n →∞ 2 2 i =1 a SEQUENCES AND SERIES Definitions, Rules and Properties APPLICATIONS OF INTEGRATION Arithmetic: • constant growth a n +1 = a n + d where d is a Area: The area between functions f(x) and g(x) (whose intersection points are a and b, and where b f (x ) > g ( x ) on (a,b)) is given by ∫ [ f ( x ) − g ( x)]dx Average Value The average value of function f(x) on the interval b [a, b] (or (a, b)) is given by . 1 Arc Length The arc length of the function f ( x ) from to x = b is given by b 2 Fibonacci: • defined recursively with a1 = 1 , a 2 = 1 , a n = a n −1 + a n − 2 for n ≥ 3 f= ∫ b−a∫ a f ( x)dx x=a 1 + ( f ′( x )) ⋅ dx a Surface Area The surface area from a to b of the solid obtained by revolving y = f (x ) around the x-axis is given by b 2π ∫ f (x ) ⋅ 1 + ( f ′( x )) ⋅ dx 2 • then the volume is V = ∫ A( x )dx a If the cross section is perpendicular to the yaxis and its area is a function of y, say A( y ) , then the volume is b V = ∫ A( y )dy Algebra of series: N Volumes of solids by revolution: • The volume of the solid obtained by rotating the graph of f x from x = a to x = b () about the y-axis is given by N ∑a ± ∑b n =1 N n ∑a n n =1 N n n is convergent, then ∞ ∑b ∞ ∑a is divergent then n that ai ≥ ai +1 ∞ ∑ (− 1) n +1 i =1 an for all i and lim a = 0 , then n n →∞ converges. Limit comparison test: a if such that lim n = L n →∞ b n a ⋅ (r M −1 − r N ) = 1− r SEQUENCES AND SERIES Convergence Tests a The volume of the solid obtained by rotating the region bounded between curves f ( x ) and g (x ) between x = a and x = b (assume g (x ) > f ( x ) for a < x < b ) about the line y = c π ∫ (c − g ( x )) − (c − f ( x )) dx 2 2 a Centroid: The moment about the y-axis is given by n b i =1 . then and n k ∞ ∑b and L≠∞, converge or diverge n k together. Ratio test: • Given a n > 0 and P-series: • A P-series • then • converges and if L > 1 , n If L = 1 , then the test fails. n →∞ If L < 1 , then ∞ a n = f (n ) where f is continuous, n Root test: • Given a n > 0 and lim(a )1 n = L . n ∞ ∑a i =1 then If converges and if L > 1 , n ∞ ∑ a diverges. L = 1 , then the test fails. i =1 • function ∞ ∑a . a n+1 =L an diverges. ∞ ∑a i =1 1 is convergent if p > 1 , and is ∑ np n =1 divergent if p ≤ 1 . i =1 If L < 1 , then i =1 • Integral test: • let ∞ be a series such that there exists a ∑ ai lim n →∞ n Absolute and conditional convergence: • A series of the form a converges ∑ a M y = lim ∑ ρ xi f ( xi )∆x = ρ ∫ xf ( x)dx ∞ ∑a L≠0 n =1 N ⋅ (N + 1) 2 n =1 N N ⋅ ( N + 1) ⋅ ( 2 N + 1) 2 ∑n = 6 n =1 n=M is is divergent. n ∑n = ∑a ⋅r n Alternating series test: • for {a } a sequence of positive numbers such n n =1 c −1 N n −1 ∞ ∑a i =1 i =1 N n =1 2π ∫ x ⋅ f ( x ) ⋅ dx n →∞ ∞ ∑b convergent. If = ∑ an − ∑ an b [ If i =1 = ∑ (a n ± bn ) Summation formulas: N a b i =1 for all n. {a n }⋅ {bn } = {a n ⋅ bn } n=c b is given by such that 0 < a n ≤ bn n w ww.prep101.com i =1 Algebra of sequences: {a n } ± {b } = {a n ± bn } c ⋅ {a n } = {c ⋅ a n } ∞ ∑b and n • Volume: • If the cross section is perpendicular to the xaxis and its area is a function of x, say A(x ) , • ∞ ∑a for i =1 a • ∫ f (x ) ⋅ dx Comparison test: 1 n converges, then the series ∞ converges and if it diverges, then the series diverges. Geometric: • proportional growth g = rg where r is a n +1 n constant, arbitrary terms are given by g = r n −1 ⋅ g . a If 1 • constant . • n absolutely if the series of absolute values (of the form an ) is convergent. positive and decreasing. ∑ • If it converges absolutely, then the original series a will also converge. ∑ n Our Course Booklets - free at prep sessions - are the “Perfect Study Guides.” ...
View Full Document

Ask a homework question - tutors are online