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26 Cards in this Set
- Front
- Back
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A ________ is a line passing through two points on a curve. The ______________ of the curve from point P to point Q is the same as the ______ of the secant line through those two points. m = ____________ sec m = ____________ sec
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Secant average rate of change slope ƒ(c+Ãx) – ƒ(c) m = —–————–– sec (c+Ãx) – c ƒ(c+Ãx) – ƒ(c) m = ——————– sec Ãx
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Second dummy question
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Second dummy answer
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Third dummy question
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Third dummy answer
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A ________ is a line that touches a curve at a given point and runs along aside that curve. As Ãx approaches zero, the slope of the secant line approaches the slope of the tangent line at point Ñ. The slope of the tangent line is equivalent to the slope (or ___________) of the curve at that point.
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tangent line instantaneous rate of the change
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Imagine the graph of a function ƒ(x). If the y-coordinate (ƒ(x)) becomes arbitrarily close to a single number, L, as x approaches a value c from either side, then the limit of ƒ(x) as x approaches c is equal to L. lim ƒ(x) = L x->c It is important to note that the existence of the function at x=c has no bearing on the existence of the limit of the function as x approaches c. It is possible for the function not to exist at x=c but the limit does exist and it is possible for the function to exist but the limit does not exist. Nonexistence of a Limit These are common types of behavior associated with the nonexistence of a limit: 1. 2. 3.
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1. ƒ(x) approaches a different number from the right side of c than it approaches from the left side. 2. ƒ(x) increases or decreases without bound as x approaches c (i.e. a vertical asymptote). 3. ƒ(x) oscillates between two fixed values as x approaches c.
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Definition of a Limit Let ƒ be a function defined on an 1.__________ containing c and let L be a real number. The statement LIM ƒ(x) = L means that for each 2. ______there exists a 3. ______ x->c such that if 4.________________ then ____________________ The goal in using the 5.______ definition to prove that the limit L is to establish a relationship between 6.__ and 7.__ using only 8.______. The choice of the 9._____is somewhat arbitrary as long as the definition holds and can be proven.
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1. open interval 2. ä>0 3. ã>0 4. if 0 < ûx–cû < ã then ûƒ(x)–Lû < ä 5. ä–ã 6. ä 7. ã 8. constants 9. constants
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Find L = LIM (x² + 4). use the ä – ã definition to prove that the limit is L. x->5 If we graph the function y = x²+4, we can see that this limit looks to equal 29. Now, we need to establish the definition: If 1.___________, then ___________
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1. If 0 < ûx–5û < ã, then ûx² + 4 – 29û < ä. ûx – 5û < ã Since â-->5, we can create an interval around 5, ûx + 5û • ûx – 5û < ûx + 5û • ã say 4 < x < 6. Since this is true, the û(x + 5)(x – 5)û < ûx + 5û • ã absolute value of x+5 must be in the interval 9 < ûx+5û < 11. We now have the following ûx²–25û < ûx+5û • ã statement: ûx² – 25û < ûx + 5û • ã < 11 • ã 1 1 If we allow ã = —ä, then ûx² – 25û < 11( —ä ) which reduces to ûx² – 25û < ä 11 11 Since this is the statement we needed to prove, we have established that the limit is indeed equal to 29.
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Let b and c represent real numbers and let n be a positive integer. 1. LIM b = ___ x->c 2. LIM x = ____ x->c 3. LIM x^n = ____
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1. b 2. c 3. c^n
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Properties of Limits Let b and c represent real numbers and let n be a positive integer. The limit of f (x) as x approaches c is equal to L. The limit of g(x) as x approaches c is equal to K. 1. LIM [bƒ(x)] = ____ x->c 2. LIM (ƒ(x) ± g(x)) = _____ x->c 3. LIM (ƒ(x) • g(x)) = ______ x->c ƒ(x) 4. LIM —— = _________ x->c g(x) n 5. LIM [ƒ(x)]^ = _______ x->c
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1. bL 2. L ± K 3. L • K L 4. – provided that K ¥1 K 5. L^n
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By combining the basic limits, which allow for direct substitution, and the properties of limits, we can establish the following for polynomial and rational functions. Limits of Polynomial and Rational Functions 1. If P(x) is a polynomial function, then LIM P(x) = ____ x->c 2. If R(x) is a rational function, then LIM R(x) = ____ = _____ x->c
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1. P(c) P(c) 2. = R(c) = ——– [provided that Q(c) ¥ 0 Q(c)
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Limits of Radical and Composite Functions n /¯ 1. LIM ¯\/ x = ______ x->c 2. LIM ƒ(g(x)) = __________ x->c
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n/¯ 1. ¯\/c 2. ƒ ( LIM g(x) ) x->c
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Assume that ƒ(x) = x + 7 and g(x) = x² . Find the following limits. a) LIM ƒ(x) x->–3 b) LIM g(x) x->4 c) LIM g(ƒ(x)) x->–3
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a) = (–3) + 7 = 4 b) = (4)² = 16 c) = g(4) = 16
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Limits of Trigonometric Functions Let c be a real number in the domain of the given trigonometric function. 1. LIM sin x = ____ x->c 2. LIM cos x = ____ x->c 3. LIM tan x = ____ x->c 4. LIM csc x = ____ x->c 5. LIM sec x = ____ x->c 6. LIM cot x = ____ x->c
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1. sin c 2. cos c 3. tan c 4. csc c 5. sec c 6. cot c
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So, now what happens when direct substitutiond oes not work (yield a result that is undefined) Let ƒ(x) = g(x) for all x¥c in an open interval containing c. If LIM g(x) exists, then x->c ______ also exists, and ______ = _______ This theorem will be especially useful for ________ functions where the denominator is equal to zero when x is equal to c. If we can reduce the function in such a way that the division by zero is eliminated, then we can find the limit by ____________.
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LIM ƒ(x) x->c LIM ƒ(x) = LIM g(x) x->c x->c rational direct substitution.
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Squeeze Theorem If h(x) < ƒ(x) < g(x) for all x in an open interval containing c, and if ¯ ¯ LIM h(x) = L = ___________, then ________ = ____ x->c
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LIM g(x), then LIM ƒ(x) = L x->c x->c In other words, if we can show that the values of f (x) lie between values of two other functions, and both of those functions have limits that exist and are equal as x approaches c, then the limit of f (x) will also exist as x approaches c and will equal the same value as the other limits.
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The Squeeze Theorem can be used to prove the following two special limits. sin x 1. LIM —— = ___ x->0 x 1 – cos x 2. LIM –———– = ____ x->0 x
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1. 1 2. 0
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Find the limit. 1 – tan x LIM —————– x->ï/4 sin x – cos x
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1 – tan x LIM —————– = x->ï/4 sin x – cos x 1 – tan x cos x = LIM –————–– • ——— x->ï/4 sin x – cos x cos x cos x – sin x = LIM –————––——— x->ï/4 cos x(sin x – cos x) –1(sin x – cos x) = LIM –————––——— x->ï/4 cos x(sin x – cos x) –1 -1 -1 /¯¯ = LIM —— = ———— = —— _— = – ¯\/ 2 x->ï/4 cos x cos(ï/4) 1þ \/2
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A function ƒ is continuous at a point c if the following conditions are true: 1. ƒ(c) is ______ 2. 3. A function ƒ is said to be ___________ if the function is continuous at every point c in the interval.
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1. defined 2. LIM ƒ(x) exists x->c 3. LIM ƒ(x) = ƒ(c) x->c continuous on an open interval (a,b)
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Discontinuities fall into two categories: ___________ and _______. A discontinuity is removable if there is a way to redefine the function so that there is no such discontinuity. An example of this is a hole in the graph of a rational function. That single point can be defined separately from the rest of the function and be “filled in”. __________ are nonremovable.
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removable and nonremovable Vertical Asymptotes.
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x – 3 Discuss the continuity of the function ƒ(x) = ——– x² – 9 Not defined at x = ±3 condition(1) fails at 2 places.
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x – 3 ƒ(x) = ——– = x² – 9 x – 3 = —————— (x + 3)(x – 3) x = 3 is a removable discontinuity (because it cancels/limit still exists) x = –3 is a nonremovable discontinuity (3 didn't go away/undefined & no limit) The function is continuous on the open intervals (–ö,–3)U(–3,3)U(3,ö)
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One-Sided Limits Limit from the Right: LIM ƒ(x) x->c+ Limit from the Left: LIM ƒ(x) x->cÝ The limit of ƒ(x) as x approaches c exists and is equal to L if and only if ______=_______=____
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LIM ƒ(x) = LIM ƒ(x) = L x->c+ x->cÝ
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A function ƒ is continuous on a closed interval [a,b] if the following conditions are true: 1. 2. LIM ƒ(x) = ƒ(a) x->a+ 3. LIM ƒ(x) = ƒ(b) x->bÝ
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1. ƒ is continuous on the open interval (a,b). 2. ƒ(a) 3. ƒ(b)
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Intermediate Value Theorem If ƒ is continuous on the closed interval [a,b] and k is any number between ƒ(a) and ƒ(b), then there is at least one number c in [a,b] such that ________
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ƒ(c) = k In Algebra and Precalculus, you use this theorem to show the existence of a zero whenthe two function values have opposite signs (zero is between a negative and a positive).This is one application of the Intermediate Value Theorem but it can be applied to other values other than just zero.
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Infinite Limits 1. The statement LIM ƒ(x) = ö means that for each _____, there exists a ____ such that x->c ____ whenever ______. 2. The statement LIM ƒ(x) = –ö means that for each ___, there exists a ___ such that x->c ____ whenever ____. You have already learned from precalculus that vertical asymptotes occur when the function increases or decreases without bound as x approaches c. This means that the limit is infinite. We also know that Vertical Asymptotes occur when the denominator of a rational function is equal to zero while the numerator is not equal to zero.
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1. M > 0, ã > 0 ƒ(x) > M 0 < ûx – cû < ö 2. N < 0, ã > 0 ƒ(x) < N 0 < ûx – cû < ö
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x² – 4 Find any vertical asymptotes of the function ƒ(x) = ——————– x³ + 2x² + x + 2
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We need to factor both the numerator and denominator of the function to see if there are any common factors that can be cancelled out. (x + 2)(x – 2) (x + 2)(x – 2) x – 2 = ————————– = ——————– = ——— x² (x + 2) + 1(x + 2) (x + 2)(x² + 1) x² + 1 Since the factor x²+1 cannot equal zero, there are no vertical asymptotes.
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