МИНИСТЕРСТВО ОБРАЗОВАНИЯ И НАУКИ РОССИЙСКОЙ ФЕДЕРАЦИИ Государственное образовательное учреждение высшего профессионального образования САНКТ-ПЕТЕРБУРГСКИЙ ГОСУДАРСТВЕННЫЙ УНИВЕРСИТЕТ АЭРОКОСМИЧЕСКОГО ПРИБОРОСТРОЕНИЯ MINISTRY OF EDUCATION AND SCIENCE OF RUSSIAN FEDERATION SAINT PETERSBURG STATE UNIVERSITY OF AEROSPACE INSTRUMENTATION (SUAI) А. В. Левичев, Ю. Н. Сирота A. V. Levichev, Yu. N. Sirota ВЫСШАЯ МАТЕМАТИКА CALCULUS ПРЕДЕЛЫ, НЕПРЕРЫВНОСТЬ, ДИФФЕРЕНЦИРОВАНИЕ LIMITS, CONTINUITY, DIFFERENTIATION Custom Course Packet Санкт-Петербург 2005 УДК 517.5 ББК 22.161.8 Л34 Левичев, А. В. Сирота, Ю. Н. Л34 Высшая математика. Пределы, непрерывность, дифференцирование: учеб.-метод. пособие / А. В. Левичев, Ю. Н. Сирота; ГУАП. СПб., 2005. 38 с. Рассмотрены понятия предела последовательности и предела функции. Определяется и кратко обсуждается непрерывность функции. Подробно (с примерами и приложениями) рассмотрена важнейшая тема математического анализа – дифференцирование. Изложение следует современным учебникам математического анализа, используемым многими университетами США. В частности, нет необходимости вводить такие понятия, как “бесконечно малая” и “второй дифференциал”. Понятие дифференциала функции (т. е. дифференциальной формы степени один) предполагается ввести в следующем выпуске. Резензент доктор физико-математических наук, профессор В. Г. Фарафонов Утверждено редакционно-издательским советом университета в качестве учебно-методического пособия c ГОУ ВПО «Санкт-Петербургский государственный университет аэрокосмического приборостроения», 2005 1. Limit of the sequence Definition 1 A sequence can be thought of as a list of numbers written in a deﬁnite order y1 , y2 , . . . yn , . . . Definition 2 The number y1 is called the ﬁrst term, y2 is the second term, and in general yn is the nth term. A sequence can also be deﬁned as a function whose domain is the set of all positive integers. We usually write yn (or xn ) instead of the function notation for the (real) value at n. Notation 1 The sequence {y1 , y2 , . . .} is also denoted by {yn } or {yn }∞ n=1 . Definition 3 A sequence {xn } is called increasing if xn+1 > xn for all n 1, that is x1 < x2 < . . . < xn < . . .. It is called decreasing if xn+1 < xn for all n. Definition 4 Non-increasing means xn+1 xn , non-decreasing: xn+1 xn . Definition 5 A sequence xn is called monotonic if it is either non-decreasing or non-increasing. It is called strictly monotonic if it is either decreasing or increasing. Definition 6 A sequence {xn } is called bounded if there is such a number M that |xn | < M for all n 1. 3 Example 1 1 + (−1)n is bounded since |xn | < xn = 1/n is decreasing. xn = 2 2. The sequence xn = n is unbounded. If we let xn be the digit in the nth decimal place of the number e, the Euler number (to be deﬁned later in the text), then xn is a well-deﬁned sequence but it doesn’t have a simple deﬁning equation. Definition 7 Fibonacci sequence xn is deﬁned recursively by the conditions x1 = 1, x2 = 1, xn = xn−1 + xn−2 , n 3. Each term is the sum of the two preceding terms: {1, 1, 2, 3, 5, 8, 13 . . .} This sequence arose when the 13th-century Italian mathematician Fibonacci solved a problem concerning the breeding of rabbits. n Consider the sequence . It appears that its terms are approaching n+1 1 as n becomes large. In fact, the diﬀerence 1 − xn = 1 − 1 n = n+1 n+1 can be made as small as we like by taking n suﬃciently large. Let n us indicate this by writing lim = 1. n→∞ n + 1 In general, the notation lim = L means that the terms of the n→∞ sequence xn approach L as n becomes large. Definition 8 A sequence {xn } has the limit L and we write lim xn = L or n→∞ xn → L as n → ∞ if we can make the terms xn as close to L as we like by taking n suﬃciently large. If lim xn exists, we say the n→∞ sequence converges (or is convergent). Otherwise, we say it diverges (or is divergent). 4 To make the above even more precise: L is the limit of xn , if for an arbitrary positive ε there exists such N (which might depend on ε) that |xn − L| < ε as soon as n N . 1.1. Limit Laws for Convergent Sequences If {an } and {bn } are convergent sequences and c is a constant, then lim (an + bn ) = lim an + lim bn ; n→∞ n→∞ n→∞ n→∞ n→∞ n→∞ lim (an − bn ) = lim an − lim bn ; lim c · an = c lim an ; n→∞ n→∞ lim (an bn ) = n→∞ an = lim n→∞ bn lim an · lim bn ; n→∞ n→∞ lim an n→∞ lim bn n→∞ , if lim bn = 0, n→∞ lim c = c. n→∞ Let us now consider a few situations which involve unbounded sequences. E x a m p√ le 2 √ 1) xn = 1/ n, yn = n, xn · yn = n → ∞; xn · yn = a → a; 2) xn = a/n, yn = n, xn · yn = 1/n → 0; 3) xn = 1/n2 , yn = n, xn · yn = (−1)n , no limit. 4) xn = (−1)n /n, yn = n, In each case, we were in a 0 · ∞ indeterminate situation. A few technicalities (when working with sequences). n2 + 1 . When trying to apply the respective limit law, Let xn = 2 n −1 ∞ we get an indeterminate expression. A way to go is to divide both ∞ 5 the top and the bottom by n2 : 1 + 1/n2 an xn = = , 1 − 1/n2 bn 1 1 where an = 1 + 2 , bn = 1 − 2 . n n Now, the limit law is applicable. We conclude that lim xn = 1. n→∞ Later, when discussing the limit of a function, we will state a theorem which deals with a limit of a rational function (that is, the quotient of two polynomials). √ √ √ Consider one other example. Let xn = n( n + 1 − n). Again, the limit law is inapplicable, we are in ∞ · 0 situation. A key word here is “conjugate” (to try to get rid of one radical, at least). We multiply xn by √ √ n+1+ n 1= √ √ n+1+ n to get a new expression for the nth term: √ n xn = √ √ , n+1+ n still indeterminate expression. We now do something similar to the previous √ example: we divide both the numerator and the denominator by n to get an expression where we conclude easily about the existence of the limit: 1 1 1 n→∞ = . −−−→ √ xn = 2 1+0+1 1 + 1/n + 1 Definition 9 A sequence xn is bounded above if there is a number M such that xn < M for all n 1. It is bounded below if there is a number M such that xn > M for all n 1. 6 (As we have deﬁned earlier, if it is bounded above and below, then xn is called a bounded sequence.) Theorem 1 (Monotonic Sequence Theorem) Every bounded, monotonic sequence has a limit. To be more speciﬁc, if xn increases and is bounded from above then it converges. If it decreases and is bounded from below then it converges. Example 3 Discuss how both of the above statements follow from the just stated MS Theorem. Theorem 2 (Squeeze Theorem for Sequences) If an bn cn for and n n0 and lim an = lim cn = L then n→∞ n→∞ lim bn = L. n→∞ Another useful fact about limits of sequences is given by the following theorem, which follows from the Squeeze Theorem because −|an | an |an |. Theorem 3 If lim |an | = 0, then lim an = 0. n→∞ n→∞ Example 4 n 1 . The base tends to1 as n → ∞, the exponent Let xn = 1 + n goes to ∞. We are thus in a 1∞ indeterminate situation. To convince yourself, just apply the natural logarithm: 1 yn = ln xn = n ln 1 + n to get an indeterminate situation 0 · ∞ (which we have been already dealing with). 7 It can be shown (but it takes time and eﬀort) that xn , n 3 increases and bounded. Hence, it has a ﬁnite limit (which is denoted e in literature). Sometimes this irrational is called the Euler number, e = 2, 71828182 . . . It is an important number (like π, in trigonometry). The function y = ex is called a natural exponential function, loge x is called the natural logarithm of x and is frequently denoted ln x. 2. Limit of a Function Definition 10 We say that L is the limit of f (x) at x = a if the sequence f (xn ) converges to L (as soon as the sequence xn has been chosen to converge to a; here xn = a). x→a We use the notation lim f (x) = L or f (x) −−→ L. x→a It was the deﬁnition of limit of a function using the sequences. Here is an alternative: Definition 11 We write lim f (x) = L and say “the limit of f (x) as x tends to x→a a, equals L” if we can make the values of f arbitrarily close to L (as close to L as we like) by taking x to be suﬃciently close to a (but not equal to a). We can make the last portion even more precise: “if for every number ε > 0 there is a corresponding number δ > 0 such that |f (x) − L| < ε whenever 0 < |x − a| < δ”. The last deﬁnition might be called the “ε, δ” deﬁnition of the limit of a function. To better grasp that fundamental notion, it is strongly recommended to practice with a graphing calculator. Namely, if ε > 0 is given, then we draw the horizontal lines y = L+ε and y = L−ε and the graph of f . If lim f (x) = L, then we can ﬁnd a number δ > 0 such that if we x→a restrict x to lie in the interval (a − δ, a + δ) and take x = a, then the 8 curve y = f (x) lies between the lines y = L − ε and y = L + ε. You can see that if such a δ has been found, then any smaller δ will also work. It is important to realize that such a process must work for every positive number ε no matter how small it is chosen. Typically, if a smaller ε is chosen, then a smaller δ may be required. 2.1. Calculating Limits Using the Limit Laws We can use calculators and graphs to guess the values of limits, but such methods don’t always lead to the correct answer. In this paragraph we use properties of limits, called the Limit Laws, to calculate limits. 2.2. Limit Laws Suppose that c is a constant and the limits lim f (x) and lim g(x) x→a x→a exist. Then 1) lim [f (x) + g(x)] = lim f (x) + lim g(x), x→a x→a x→a x→a x→a x→a 2) lim [f (x) − g(x)] = lim f (x) − lim g(x), 3) lim [cf (x)] = c lim f (x), x→a x→a 4) lim [f (x)g(x)] = lim f (x) lim g(x), x→a x→a x→a lim f (x) f (x) x→a = , if lim g(x) = 0. 5) lim x→a g(x) x→a lim g(x) x→a These ﬁve laws can be stated verbally as follows: 1. The limit of a sum is the sum of the limits. 2. The limit of a diﬀerence is the diﬀerence of the limits. 3. The limit of a constant times a function is that constant times the limit of the function. 4. The limit of a product is the product of the limits. 9 5. The limit of a quotient is the quotient of the limits (provided the limit of the denominator is not 0). These laws are quite similar to the respective laws of sequences stated earlier. Here are some more statements and examples about functions. Definition 12 We write lim− f (x) = L or lim f (x) = L or f (a − 0) = L x→a−0 x→a and say the left-hand limit of f (x) as x approaches a (or the limit of f (x) as x approaches a from the left ) is equal to L if we can make values of f (x) as close to L as we like by taking x to be suﬃciently close to a and x less than a. The symbol “x → a− ” indicates that we consider only values of x that are less than a. Similarly, if we require that x be greater than a, we get “the right-hand limit of f (x) as x approaches a is equal to L” and we write lim f (x) = L or x→a+ lim f (x) = L or f (a + 0) = L. x→a+0 All main properties hold for one-sided limits, too. The following is a true statement (it might be an eﬀective tool when discussing a limit of a given function): lim f (x) = L if and only if lim f (x) = L and lim f (x) = L. x→a−0 x→a x→a+0 Theorem 4 If f (x) g(x) when x is near a (except possibly at a) and the limits of f (x) and g(x) both exist as x approaches a, then lim f (x) lim g(x). x→a x→a Theorem 5 (The Squeeze Theorem) If f (x) g(x) h(x) when x is near a (except possibly at a) and lim f (x) = lim h(x) = L then lim g(x) = L. x→a x→a x→a 10 Example 5 1 Show that lim x2 sin = 0. x→0 x The Euler number e has been earlier introduced as a limit of a certain sequence. us notice that the same number can be deﬁned Let x 1 as either lim 1 + or lim (1 + y)1/y . x→∞ y→0 x Here is an example how to use it when dealing with certain indeterminate expressions. Example 6 x Let f (x) = 1 + x3 . It is an indeterminate expression of the form 1∞ when x → ∞. To deal with it, introduce y = 3/x. Clearly, y → 0 as x → ∞. Now, 3 x 3 = lim (1 + y)3/y = lim (1 + y)1/y = e3 . lim 1 + x→∞ y→0 y→0 x Here are a few more limits to remember: sin x x→0 x logα (1 + x) lim x→0 x (1 + x)α − 1 lim x→0 x αx − 1 lim x→0 x lim 3. = 1, = logα e = 1 , ln α = α, = ln α. Continuity You can notice that the limit of a function as x approaches a can often be found simply by calculating the value of the function at a. Functions with this property are called continuous at a. We will see 11 that the mathematical deﬁnition of continuity corresponds closely with the meaning of the word continuity in everyday language. (A continuous process is one that takes place gradually, without interruption or abrupt change.) Definition 13 A function f (x) is continuous at a number a if lim f (x) = f (a). x→a If f (x) is not continuous at a, we say f (x) is discontinuous at a, or f (x) has discontinuity at a. Notice that Deﬁnition implicitly requires three things if f (x) is continuous at a: 1. f (a) is deﬁned (that is, a is in the domain of f (x)). 2. lim f (x) exists (so f (x) must be deﬁned on an open interval x→a that contains a). 3. lim f (x) = f (a). x→a The deﬁnition says that f (x) is continuous at x0 if f (x) approaches f (x0 ) as x approaches x0 . Some texts introduce the increment of x as x−x0 . The corresponding change in y = f (x) is ∆y = f (x) − f (x0 ). Assume that y = f (x) is deﬁned in some open interval containing x0 . Clearly, it is continuous at x0 if lim ∆y = 0 or, which is the ∆x→0 same, lim [f (x0 + ∆x) − f (x0 )] = 0. ∆x→0 We can distinguish between diﬀerent kinds of discontinuity at x0 . Example 7 Give an example (graphically, or by a formula, or both) of a function which 12 1) has a removable discontinuity; at x0 . 2) has an inﬁnite discontinuity at x0 ; 3) has a jump discontinuity at x0 . The following statement is often helpful. Theorem 6 Each elementary function is continuous at every number in its domain. Remark 1 There are main elementary functions which include root functions y = xa , a is a real number; exponential functions y = bx , b > 0, logarithmic functions y = loga x, positive a = 1, trigonometric functions y = sin x, y = cos x, y = tan x, y = cot x or y = ctgx, y = sec x, y = csc x or y = cosecx; and inverse trigonometric functions y = arcsin x, y = arccos x, y = arctgx, y = arcctg x, y = arcsecx, y = arccosecx. As it has been just made clear, the Western notation for some of trigonometric and inverse trigonometric functions might diﬀer slightly from the Russian one. Also, one can write sin−1 x for arcsin x, etc. This is due to a standard way to denote by f −1 an inverse function for f . y = f (x) is called an elementary function if it can be obtained from a ﬁnite number of main elementary functions (and, possibly, constants) by applying (a ﬁnite number of) operations of addition, subtraction, multiplication, division, and composition. We also distinguish between algebraic (they are necessarily elementary functions) and trancedental (or non-algebraic) functions. Having made the above reminder (about diﬀerent classes of functions), we now return to continuity. Definition 14 A function f (x) is continuous from the right at x = a if lim f (x) = f (a), x→a+ or 13 f (a + 0) = f (a) and f (x) is continuous from the left at x = a number if lim f (x) = f (a), x→a− or f (a − 0) = f (a) (as you have noticed earlier, the Western notation for one-sided limits diﬀers slightly from the Russian one). Definition 15 A function f (x) is continuous on an interval if it is continuous at every number in the interval. (At an endpoint of the interval we understand continuous to mean continuous from the right or continuous from the left. Example 8 1 1. A function f (x) = arctan is not deﬁned at 0. We conclude x that it is discontinuous at 0. We cannot remove this discontinuity since lim arctan x→0− 1 π =− , x 2 lim arctan x→0+ 1 π = . x 2 sin x is not deﬁned at 0. This is a removable 2. A function y = x discontinuity since sin x = 1. x→0 x f (−0) = f (+0) = lim 1 has a jump discontinuity at x = 1. x−1 Clearly, y > 0 if x > 1 and y < 0 if x < 1 from where we conclude about one-sided limits f (1 + 0) = +∞, f (1 − 0) = −∞, and, ﬁnally, about the jump discontinuity at x = 1 (which cannot be removed). 3. A function f (x) = 1 4. A function f (x) = sin has one other type of discontinuity at x 0. 14 4. Derivatives 4.1. Velocities Suppose an object moves along a straight line according to an equation of motion s = f (t), where s is the displacement (directed distance) of the object from the origin at time t. The function s is called the position function of the object. In the time interval from t to t + ∆t the change in position is ∆s = f (t + ∆t) − f (t). The average velocity over this time interval is average velocity = f (t + ∆t) − f (t) ∆s displacement = = . time ∆t ∆t (If we draw the curve s = f (t) in the t, s–plane, then this number is the slope of the secant P Q, with P = (t, f (t)), Q = (t + ∆t, f (t + ∆t)). Suppose now that we compute the average velocities over shorter and shorter time intervals [t, t + ∆t]. In other words, we let ∆t approach 0. We deﬁne the velocity (or instantaneous velocity) v(t) at time t to be the limit of these average velocities: f (t + ∆t) − f (t) . ∆t→0 ∆t v(t) = lim This means that the velocity at time t is equal to the slope of the tangent line at P , see Section 4.3. Example 9 Suppose an object moves vertically (the Galileo’s Law reads: s = 1/2gt2 here a constant g equals 9.8m/sec2 ). Find its velocity at t; at t = 2. 15 Solution Find the s -increment: 1 1 1 ∆s = g(t + ∆t)2 − gt2 = gt∆t + g(∆t)2 . 2 2 2 ∆s 1 Arrange for the diﬀerence quotient: = gt + g∆t. Find the ∆t 2 limit of the diﬀerence quotient as ∆t goes to zero: ∆s = gt. ∆t→0 ∆t v = lim A n s w e r: v = gt; the value of this function (at t = 2) is 19.6m/sec. 4.2. Deﬁnition of the Derivative Definition 16 The derivative of a function y = f (x) at x = a, denoted by f (a) is f (a + h) − f (a) h→0 h f (a) = lim ∆y ∆x→0 ∆x or f (a) = lim if this limit exists. One can use x for a, and ∆x for h to get f (x), derivative at x. In that case, f (x + ∆x) − f (x) ∆y = lim . ∆x→0 ∆x→0 ∆x ∆x f (x) = lim Clearly, f (x) is, in itself, a function of x. It should always be clear from the context whether we look for a number f (a) (in the case x = a is a chosen number) or for a function f (x) (when we allow a to vary). Also, f (a) is read: ”f prime of a“ (Newton’s notation for the derivative). 16 4.3. Tangents If a curve C has equation y = f (x) and we want to ﬁnd the tangent to C at the point P (x, f (x)), then we consider a nearby point Q(x + ∆x, f (x + ∆x)) where ∆x = 0 and compute the slope of the secant line P Q: f (x + ∆x) − f (x) . mP Q = ∆x Then we let Q approach P along the curve C by letting ∆x approach 0. If the slope mP Q approaches a number m, then we deﬁne the tangent T to be the line through P with slope m. (This amounts to saying that the tangent line is the limiting position of the secant line P Q as Q approaches P ). Remark 2 Geometrically, the slope equals tan ϕ, where ϕ is an angle between the positive x-semiaxis and the tangent T . 4.4. Other notations Some common alternative notations for the derivative are as follows: f (x) = y = d dy = f (x) = Dx f (x). dx dx d The symbols Dx and are called diﬀerential operators because dx they indicate the operation of diﬀerentiation, which is the process dy of calculating a derivative. The symbol which was introduced dx by Leibniz, should not be regarded as a ratio (for the time being); it is simply a synonym for f (x). Nonetheless, it is a very useful and suggestive notation, especially when used in conjunction with increment notation. 17 dy If we want to indicate the value of a derivative in Leibniz dx dy dy notation at a speciﬁc number a, we use the notation or dx dx x=a x=a which is a synonym for f (a). Definition 17 A function f (x) is diﬀerentiable at a if f (a) exists. It is diﬀerentiable on an open interval (a, b) [or (a, +∞) or (−∞, a) or (−∞, +∞)]. if it is diﬀerentiable at every number in the interval. Both continuity and diﬀerentiability are desirable properties for a function to have. The following theorem shows how these properties are related. Theorem 7 If f is diﬀerentiable at a, then f is continuous at a. Proof Clearly, the diﬀerence f (x) − f (x0 ) tends to zero (where x tends to x0 ) since the limit of the diﬀerence quotient exists. So, f (x) tends to f (x0 ) as x goes to x0 . This means continuity of f at x0 . How can a function fail to be diﬀerentiable? The above theorem implies that if f is not continuous at a, then f is not diﬀerentiable at a. So at any discontinuity f fails to be diﬀerentiable. In the examples (below) we discuss two situations where a function is continuous at a given number but it fails to be diﬀerentiable there. E x a m p l e 10 Consider the following function: x, if 0 x 1, f (x) = 2x − 1, if 1 < x 2. Let us show that y = f (x) is continuous at 1, but it is not diﬀerentiable there. 18 If we stay with positive ∆x, then ∆y = f (1+∆x)−f (1) = 2(1+∆x)−1−1 = 2∆x, ∆y ∆x→0 = 2 −−−→ 2. ∆x If ∆x < 0, ∆y = f (1 + ∆x) − f (1) = 1 + ∆x − 1 = ∆x, ∆y ∆x→0 = 1 −−−→ 1 ∆x We have proven that the derivative at 1 does not exist. However, one-sided lim f (x), lim f (x) limits are both equal to the value x→1+0 x→1−0 (being 1) of f at x = 1, which means continuity. E x a m p l e 11 √ One other example is provided by the function y = 3 x at x = 0. It turns out that the limit of the diﬀerence quotient at x = 0 is ∞, which means our function is not diﬀerentiable at 0. Clearly, it is continuous everywhere. 4.5. Other Rates of Change Suppose y is a quantity that depends on x : y = f (x). If x changes from x1 to x2 , then the change in x (or the increment of x) is ∆x = x2 − x1 and the corresponding change in y is ∆y = f (x2 ) − f (x1 ). The diﬀerence quotient ∆y f (x2 ) − f (x1 ) = ∆x x2 − x1 is called the average rate of y with respect to x over the interval [x1 , x2 ] and can be interpreted as the slope of the respective secant line (if we graph the function). By analogy with velocity, we consider the average rate of change over smaller and smaller intervals by letting x2 approach x1 and therefore letting ∆x approach 0. 19 The limit of these average rates of change is called the (instantaneous) rate of change of y with respect to x at x = x1 , which is interpreted as the slope of the tangent to the curve y = f (x) at P (x1, f (x1 )): ∆y f (x2 ) − f (x1 ) = lim . x2 →x1 ∆x→0 ∆x x2 − x1 inst.rate of change = lim We recognize this limit as being the derivative of f (x) at x1 , that is f (x). This gives one more interpretation of the derivative: The derivative f (a) is the instantaneous rate of change of y = f (x) with respect to x when x = a. The velocity of a particle is the rate of change of displacement with respect to time. Physicists are interested in other rates of change as well – for instance, the rate of change of work with respect to time (which is called power). Chemists who study a chemical reaction are interested in the rate of change in the concentration of a reactant with respect to time (called the rate of reaction). A steel manufacturer is interested in the rate of change of the cost of producing x tons of steel per day with respect to x (called the marginal cost). A biologist is interested in the rate of change of the population of a colony of bacteria with respect to time. In fact, the computation of rates of change is important in all of the natural sciences, in engineering, and even in social sciences. 4.6. Derivatives of elementary functions Derivative of a constant is zero: The Power Rule: d (c) = 0. dx d n (x ) = nxn−1 , here n is any real number. dx The Constant Multiple Rule: d dy (cy) = c , y = f (x). dx dx The sum rule: [f (x) + g(x)] = f (x) + g (x). 20 The Diﬀerence Rule: [f (x) − g(x)] = f (x) − g (x). Derivative of the Natural Exponential Function: (ex ) = ex . The Product Rule: [f (x)g(x)] = f (x)g(x) + f (x)g (x). f (x)g(x) − f (x)g (x) f (x) = . The Quotient Rule: g(x) g 2 (x) Derivatives of Trigonometric Functions: (cos x) = − sin x, (sin x) = cos x, (cot x) = − csc2 x, (tan x) = sec2 x, (csc x) = − csc x cot x, (sec x) = sec x tan x. 4.7. The Chain Rule If f (x) and g(x) are both diﬀerentiable and h(x) = f (g(x)), then h is diﬀerentiable and h is given by the product h (x) = f (g(x)) × g (x). 4.8. Implicit Diﬀerentiation Some functions are deﬁned implicitly by a relation between x and y such as (1) x2 + y 2 = 25 or x3 + y 3 = 6xy. (2) In some cases it is possible to solve such an equation for y as an explicit function (or several √ functions) of x. For instance, if we solve (1) for y, we get y = ± √25 − x2 so two functions √ determined by the implicit (1) are f (x) = 25 − x2 and g(x) = − 25 − x2 . It’s not easy to solve (2) for y explicitly as a function of x by hand. Nonetheless, 2 is the equation of a certain curve and it implicitly 21 deﬁnes y as several functions of x. When we say that f is a function deﬁned implicitly by (2), we mean that the equation x3 + (f (x))3 = 6xf (x) is true for all values of x in the domain of f . Fortunately, it is not necessary to solve an equation for y in terms of x in order to ﬁnd the derivative of y. Instead we can use the method of implicit diﬀerentiation. This consists of diﬀerentiating both sides of the equation with respect to x and then solving the resulting equation for y . E x a m p l e 12 dy . If x2 + y 2 = 25, ﬁnd dx Diﬀerentiating both sides of the equation d 2 d (x + y 2 ) = (25). dx dx Remembering that y is a function of x and using the Chain Rule, we have 2x + 2yy = 0. (x2 ) + (y 2 ) = 0, x Now we solve for y : y = − . y E x a m p l e 13 Find the equation of the tangent to the circle x2 + y 2 = 25 at the point (3, −4). S o l u t i o n 1. At the point (3, −4) we have x = 3 and y = −4, so y = 3/4. An equation of the tangent to the circle at (3, −4) is therefore 3 y + 4 = (x − 3) or 3x − 4y = 25. 4 S√ o l u t i o n 2. Solving the equation x2 + y 2 = 25, we get y = − x2 . The point (3, −4) lies on the lower semicircle√ ± 25√ y = − 25 − x2 and so we consider the function f (x) = − 25 − x2 . 22 Diﬀerentiating f (x) using the Chain Rule, we have 1 1 −2x f (x) = − (25 − x2 )−1/2 (25 − x2 ) = − √ = 2 2 25 − x2 3 x , f (3) = . =√ 4 25 − x2 and, as in Solution 1, the equation of the tangent is 3x − 4y = 25. Remark 3 Examples 1, 2 illustrate that even when it is possible to solve an equation explicitly for y in terms of x, it may be easier to use implicit diﬀerentiation. Remark 4 x The expression y = − gives the derivative in terms of both x y and y. It is correct no matter which function y is determined by the given equation. Exercise 1 • Find y if x3 + y 3 = 6xy. • Find the tangent to this curve at the point (3, 3). • At what points on the curve is the tangent line horizontal or vertical? E x a m p l e 14 We can use the Chain Rule to diﬀerentiate an exponential function with any base a > 0. Recall that a = eln a . So ax = (eln a )x = e(ln a)x and the Chain Rule gives (ax ) = (e(ln a)x ) = e(ln a)x [(ln a)x] = e(ln a)x ln a = ax ln a because ln a is a constant. So we have the formula d x (a ) = ax ln a. dx 23 4.9. Diﬀerentiation of the Inverse Function Let the domain of the function y = f (x) be an open interval (a, b). Assume, for certainty, that it is an increasing function (for a decreasing function the presentation will be quite similar). Such a function performs a one-to-one correspondence between the domain and the range. It is a criterion for the inverse function to exist. Definition 18 Let f be a one-to-one function with domain A and range B. Then its inverse function h has domain B and range A and is deﬁned by h(y) = x ⇔ f (x) = y for any y in B. Remark 5 Frequently, the symbol f −1 is used to denote the inverse function. The latter “undoes”, so to say, the eﬀect of the original function. Notice that domain of f −1 = range of f , range of f −1 = domain of f. Also, the cancellation equations hold: f −1 (f (x)) = x for every x in A, f (f −1 (x)) = x for every x in B. Remark 6 How to Find the Inverse Function to a One-To-One Function f. Step 1. Write y = f (x). Step 2. Solve this equation for x in terms of y (if possible). Step 3. To express f −1 as a function of x, interchange x and y. The resulting equation is y = f −1 (x). 24 E x a m p l e 15 Find the inverse function of f (x) = x3 + 2. According to the previous Remark we ﬁrst write y = x3 +2. Then √ we solve this equation for x : x3 = y − 2, x =√ 3 y − 2. Finally, we interchange x√and y : y = 3 x − 2. Therefore, the inverse function is f −1 (x) = 3 x − 2. Remark 7 The graph of f −1 is obtained by reﬂecting the graph of f about the line y = x. Theorem 8 If there is an inverse function x = h(y) for a given function y = f (x), and if the derivative f (x) is not zero at a given number x, then the inverse function h is diﬀerentiable at the number y = f (x) and 1 h (y) = . f (x) E x a m p l e 16 Derivatives of Inverse Trigonometric Functions Recall that y = sin−1 x means sin y = x and −π/2 y π/2. Diﬀerentiating sin y = x implicitly with respect to x, we obtain cos y dy = 1 or dx dy 1 = . dx cos y Now cos y 0, since −π/2 y π/2, so √ 2 cos y = 1 − sin y = 1 − x2 . Therefore 1 1 dy = =√ , 2 dx cos y 1−x d 1 (sin−1 x) = √ . 2 dx 1−x The formula for the derivative of the inverse tangent function is derived in a similar way. If y = tan−1 x, then tan y = x. Diﬀerentiating 25 this latter equation implicitly with respect to x, we have sec2 y dy 1 1 1 = = = , dx sec2 y 1 + tan2 y 1 + x2 dy =1 dx d 1 (tan−1 x) = . dx 1 + x2 Here are the remaining diﬀerentiation formulas d −1 (cos−1 x) = √ , 2 dx 1−x 4.10. −1 d (arcctg x) = . dx 1 + x2 Parametric Curves Reminder The Vertical Line Test (or VLT): a curve in the xy−plane is the graph of a function of x if and only if no vertical line intersects the curve more than once. Imagine that a particle moves along a curve C in the xy−plane. It is, sometimes, impossible to describe C by an equation of the form y = f (x) because C fails the Vertical Line Test. But the x− and y−coordinates of the particle are functions of time and so we can write x = x(t) and y = y(t). Such a pair of functions is often a convenient way of describing a curve and gives rise to the following deﬁnition. Suppose that x and y are both given as continuous functions of a third variable t (called a parameter ) by the equations x = x(t), y = y(t) (called parametric equations). Each value of t determines a point (x, y), which we can plot in a coordinate plane. As t varies, the point (x(t), y(t)) varies and traces out a curve C, which we call a parametric curve. The parameter t does not necessarily represent time. Other letter can be used for the parameter. However, in many applications of parametric curves, t does denote time and we can interpret (x, y) = (x(t), y(t)) as the position of a particle at time t. 26 Remark 8 Sometimes, new letters are introduced to denote the two functions of the parameter t. Like, (x, y) = (f (t), g(t)) which means x = f (t) and y = g(t), two given functions of t. To simplify, one just stays with x(t), y(t). E x a m p l e 17 Sketch and identify the curve deﬁned by the parametric equations x = t2 − 2t, y = t + 1. Each value of t gives a point on the curve (it is advised to arrange for the respective table of inputs and outcomes). For instance, if t = 0, then x = 0, y = 1 and so the corresponding point is (0, 1). We plot the points (x, y) determined by several values of the parameter and we join them to produce a curve. E x a m p l e 18 You are advised to plot the circle centered at the origin in order to convince yourself that parametric equations x = R cos t, y = R sin t; 0 t 2π deﬁne such a circle (with radius R). The parameter t is the angle (in radians) between the positive x−semiaxis and the vector OM , where M = (x, y). One can convert the above parametric equations into a general equation of a circle by squaring parametric expressions and adding them: x2 + y 2 = R2 (cos2 t + sin2 t) or x2 + y 2 = R2 . E x a m p l e 19 Let an ellipse be given by its standard equation x2 y 2 + 2 = 1. a2 b Introduce x = a cos t, y = b sin t; 0 t 2π. These are the ellipse parametric equations. 27 4.11. Tangents to Parametric Curves Let a parametric curve be given by equations x = x(t), y = y(t). Suppose both functions are diﬀerentiable and we want to ﬁnd the tangent line at a point on the curve where y is alsoa diﬀerentiable dy dx dy = . If function of x. Then the Chain Rule gives dt dx dt dy dy dy dx dx dx = 0, we can solve for : = : if = 0. dt dx dx dt dt dt E x a m p l e 20 Consider a parametric circle (part of) given by x = a cos t, y = dy π a sin t; 0 < t < π. Find : 1) at an arbitrary t; 2) at t = . dx 4 (a sin t) a cos t 1) yx = = − ctg t; = (a cos t) −a sin t π 2) yx t=π/4 = − ctg = −1. 4 E x a m p l e 21 Find an equation of the tangent line to the parametric curve √ x = 2 sin 2t, y = 2 sin t at the point ( 3, 1). Where does this curve have horizontal or vertical tangent? At the point with parameter value t, the slope is dy dy/dt 2 cos t cos t = = = . dx dx/dt 2(cos 2t)2 2 cos 2t √ π The point ( 3, 1) corresponds to the parameter value t = , so 6 the slope of the tangent at that point is √ √ cos π6 dy 3/2 3 = = = . dx t=π/6 2 cos π3 2 2 · 12 An equation of the tangent line is therefore √ √ √ 3 3 1 y−1= (x − 3) or y = x− . 2 2 2 28 dy = 0, which occurs when dx π π or 3 . Thus, the cos t = 0 (and cos 2t = 0), that is, when t = 2 2 curve has horizontal tangents at the points (0, 2) and (0, −2). dx = 4 cos 2t = 0 (and cos t = 0), The tangent is vertical when dt π π π π that is, when t = , 3 , 5 , or 7 . The corresponding four points 4 √4 4 √ 4 √ √ on the curve are (2, 2), (−2, 2), (2, − 2), (−2, − 2). The tangent line is horizontal when 4.12. Hyperbolic Functions Certain combinations of the exponential functions ex and e−x arise so frequently in mathematics and its applications that they deserve to be given special names. Here we deal with hyperbolic functions: ex + e−x ex − e−x , cosh x = , sinh x = 2 2 1 sinh x , sechx = . cosh x cosh x The reason for the names of these functions is that they are related to the hyperbola in much the same way that the trigonometric functions are related to the circle. Texts in Russian use notation ch x sh x, ch x, thx, cthx = . Here are some identities similar to trish x gonometric ones: tanh x = • ch2 x − sh2 x = 1, • ch(a + b) = ch a ch b + sh a sh b, • sh(a + b) = sh a ch b + ch a sh b. The proof is straightforward. These functions can be used to parameterize a hyperbola: x = ch t, y = sh t. These equations are similar to parametric equations of a unit circle (introduced earlier). 29 4.13. The Second Derivative The second derivative f of a function f is deﬁned as (f ) . Using Leibniz notation, we write the second derivative of y = f (x) as d dy d2 y = 2. dx dx dx In general, we can interpret a second derivative as a rate of change of a rate of change. The most famous example of this is acceleration, which we deﬁne as follows. If s = s(t) is the position function of an object that moves in a straight line, we know that its ﬁrst derivative is the velocity v(t) of the object ds v(t) = s (t) = . dt The instantaneous rate of change of velocity with respect to time is called the acceleration a(t) of the object. Thus, the acceleration function is the derivative of the velocity function and is therefore the second derivative of the position function: a(t) = v (t) = s (t) dv d2 s or, in Leibniz notation, a = = 2. dt dt 5. 5.1. More Applications of the Derivative Certain Theorems About Diﬀerentiable Functions Theorem 9 (Rolle’s Theorem) If f (x) is continuous on [a, b], and it is diﬀerentiable on (a, b), and f (a) = f (b) = 0, then there is at least one number c inside (a, b) with the property f (c) = 0. Remark 9 It is enough to assume f (a) = f (b). 30 Theorem 10 (Lagrange’s Theorem) If f (x) is continuos on [a, b] and it is diﬀerentiable in (a, b), then there is at least one c inside (a, b) with the property f (b) − f (a) = f (c)(b − a). Theorem 11 (Cauchy’s Theorem) Let f (x), g(x) be diﬀerentiable in (a, b) and they are continuous on [a, b]. Assume that g (x) is never zero for x from (a, b). Then there exists c inside (a, b) with the property f (b) − f (a) f (c) = . g (c) g(b) − g(a) 5.2. Limits Involving Inﬁnity Definition 19 Let f (x) be a function deﬁned on both sides of a, except possibly at a. Then lim f (x) = ∞ means that for every number M there exists x→a such δ > 0 that f (x) > M whenever 0 < |x − a| < δ. Notation 2 Typically, we write just ∞ for +∞; the notion lim f (x) = −∞ x→a is deﬁned similarly to the just deﬁned lim f (x) = +∞. x→a 5.3. Indeterminate Forms and L’Hospital’s Rule Suppose we are trying to analyze the behavior of the function h(x) = ln x . Although h is not deﬁned when x = 1, we need to know how x−1 h behaves near 1. In particular, we would like to know the value of the limit ln x lim . (3) x→1 x − 1 But we can’t apply the respective Law of limits (the limit of a quotient is the quotient of the limits) to (3) because the limit of the 31 denominator is 0. In fact, although the limit in (3) exists, its value is not obvious because both numerator and denominator approach 0 and 0/0 is not deﬁned. f (x) In general, if we have a limit of the form lim where both x→a g(x) f (x) → 0 and g(x) → 0 as x → a, then this limit may or may not exist and is called an indeterminate form of type 0/0. You could have met some limits before. For rational functions, we can cancel common factors: x2 − x x(x − 1) x 1 = lim = lim = . lim x→1 x − 1 x→1 (x − 1)(x + 1) x→1 x + 1 2 sin x = 1. x→0 x But these methods do not work for limits such as (3), so we now introduce a systematic method, known as l’Hospital’s Rule, for the evaluation of indeterminate forms. Another situation in which a limit is not obvious occurs when we look for a horizontal asymptote of f (x) and need to evaluate the limit ln x lim . (4) x→∞ x − 1 It is not obvious how to evaluate this limit because both numerator and denominator become large as x → ∞. There is a struggle between numerator and denominator. If the numerator wins, the limit will be ∞; if the denominator wins, the answer will be 0. Or there may be some compromise, in which case the answer may be some ﬁnite positive number. In general, if we have a limit of the form A geometric argument can be used to show lim f (x) , x→a g(x) lim where both f (x) → +∞ (or −∞) and g(x) → +∞ (or −∞), then the limit may or may not exist and is called an indeterminate form 32 ∞ of type . For certain functions, this type of limit can be evaluated ∞ by dividing numerator and denominator by the highest power of x that occurs. For instance, x2 − 1 1 − 1/x2 1−0 1 = = lim = . lim x→∞ 2x2 + 1 x→∞ 2 + 1/x2 2+0 2 This method does not work for limits such as (4), but l’Hospital’s Rule also applies to this type of indeterminate forms. L’Hospital’s Rule. Suppose f and g are diﬀerentiable and g (x) = 0 near a (except possibly at a). Suppose that lim f (x) = 0 and x→a lim g(x) = 0 or that lim f (x) = ±∞ and lim g(x) = ±∞. (In other x→a x→a 0 ∞ words, we have an indeterminate form of type or .) Then 0 ∞ x→a f (x) f (x) = lim , lim x→a g(x) x→a g (x) if the limit on the right side exists (or is ∞ or −∞). R e m a r k 10 L’Hospital’s Rule says that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives. It is important to verify the conditions regarding the limits of f and g before using l’Hospital’s Rule. R e m a r k 11 L’Hospital’s Rule is also valid for one-sided limits and for limits at inﬁnity or negative inﬁnity; that is, “x → a” can be replaced by any of the following symbols: x → a+ , x → a− , x → ∞, x → −∞. For the special case in which f (a) = g(a) = 0, f and g are continuous, and g (a) = 0, it is easy to prove the l’Hospital’s Rule. 33 We have: f (x) − f (a) f (x) f (x) − f (a) f (x) f (a) x→a x−a = = = lim . lim = lim x→a g (x) x→a g(x) − g(a) x→a g(x) g(x) − g(a) g (a) lim x→a x−a lim It is more diﬃcult to prove the general version of l’Hospital’s Rule. E x a m p l e 22 ln x . x→1 x − 1 Since lim ln x = ln 1 = 0 and lim (x − 1) = 0, we can apply x→1 x→1 l’Hospital’s Rule: Find lim ln x 1/x 1 = lim = lim = 1. x→1 x − 1 x→1 1 x→1 x lim E x a m p l e 23 sin 5x . x→0 3x Find lim sin 5x (sin 3x) 5 cos 5x 5 lim = lim = lim = . x→0 3x x→0 (3x) x→0 3 3 E x a m p l e 24 (l’Hospital’s Rule applied more than once) ex Find lim 2 . x→∞ x ex ex ex lim = lim = lim = ∞. x→∞ x2 x→∞ 2x x→∞ 2 34 6. Linear Approximations An equation of the tangent line to the curve y = f (x) at (a, f (a)) is y = f (a) + f (a)(x − a) Definition 20 The approximation f (x) ≈ f (a) + f (a)(x − a) is called the linear approximation or tangent line approximation of f at a, and the function L(x) = f (a) + f (a)(x − a) (whose graph is the tangent line) is called the linearization of f at a. The linear approximation is a good approximation when x is near a. E x a m p l e 25 √ Find the linearization of the function f (x) = x + 3 at x = 1 √ √ and use it to approximate the√numbers 3.98 and 4.05 The derivative of f (x) = x + 3 is 1 1 f (x) = (x + 3)−1/2 = √ 2 2 x+3 and so we have f (1) = 2 and f (1) = 1/4. Putting these values into the expression for L(x), we see that the linearization is 7 x 1 L(x) = f (1) + f (1)(x − 1) = 2 + (x − 1) = + . 4 4 4 √ 7 x The corresponding linear approximation is x + 3 ≈ + . In 4 4 particular, we have √ √ 7 0.98 7 1.05 3.98 ≈ + 4.05 ≈ + = 1.995 and = 2.0125. 4 4 4 4 35 Bibliography 1. Пискунов Н. С. Дифференциальное и интегральное исчисление. М.: Наука, 1975. Т. 1. 2. Кремер Н. Ш. и др. Высшая математика для экономистов. М.: ЮНИТИ, 2000. 3. Виленкин Н. Я., Мордкович А. Г. Производная и интеграл. М.: Просвещение, 1976. 4. Фихтенгольц Г. М. Курс дифференциального и интегрального исчисления. СПб.: Лань, 1997. 36 Contents 1. Limit of the sequence 1.1. Limit Laws for Convergent Sequences . . . . . . . . . 3 5 2. Limit of a Function 2.1. Calculating Limits Using the Limit Laws . . . . . . . 2.2. Limit Laws . . . . . . . . . . . . . . . . . . . . . . . 8 9 9 3. Continuity 11 4. Derivatives 4.1. Velocities . . . . . . . . . . . . . . . . 4.2. Deﬁnition of the Derivative . . . . . . 4.3. Tangents . . . . . . . . . . . . . . . . . 4.4. Other notations . . . . . . . . . . . . . 4.5. Other Rates of Change . . . . . . . . . 4.6. Derivatives of elementary functions . . 4.7. The Chain Rule . . . . . . . . . . . . . 4.8. Implicit Diﬀerentiation . . . . . . . . . 4.9. Diﬀerentiation of the Inverse Function 4.10. Parametric Curves . . . . . . . . . . . 4.11. Tangents to Parametric Curves . . . . 4.12. Hyperbolic Functions . . . . . . . . . . 4.13. The Second Derivative . . . . . . . . . . . . . . . . . . . . . . 15 15 16 17 17 19 20 21 21 24 26 28 29 30 5. More Applications of the Derivative 5.1. Certain Theorems About Diﬀerentiable Functions . . 5.2. Limits Involving Inﬁnity . . . . . . . . . . . . . . . . 5.3. Indeterminate Forms and L’Hospital’s Rule . . . . . . 30 30 31 31 6. Linear Approximations 35 Bibliography 36 37 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Учебное издание Левичев Александр Владимирович Сирота Юрий Наумович ВЫСШАЯ МАТЕМАТИКА Пределы, непрерывность, дифференцирование Учебно-методическое пособие Редактор Г. Д. Бакастова Подписано к печати 5.08.05. Формат 60×84 1/16. Бумага офсетная. Печать офсетная. Усл. печ. л. 2,2. Уч.-изд. л. 1,63. Усл.кр-отт. 2,33. Тираж 300 экз. Заказ № Редакционно-издательский отдел Отпечатано с авторского оригинал-макета Отдел оперативной полиграфии ГУАП 190000, Санкт-Петербург, ул. Б. Морская, 67

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