Sec 4.1 Limits, Informally. When we calculated f (x), we first started with the difference quotient. f(x + h) f(x) h

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1 1 Sec 4.1 Limits, Informally When we calculated f (x), we first started with the difference quotient f(x + h) f(x) h and made h small. In other words, f (x) is the number f(x+h) f(x) approaches as h gets smaller and h smaller. In fact this process involves the concept of limit. In this chapter, we will study basics of limits and develop various rules of differentiation. Definition Let f be a function and c be a real number. Suppose the value of f(x) approaches l as x approaches c. In this case, l is called the limit of f as x approaches c, and we write lim f(x) = l. x c If there is no such l, then we say that the limit does not exist. Example 1 Find lim x 1 (x + 5). Example 2 Find lim x 2 (x2 + 2x + 3). 1 Example 3 Find lim x 3 x. x 2 + 4x 5 Example 4 Find lim. x 1 x 1

2 2 2x 2 3x 2 Example 5 Find lim. x 2 x 2 Theorem 1 A polynomial p(x) has a factor (x a) if and only if p(a) = 0. Example 6 Find lim x 1 x 3 + 3x 2 x 3. x + 1 x 2 x 6 Example 7 Find lim x 2 2x 3 + 4x 2 + x + 2. x Example 8 Find lim x 0 x. Example 9 Find lim x 1 1 (x 1) 2. x 3 Example 10 Find lim. x 3 x + 1 2

3 3 Example 11 Find lim x 0 sin x x. We have the following properties of limits: Theorem 2 Suppose that f, g, and h are functions and a is a real number. 1. If f(x) = c for all x, then lim x a f(x) exists and equals c. 2. if both lim x a f(x) and lim x a g(x) exist (say lim x a f(x) = l and lim x a g(x) = k), then (a) lim x a cf(x) (c is a constant) also exists and equals cl. (b) lim x a (f(x) ± g(x)) also exists and equals l ± k. (c) lim x a (f(x) g(x)) also exists and equals l k. f(x) (d) If in addition k 0, then lim x a g(x) also exists and equals l. k 3. If f(x) h(x) g(x) for all x a and if l = k, then lim x a h(x) also exists and equals l(= k). Remark The last result is called the Squeeze Lemma. Example 12 If limf(x) exists and equals a nonzero number l, then lim x a f(x). x a 1 also exists and equals Example 13 Redo Example 11 using the Squeeze Lemma. 1 cos x 1 cos x tan x Example 14 Compute lim, lim, and lim x 0 x 2 x 0 x x 0 x.

4 4 sin 2x Example 15 Compute lim. In general, compute lim x 0 x x 0 sin ax, where b 0. bx cos 3x 1 Example 16 Compute lim. x 0 2x ax Example 17 Find lim, where b 0. x 0 tan bx

5 5 Sec 4.2 Differentiating Polynomials Now we understand that the derivative f (x) of a function f(x) is defined to be f(x + h) f(x) lim, h 0 h provided the limit exists. In this section, we will learn how to differentiate a polynomial. To this end, we need to develop some properties of differentiation, which are based on the properties of limits. Let us recall a result from the previous chapter. Power Rule If f(x) = x n, n = 1, 2, 3,..., then f (x) = nx n 1. We collect some other facts on the differentiation. Constant Multiple Rule If c is a constant and f(x) is differentiable, then the constant multiple cf(x) of f(x) is also differentiable and the derivative is given by (cf(x)) = cf (x). Proof Example 1 Differentiate f(x) = 3x 2. Sum Rule If f(x) and g(x) are differentiable, then their sum f(x) + g(x) is also differentiable with the derivative given by (f(x) + g(x)) = f (x) + g (x). Proof

6 6 Remark 1. With the same assumption as above, we get (f(x) g(x)) = f (x) g (x). 2. The Sum Rule extends to the case with more than two functions. For example, (f(x) + g(x) + h(x)) = f (x) + g (x) + h (x). Example 2 Find (x 4 + x 7 ). Example 3 Calculate the derivative of 1 2 x4 1 7 x14. Example 4 Differentiate f(x) = x 2 + 2x 3 x Example 5 It costs 25 cents per square inch to manufacture metal jar tops. Therefore it costs C(r) = 25πr 2 cents to make a top with radius r inches. What s the rate of increase of cost with respect to radius when the radius is 3 inches?

7 7 Sec 4.3 The Product Rule Product Rule If f(x) and g(x) are differentiable, then their product f(x)g(x) is also differentiable with the derivative given by (f(x)g(x)) = f (x)g(x) + f(x)g (x). Proof Example 1 Compute the derivative of f(x) = x 3 (x + 2) 2. For more examples, we will study the derivative of some trigonometric functions. We begin with the trigonometric addition formulas. 1. sin(x + y) = sin x cos y + cos x sin y 2. sin(x y) = sin x cos y cos x sin y 3. cos(x + y) = cos x cos y sin x sin y 4. cos(x y) = cos x cos y + sin x sin y 5. tan(x + y) = 6. tan(x y) = Remark tan x + tan y 1 tan x tan y tan x tan y 1 + tan x tan y 1. Since sin( x) = sin x, cos( x) = cos x, identities 2, 4, 6 above come from 1, 3, It follows that sin 2x = 2 sin x cos x, cos 2x = cos 2 x sin 2 x = 2 cos 2 x 1 = 1 2 sin 2 x. 3. It also follows that sin 2 x 2 = 1 cos x 2, cos 2 x 2 = 1+cos x 2.

8 8 Example 2 Compute the derivative of f(x) = sin x. Example 3 Compute the derivative of g(x) = cos x. Example 4 Compute the derivative of h(x) = sin 2x. Example 5 Compute the derivative of k(x) = x 2 sin x. Example 6 Compute the derivative of l(x) = (x 3 2x ) sin 2x cos x. 3

9 9 Sec 4.4 The Quotient Rule Quotient Rule If f(x) and g(x) are differentiable, then the quotient f(x) g(x) with the derivative given by ( ) f(x) = f (x)g(x) f(x)g (x), g(x) (g(x)) 2 provided that g(x) 0. is also differentiable Proof Example 1 Find ( cos x 3x 5). Example 2 Let n be a natural number. Find ( 1 x n ). So we can improve the Power Rule. Power Rule, Improved Version If f(x) = x n and n is an integer, then f (x) = nx n 1. Recall the definition of the following trigonometric functions: Definition 1. sec x = 1 cos x 2. csc x = 1 sin x 3. cot x = 1 tan x

10 10 Remark We have the following identities: 1. sin 2 x + cos 2 x = tan 2 x = sec 2 x cot 2 x = csc 2 x Example 3 Find the derivative of tan x. Example 4 Find the derivative of cot x. Example 5 Find the derivative of csc x. Example 6 Find the derivative of sec x.

11 11 Sec 4.5 The Chain Rule We know how to differentiate the function f(x) = sin x. But how about h(x) = sin(x 2 + 3x + 4)? Example 1 Differentiate h(x) = sin(x 2 + 3x + 4). Chain Rule If h(x) = f(g(x)), then h (x) = f (g(x)) g (x). Proof Example 2 In Example 1, f(x) = and g(x) =, so h (x) =. Example 3 Find the derivative of (x 2 3x) 2 by 1. expanding the function and differentiating the resulting polynomial 2. applying the Chain Rule. Example 4 Find (sin(8 x 3 )).

12 12 Example 5 Calculate the derivative of x. Using this, compute the derivative of x 2 + 4x + 5. Remark Note that x = x 1/2 and ( x) = 1 2 x = 1 2 x 1/2. In fact, the Power Rule still holds with any real number. That is to say, we have the following: Power Rule, Full Version If f(x) = x r and r is a real number, then f (x) = rx r 1. Example 6 Compute the derivative of sin(cos(x 2 tan x)).

13 13 Sec 4.6 Differentiating Complicated Functions Differentiate the following functions 1. f(x) = x 3x 2. f(m) = 1+m2 1 m 3. f(j) = 5 (j 34) 9 4. f(x) = (x+4)(2x 3) 2x+5x 2 5. f(z) = z z 3 6. f(z) = tan(z 1 z )

14 14 7. f(t) = t 7 cos(2 + t) 8. f(x) = x sec 8x x+3 9. f(x) = (2x + 5)(tan(x 2 )) f(x) = 15x4 (x 2 12) csc(2x+4) 11. f(u) = 1 cos 2 u. 12. f(n) = 1 3n 2 n 2

15 f(x) = ( ) (7x+11) 8 3 x f(x) = sin(2x 2 + 5x) cos 4 (x 7 ) 15. f(y) = y sin y 16. f(z) = z sin z cos z 17. f(x) = 3x(cot(3 + 4x 3 )) 9

16 16 Sec Digression to Sequences and Their Limits Definition A sequence is a list of real numbers indexed by positive integers. Example 1 1, 4, 9, 16, is a sequence. Let a denote the sequence, then we will use notations a 1 = 1, a 2 = 4, a 3 = 9, and so on. Each a i is called a term. In this example, a n =. Remark We sometimes use a = (a n ) to denote a sequence whose nth term is given by a n. Example 2 Consider the sequence b = ( 1 ). Write out first few terms. n Example 3 Consider the sequence 1, 1, 2, 3, 5, 8, 13,. Can you read off the pattern? What is the next term? Example 4 Find a n when a is 1. 4, 7, 10, 13, 16, 19, 2. 2, 4, 6, 8, 10, 12, 3. 1, 3, 5, 7, 9, 11, 4. 1, 2, 4, 8, 16, 32, 5. 1, 1, 1, 1, 1, 1, 6. 2, 0, 2, 0, 2, 0, 7. 0, 1, 0, 1, 0, 1, , 2 3, 3 4, 4 5, 5 6, 6 7, 9. 10, 100, 1000, 10000, , , 10. 1, 11, 111, 1111, 11111, , Definition Let a, b be sequences and k be a constant. The constant multiple of a by k is a sequence whose terms are ka 1, ka 2, ka 3,. The sum a + b of the sequences a and b is defined to be the sequence a 1 + b 1, a 2 + b 2, a 3 + b 3,. The product ab of the sequences a and b is the sequence a 1 b 1, a 2 b 2, a 3 b 3,. If b n 0 for all n, the quotient a is the sequence a 1 b b 1, a 2 b 2, a 3 b 3,. Consider the sequence given by a n = 1. The first few terms are 1, 1, 1,. As n gets bigger and bigger, a n gets closer and closer to. This observation leads to the concept of the limit of a n 2 3 sequence. Definition Let a be a sequence. Suppose a approaches a number l as n gets bigger and bigger. In this case, we say that the limit of (a n ) is l (or (a n ) converges to l) and write lim n a n = l or a n l. If there is no such l, we say that the limit does not exist (or (a n ) diverges).

17 17 Example 5 What is the limit of following sequences? 1. 2, 3, 4, 5, 6, , 3, 5, 7, 9, , 1, 1, 1, 1, 1, 4. 1, 2, 3, 4, 5, 6, Example 6 Compute the limit of the following sequences: 1. a n = 2 2. a n = 2 n 3. a n = 1 n 4. a n = 1 n 5. a n = 1 2 n 6. a n = ( 2 3 )n 7. a n = 3n+1 n 8. a n = sin 2 n We have the following collection of useful facts about sequences: Theorem Suppose that lim n a n = l and lim n b n = k. 1. If k is a constant, then lim n ka n = kl. 2. lim n (a n ± b n ) = l ± k. 3. lim n a n b n = l k. a 4. If in addition b n 0 for all n and k 0, then lim n n bn = l. k 5. If a n c n b n for all n and if l = k, then lim n c n also exists and equals l(= k). Example 7 Compute the following limits: ( ( ) n ) 2n lim + n n 3 2. lim n 3n 2 2n + 1 n n

18 18 Example 8 Compute the following limits: 100n lim n n n 2. lim n n + 2 4n 1 3. lim n ( n + 1 n ) ( 1) n 4. lim n n 5. lim n sin n n

19 19 Sec Digression to Series and Their Limits m Definition Let n m be positive integers. The expression a k is shorthand for the sum k=n a n + a n+1 + a n a m 1 + a m. Remark m 1. k in the expression a k is just an index, so we can (and will) also use expressions like or k=n m a j to denote the same sum. j=n m i=n a i 2. The sum of a sequence is called a series. 3. In most of cases, we are interested in the series of the form m m n 1 a k can expressed by a k a k. k=n k=1 k=1 n a k, because the general sum k=1 Example 1 Evaluate each of the following 1. 4 k=1 k j j=2 10 k=1 3 We now study some properties of the series. Theorem 1 Let a = (a k ) and b = (b k ) be sequences. n 1. For a constant c, c = nc. k=1 n n 2. For a constant c, ca k = c a k. k=1 k=1 n n n 3. (a k ± b k ) = a k ± b k. k=1 k=1 k=1

20 20 Example 2 Verify Theorem 1 using the following series k 2 k=1 4 (k 2 + 2k) k=1 We now study some special summation formulas. Theorem 2 n 1. k = k=1 n k 2 = k=1 n k=1 n(n + 1). 2 n(n + 1)(2n + 1). 6 k 3 = n2 (n + 1) 2. 4 Proof

21 21 Example 3 Evaluate and Example 4 Explain why the sum of first n positive odd integers is n 2. n 1 n 1 n 1 Example 5 Develop formulas for k, k 2, and k 3. k=1 k=1 k=1 1 Example 6 Find lim n n 3 n k 2. k=1

22 22 Sec 4.7 Integrating Elementary Functions Recall that b a f(x)dx represents (in a generalized sense) area of the region enclosed by the x- axis, lines x = a and x = b, and the graph y = f(x). If the graph of y = f(x) is a portion of a line or a circle, then we are able to compute the integral explicitly. But how about integrals like 2 1 x 2 dx? Mensuration by Parts Let s consider a circle with radius r. We all know that the area of the circle is equal to πr 2. But why is that?

23 23 Example 1 Calculate 2 1 x 2 dx. Example 2 Calculate 2 1 x 3 dx.

24 24 Definition Suppose that g(x) is the derivative of a function f(x) (that is, f (x) = g(x)), then we say that f(x) is an antiderivative of f(x). Remark If f(x) is an antiderivative of g(x), then so is f(x) + C for any constant C. every antiderivative should be of this form. In fact, Now we are ready to state one of the most important results in Calculus. The Fundamental Theorem of Calculus Let g(x) be a continuous function. Then 1. g(x) has an antiderivative. 2. If f(x) is an antiderivative of g(x), then Sketch of Proof b a g(x) dx = f(b) f(a). Example 3 Calculate π 2 0 cos x dx and π 0 cos x dx.

25 25 b Remark Previous examples show that finding g(x) dx is pretty much the same as finding an a antiderivative of g(x). We write g(x) dx for the collection of all antiderivatives of g(x). This collection is called the indefinite integral of g. If f(x) is an antiderivative of g(x), functions of the form f(x) + C, and only they, are antiderivatives of g(x). For this reason, we also write g(x) dx = f(x) + C. Example 4 Find the indefinite integral x 3 dx and using this compute 3 2 x 3 dx. Example 5 Find the indefinite integral sin x dx.

26 26 Sec 4.8 Integrating Complicated Functions So, finding the indefinite integral is important. This section is devoted to making a list of the indefinite integrals of some elementary functions. We begin with general properties. Theorem 1 Suppose f(x) and g(x) have antiderivatives. 1. For any constant k, kf(x) dx = k f(x) dx. 2. (f(x) + g(x)) dx = f(x) dx + g(x) dx. Remark We also have (f(x) g(x)) dx = f(x) dx g(x) dx. Theorem 2 If r 1, Example 1 Find x 5 dx. x r dx = 1 r + 1 xr+1 + C. Example 2 Find 3x 5 dx. Example 3 Find (17x 31 6x ) dx. Example 4 Calculate x dx.

27 27 Example 5 Find t 2t dt. Example 6 Find (2x + 7) 6 dx. Example 7 Find sin 2x dx. We close this section with some examples of initial value problems. Example 8 Suppose f (t) = t and f(0) = 1. What is f(3)? Example 9 Suppose f (x) = (x + 2) 2 and f(3) = 0. What is f(x)?

28 28 Sec 4.9 Exponential Growth and Decay Example 1 Draw the graph of y = 2 x. Example 2 Draw the graph of y = 3 x. Example 3 Draw the graph of y = ( 1 2 )x. Remark Let a > 0 and a 1, then 1. The graph of y = a x always passes through (0, 1). 2. The graph of y = a x is always above the x-axis. 3. If a > 1, then y = a x is increasing. If 0 < a < 1, then y = a x is decreasing.

29 ( Definition It is well-known that lim n exists. In fact this limit is approximately n n) and known to be irrational. We will denote the limit by e. Remark 1. e is sometimes called Euler s number. 2. It is also known that e = ! + 1 3! + 1 4! We want to study the function f(x) = e x. Since e > 1, this functions is always positive and increasing. Moreover, the following is true. Fact lim x 0 e x 1 x = 1. Now we are ready to study the derivative of e x. Theorem 1 The derivative of the exponential function y = e x is itself. That is, (e x ) = e x. Proof So f(x) = e x has a very interesting property that f(x) = f (x). A simple calculation shows that any constant multiple of e x also has this property. A natural question arises: is there any other function f satisfying f(x) = f (x)? Theorem 2 If f(x) = f (x), then f(x) must be of the form f(x) = ke x for some constant k. Proof

30 30 Example 4 Find the derivative of e 4x. Example 5 Show that for any constant k, f(x) = ke 4x satisfies the differential equation f (x) = 4f(x). Conversely, if f satisfies f (x) = 4f(x), then f must be of the form f(x) = ke 4x for some constant k. The above observation can be generalized: Theorem 3 Let a be a constant. f(x) = ke ax for some constant k. If f satisfies the differential equation f (x) = af(x), then Example 6 Solve the following initial value problem: given y = 2y and y(2) = 4, find y(1). We close this section with differentiation involving the exponential function. Example 7 Differentiate e x3. Example 8 Differentiate e 4x cos x.

31 31 Sec 4.10 The Natural Logarithm For real numbers a, b, c with a > 0 and a 1, suppose c = a b. Then a = c 1 b. Then how do we express b in terms of a and c? Definition For a real number a with a > 0, a 1, if c = a b, then we write b = log a c. Example 1 Compute log Example 2 Suppose log a 16 = 2. Determine a. Example 3 log a 1 = and log a a = for all a with a > 0, x 1. Remark When we write y = log a x, 1. a > 0 and a 1, 2. x > 0, and 3. y could be any real number. Theorem Let a > 0 and a 1. Let x, y > log a xy = log a x + log a y. 2. log a x y = log a x log a y. 3. For every real numbers s( 0) and t, log a s x t = t s log a x. 4. y log a x = x log a y. 5. If b > 0 and b 1, then log a x = log b x log b a. In particular, log a b = 1 log b a. Proof

32 32 Example 4 Let a > 0 and a 1. Let x, y > 0. Express each of the following in terms of log a x and log a y: 1. log a x 4 y 3 2. log a x 2 y 3. log a ax 4. log 1 a 2 x a Example 5 Let a > 0 and a 1. Let x > 0. Simplify each of the following: 1. a log a x 2. a 3 log a x Definition Consider the function defined by f(x) = log e x, where e is Euler s number. This functions is called the natural logarithm function and is denoted by f(x) = ln x. So ln is a function defined on the set of positive real numbers. To study it s derivative, let g(x) = e x, f(x) = ln x, and consider the composition g f defined on the set of all positive real numbers. First of all, note that g f(x) = e ln x =. Differentiating both sides with respect to x, we get g (f(x))f (x) = 1 and hence Therefore, e f(x) f (x) = 1. f (x) = 1 e f(x) = 1 e ln x = 1 x. To summarize, for x > 0, we get (ln x) = 1 x. Remark What if x < 0? If x < 0, then ln x is not defined. To define ln for negative values, we consider h(x) = ln x. For x > 0, h(x) = ln x and hence h (x) = 1. For x < 0, h(x) = ln( x) x and using the chain rule, we get h (x) = 1 x ( 1) = 1 x. Conclusion: ln x is defined for all real numbers except for 0 and (ln x ) = 1 x.

33 33 Example 6 Differentiate ln(x 2 + 1). Example 7 Differentiate ln 2x, ln 3x, and ln 100x. Example 8 Differentiate e x2 1 tan(x + ln x ). Example 9 Find x dx. Example 10 Find x dx. We close this section with differentiation of general exponential functions and logarithmic functions. Example 11 Find the derivative of y = 2 x. Example 12 Find the derivative of y = log 2 x. In general, for a > 0, a 1, (a x ) = a x ln a and (log a x) = 1 x ln a.

34 34 Sec Applications Radioactive Dating Half-life is the period of time it takes for a substance undergoing decay to decrease by half. For example, the half-life of carbon 14 is 5,730 years. Example 1 Let C(t) (in grams) be the amount of carbon 14 in a certain material at time t (in years). If C(0) = 6, then C(5, 730) = and C(11, 460) =. Example 2 It is known that C(t) satisfies the differential equation C (t) = ac(t) for some constant a. Describe C(t) in terms of C(0). Example 3 We want to date a piece of cloth that is made of cotton. The amount of C 14 in the cloth was about 92% of the amount in living plants today. The amount of C 14 in living plants today is assumed to be the same as the amount of C 14 in living plants when the cloth was made. Estimate the age of the cloth. Gravity Suppose we let h(t) be the height of an object above the ground at time t. Then h (t) is the velocity of the object, and h (t) is its rate of change of velocity, also called acceleration. It s common to use v for velocity and a for acceleration. So we have at time t: Height h(t) Velocity v(t) = h (t) Acceleration a(t) = v (t) = h (t) It is known that, without external force, the acceleration is always constant. The constant is known to be approximately 9.8m/sec 2 is called the gravity. Example 4 An object is dropped from a building 100 meters tall. When does the object hit the ground?

35 35 Example 5 Standing on the ground, I toss an object upward with a velocity of 15 meters per second. When does it reach the highest point? What is the maximum height? How fast is it going when it hits the ground again? Example 6 Standing on a building 100 meters tall, I toss an object upward with a velocity of 10 meters per second. When does it reach the highest point? What is the maximum height? How fast is it going when it hits the ground again? Example 7 Standing on the ground, I toss an object upward at an angle of 45 degrees and with a velocity 15 meters per second. How long does it take for the object to hit the ground? Example 8 Standing on the ground, I toss an object upward at an angle θ measured from the ground and with a velocity v 0 meters per second. Which θ gives the largest horizontal displacement?