1.7 Sums of series Example 1.7.1: 2. Real functions of one variable Example 1.7.2: 2.1 General definitions Example 2.1.3: Example 2.1.

Transcription

1 7 Sums of series We often want to sum a series of terms, for example when we look at polynomials As we already saw, we abbreviate a sum of the form For example and u + u + + u r by r u i i= a n x n + a n x n + + a x + a = (a + b n = i= ( n a n i b i i a i x i i= Suppose that u i = a + (i d, so that u i with i form an arithmetic progression (AP with initial value a and common difference d Then n i= u i = a + (a + d + + (a + (n d = na + d + d + + (n d = na + d n(n = n(a + (n d Next suppose that u i = ar i, so that u i with i form a geometric progression (GP with initial value a and common ratio r Then { na if r = u i = a + ar + + ar n = a( r n i= r if r (To verify the second case, rearrange the expression for n r n given in Section 5 of lecture Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5 Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5 We can also sum certain series of powers of consecutive integers: i = i= i = i= i= n(n + n(n + (n + i = n (n + Example 7: The fourth term in a geometric progression is 7 and the seventh is Find the sum S 8 of the first eighteen terms We have u = ar = 7 and u 7 = ar = ar ar = 7 and so r = ( 7 Substituting into the expression for u we deduce that a = 9 Then S 8 = ( 9 ( 7 8 ( 7 ( 9 ( 7 = ( 7 Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5 Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5 Example 7: Find the sum S n of the squares of the first n even integers greater than zero S n = (n = (k = k = k k= k= = n(n + (n + k= Real functions of one variable General definitions A real function is a rule that assigns to each real number in some set another real number, in a unique fashion The set of inputs is called the domain of the function, and the set of outputs is called the range or image Usually we talk about a function going from one set to another without guaranteeing that every value in the latter set occurs as an output of the function We refer to such a target set as the codomain Thus the range is a subset of the codomain Andreas Fring (City University London AS5 Lecture 5-8 Autumn 5 / 5 Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5 Let D f be the domain of f, with codomain C f and range R f We write this as f : D f C f or f : x f (x where x D f (and f (x C f This has the advantage over the form f (x = that we do not need to give an explicit formula for f, which is useful when we talk in general terms Example : Let f (x = x with x R This has domain R, ie < x <, and range the set of y with y Example : Take f as in the preceding example, but with x This has domain x and range y The graph of a function is the set {(x, y : y = f (x, x D f } which is a subset of the plane R We often represent this graphically Example : The graph for Example is {(x, x : x } Andreas Fring (City University London AS5 Lecture 5-8 Autumn 7 / 5 Andreas Fring (City University London AS5 Lecture 5-8 Autumn 8 / 5

2 lecture If the domain of a function is not specified, we assume that it is the largest set of real numbers on which the function is defined Example : Specify the domain and range of f (x = x Domain: Any real number except Range: We need to solve y = x This is not possible if y = If y then y = x and x = + y the range is all real numbers except zero The composition of two functions f and g, written f g, or just fg, is the function defined by (f g(x = f (g(x This only makes sense if g(x is contained in the domain of f, so the domain of f g is the set of all x D g such that g(x D f Example 5: Let f (x = x x + x with x and g(x = x + with x R (f g(x = f (x + = (x + (x + + x + which has domain x (g f (x = g(x x + x = (x x + x + which has domain x Andreas Fring (City University London AS5 Lecture 5-8 Autumn 9 / 5 Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5 A function f is one-to-one ( or injective if x y implies that f (x f (y Example : f (x = x + with x R is injective as if f (x = f (y then x + = y + so x = y f (x = x with x R is not injective, as f ( = f ( An injective function f has an inverse f For each b in the image of f, we set f (b to be the unique element a in the domain of f such that f (a = b So D f = R f and R f = D f Also The graph of f is the reflection of the graph of f in the line y = x f f f f (x = x and f f (x = x Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5 Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5 Example 7: Let f (x = x x+ = x+ for x Set y = f (x, so (x + y = x Rearranging we get that x = + y y and hence f (x = +x x with x Check: f f (x = +x x +x x + = + x + x + x + x = x = x Note that it is not possible to talk about the inverse of a non-injective function For example, consider f (x = x with x R If f ( exists, is it or? However, f (x = x with x does have an inverse: f (x = x This is one reason why we may restrict the domain of a function Try also f f (x Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5 Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5 Special functions We have already considered certain special classes of functions: polynomials, and rational functions Here are a few more The square root function f (x = x where x (Recall that we have already defined this function in Section Example : Find the domain and range of x x Set y = h(x = x x = f g(x where g(x = x x and f (x = x The domain is x x, ie (x + (x So x or x The range is y { x if x The modulus function f (x = x = x if x < Example : Sketch the graph of f (x = x x f (x = x x if x x + x + if < x < x x if x Andreas Fring (City University London AS5 Lecture 5-8 Autumn 5 / 5 Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5

3 Example : Solve x = x lecture 7 Trigonometric functions x x From the graph we see that the solution occurs when x < we need x = x with x <, ie x = We could also solve x = x, which gives x = However, this does not make sense in x = x and we therefore have to discard this solution θ P=(a,b We define for θ R, and sin θ = b tan θ = b a cos θ = a for θ R with θ ( n + for some n Z Note: (i tan θ = sin θ cos θ (ii We use radians for angles radians equals degrees (iii Positive angles are measured anticlockwise Andreas Fring (City University London AS5 Lecture 5-8 Autumn 7 / 5 Andreas Fring (City University London AS5 Lecture 5-8 Autumn 8 / 5 The graphs of these functions are: y = sin θ y = tan θ y = cos θ We define cosec θ = sin θ sec θ = cos θ cot θ = tan θ wherever these functions are defined, and set cot = Andreas Fring (City University London AS5 Lecture 5-8 Autumn 9 / 5 Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5 A function is periodic of period t if f (x + t = f (x for all x D f and t is the least positive number for which this occurs A function is even if f ( x = f (x for all x D f and odd if f ( x = f (x for all x D f Here is a summary of the basic properties of trigonometric functions Function Domain Range Period Zeros Odd/Even sin R y ( n O cos R y n+ E tan θ ( n+ R n O cosec θ n y O sec θ ( n+ y ( E cot θ n R n+ O Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5 Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5 You must memorise the following values: sin θ cos θ tan θ You must also know all of the following identities: sin(x = cos x cot(x = tan x cos x + sin x = cot x + = cosec x + tan x = sec x sin(x + y = sin x cos y + cos x sin y cos(x + y = cos x cos y sin x sin y tan x+tan y tan(x + y = tan x tan y (From these you can work out sin(x y etc Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5 Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5

4 Special cases of these last equations which should also be known are: sin(x = sin x cos x cos(x = cos x sin x tan(x = tan x tan x You should also know: sin x + sin y = sin ( x+y ( cos x y cos x + cos y = cos ( x+y ( cos x y This last pair of equations can be derived from the preceding sets For example, let x = p + q and y = p q Then The righthand side equals sin x + sin y = sin(p + q + sin(p q sin p cos q + cos p sin q cos p sin q which equals ( x + y sin p cos q = sin cos + sin p cos q ( x y Andreas Fring (City University London AS5 Lecture 5-8 Autumn 5 / 5 Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5 Example : Express sin θ in terms of sin θ sin θ = sin(θ + θ = sin θ cos θ + cos θ sin θ = sin θ(cos θ sin θ + cos θ sin θ cos θ = sin θ cos θ sin θ = sin θ( sin θ sin θ = sin θ sin θ lecture 8 When solving any trigonometric equation, we ultimately reduce this to solving some equation of the form f (θ = a where f is a trigonometric function such as cos, sin, or tan Thus we must know the general solution to such equations As the functions are periodic of period (respectively for cos and sin (respectively tan, it is enough to find all solutions in some period (respectively period For sin, if θ is a solution then so is θ For cos if θ is a solution then so is θ Tan is injective on the domain < θ <, so has only one solution in each period Andreas Fring (City University London AS5 Lecture 5-8 Autumn 7 / 5 Andreas Fring (City University London AS5 Lecture 5-8 Autumn 8 / 5 In summary, the general solutions (which are to be memorised in terms of a particular solution θ are: Example : Find all solutions to sin θ = One solution is θ =, and so the general solution is with θ sin θ + n or θ + n with n Z cos ±θ + n with n Z tan θ + n with n Z θ = + n or θ = + n with n Z Example : Find the general solution to cos θ = One solution is θ =, so the general solution is θ = ± + n with n Z θ = + n or θ = + n with n Z In the required range θ takes the values, 5,, 7,,, 5, 8 Andreas Fring (City University London AS5 Lecture 5-8 Autumn 9 / 5 Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5 Example : Solve cos θ sin θ = for θ and so we require cos θ sin θ = ( sin θ sin θ ( sin θ (sin θ + = This has solutions sin θ = and We want θ For sin θ = have θ =, 5,, 7 and for sin θ = have θ =, 7 θ =, 5,, 7,, 7 Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5 A function of the form a cos θ + b sin θ can be rewritten in either of the forms R cos(θ α or R sin(θ + α for suitable choices of R and α < Suppose a cos θ + b sin θ = R cos(θ α = R cos θ cos α + R sin θ sin α Comparing coefficients we have and so R = a + b Then and so α = tan ( b a a = R cos α and b = R sin α R (cos α + sin α = R = a + b R sin α R cos α = tan α = b a Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5

5 Similarly ( a cos θ + b sin θ = a + b sin θ + tan ( a b Example 5: Find the general solution of the equation cos x + sin x = Let cos x + sin x = R cos(x α with R > and < α < By the above we have R = + and tan α = There is a simple method for solving an equation of the form cos aθ = cos bθ By the general form of the solution to cos we must have and so θ = n a ± b aθ = n ± bθ with n Z which implies that R = and α = Thus we have to solve ( cos x = This has general solution x = ± + n with n Z Similar results hold for and sin aθ = sin bθ tan aθ = tan bθ Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5 Andreas Fring (City University London AS5 Lecture 5-8 Autumn / 5 This method works when both sides of the equation involve the same function Sometimes we will have to first rearrange to ensure this Example : Find the general solution of cos θ = sin θ sin θ = cos θ and so cos(θ = cos θ θ = n ± θ with n Z Rearranging, we find that θ = n + or θ = n with n Z Andreas Fring (City University London AS5 Lecture 5-8 Autumn 5 / 5