UNIT 8 TAYLOR9$ THEOREM Structure 8.1 Introduction 5 Objectives 8.2 Taylor's The~rem 5 Taylor's Theorem for Functions of One Variable Taylor's Theorem for Functions of Two Variables 8.3 Maxima and Minima 15 . LocalExtrema Second Derivative Test for Local Extrema 8.4 Lagrange's Multipliers 8.5 Summary 8.6 Solutions and Answers 8.1 INTRODUCTION In this unit we state, without proof, Taylor's Theorem (about approximating a function by polynomials) for real-valued functions of several variables. This theorem is the principal tool for finding out the points of relative maxima and minima for these functions. We also discuss briefly Lagrange's method of multipliers, which enables us to locate the stationary points when the variables are not free but are subject to some additional conditions. In this unit we will be dealing with functions of two variables. Even though the results are true for any number of variables, their proof involves techniques which are not easy to understand at this level. So, for the sake of simplicity, we confine our attention to the two- variable case. We start our discussion with the one variable case. Objectives After studying this unit, you should be able to find the Taylor polynomials for functions of one or two variables, ., state and apply Taylor's theorem for functions of one and two variables, I I locate the stationary points of functions, use the second derivative test to find the nature of stationary points, use the technique of Lagrange's multipliers in locating the stationary points of functions of two variables. , 8.2 TAYLOR'S THEOREM In the calculus course you have seen (Unit 6) that if we know the values of a function of one variable and its derivatives at 0, then we can find an expression for the value of the function at a nearby point. We can derive a similar expression for functions of two variables using partial derivatives. This expression was first derived by Brook Taylor, an English mathematician of the eighteenth century. We shall first discuss Taylor's theorem for functions of a single variable. Taylor (1685-1731) 8.2.1 Taylor's Theorem for Functions of One Variable You will agree when we say that polynomials are by far the simplest functions in calculus. We can evaluate the value of a polynomial at a point by using the four basic operations of addition, multiplication, subtraction and division. However, the situation in the case of functions like ex- lnx, sinx, etc., is not so simple. These functions occur so frequently in all branches of mathematics, that approximate values of these fun&ns have been tabulated I ~pplicationsof Partial ' extensively. The main tool for this purpose has been to find polynomials which approximate Derivatives these functions in a neighbourhood of the point under consideration. You are already familiar with Lagrange's mean value theorem. This theorem states that if f(x) is differentiable in some neighbourhood N of the point x,, then we have for all x such that [x,;x] or [x, x,] is contained in N. Here 5 is a point lying between x, and X. Iff is twice differentiable in N, then, again applying mean value theorem to the function f, we can go a step further and write - 1 f(x) = f(x,) + (x-x,) f (x,)+ 5 f' (6) (x-xo)l , where 6 is some point in N lying between x, and x. Thus, the constant polynomial f(x,) approximates f(x) in N in the first case, while the polynomial f(x,) + (x-x,) f (x,)approximates f(x) in N in the second case. The difference between the actual value and the approximated value is called the error term. 1 The error term in the first case is f (5) (x-x,), and in the second case it is - f" (6) (x-xJ2. We 2 can estimate these error terms iff and f" are bounded. Taylor's theorem tells us that if a function f(x) has derivatives of all orders upto n+l in a neighbourhood of x,, then we can find polynomials P,(x), ...... Pn (x) of degree 0, ...... n, respectively, such that the error term f(x) - P,(x) is a polynomial of degree less than or equal to r+l. Note that here we consider the polynomial 0 also as a polynomial of degree zero, which is not the usual practice. We have done this for the sake of uniformity of expression. In order to state the precise result, we 1 start with the following definition. I ' Definition 1 : Let f(x) be a real-valued function having derivatives upto order n 2 1 at the point x, . A polynomial P(x) is said to be the rth Taylor polynomial of f(x) at x,, if i) the degree of P(x) lr, r I n ii) P(j)(x,) = f(j) (q)for 0 lj lr, where Po)(x,) = P(x,) and fro, (x,) = f(x,). Recall that a polynomial P(x) is an expression that can be written as where c,, c,, ....... cn are real numbers. Apart from these there are expressions like P(x) = c, + c, (x-x,) + ...... + c, (x-x,)" . ... (2) where x,, c,, c,, .... are real numbers and x, # 0, which are also called polynomials. You can easily see that (2) can be rewritten in the form (1) by expanding the powers (x-x,)~, .... (x-x,)". We also call the expression in (1) a polynomial it zero and.that in (2), a polynomial at x,. Now we state and prove a theorem which tells us that Taylor plynomials of a given function are unique. It also tells us how to find out the Taylor polynomials of a given function. Theorem 1 : Let %. .... a, be any r + 1 real numbers. Then there exists a unique polynomial P(x) such that i) The degree of P(x) Ir ii) Pc~)(x,)=a~,OSjIr, where x, is any fixed real number. Taylor's Theorem 1 Moreover. P(x) = C,'a, -(x-xJrn. m! I Proof :We can write a polynomial at x, as where bo . ....., b, are real numbers. Now we have to determine b,, ....., br such that P")(x,) = a, for 0 I jI r. If we differentiate the expression in (3) jtimes, then we get , r Fj)(x) =zk (k- 1) .... (k -j + 1,) 4, (x-x~)~-~,1 I j 5 r, k=j and therefore, Thus. Also, Hence, PC J ) (x,) "j= j! forOljlr Substituting for bj s in (3) , we get Note that the polynomial P(x) will be of degree r if and only if a, # 0. Now by (4) we can conclude that the polynomial is unique. The following corollary of Theorem 1 tells us how to find the Taylor polynomials of a given function. Corollary 1 :If f(x) is a real-valued function having derivatives of all orders upto n (n 2 1)' then the mh Taylor polynomial of f(x) at x, is given by Proof : Let us take a, = fi) (x,), 0 5 k .5 m, in Theorem 1. Then the mh Taylor polynomial off, if it exists; must be in the form of Equation (5). Thus, \ The above discussion shows that the Taylor polynomials of a given function can be found step by step using. the relation 1 MOrebver, if P,,,(x) is the mb Taylor polynomial of f(x) at x, ,then you can check that the derivative of Pm(x) at x, is the (m-I)* Taylor polynomial of f(x) at x, . Let us consider some examples now. Example 1 : Let us find the Taylor polynomials of We apply Theorem 1 with x, = 3. Applications of Partial since eO)(3) = f(3) = 22, Derivatives e') (3) = 19, en (3) = 6 and 19 Po (x) = 22, P, (x) = 22 + (x - 3), 19 14 6 P3 (x) = 22 + (x-3) + -2! (x-3)' + -3! (x-3)l and P, (x) = P3 (x) for all r > 3. b Example 2 :Let us find the fourth Taylor polynomial 1 1 We have f(x) =- (1 + x)-' 2 1 1 3 15 Therefore, f(0) = 1, f (0) = , f" (0) = - (0) = - ti4)(0) = - - 2 -4 ' e3) 8 ' 16 - The desired polynomial is f (0) f"(0) (0) x3 + ff4I (0) 4 T4(x) = f(0) +--- x+- x2+ l! 2! 3! 4! Example 3 : Let us find T, (x) for cos x at x, = z. Now cos x = - 1 and the first eight derivatives of cos x at x are Dropping the terms with coefficients 0, we have the polynomial 1 .Example 4 :Let us find T5 (x) at x, = 0 for f, where f(x) = -= (1 - x)-' 1-x Computing the derivatives, we obtain f. (x) = (1 - x)-~,f' (x) = 2(1-~)-~. r3)(x) = 3.2 (I-X)~, f4' (x) = 4 ! ( l-x)" P" (x) = 5! (1 - x)~ Thus, the successive derivatives off at 0 , in order, are Since f(0) = 1, we obtain Taylor's Theorem = 1 +x+x2+x3+x4+x5 Now you can try these exercises. El) Find the nthTaylor polynomial of the function ex at x = 2. E2) Find the 6" Taylor polynomial of sin x at x = 0. E3) Find the rh Taylor polynomials of the following functions at the indicated point and for the indicated value of r. E4) Find a polynomial f(x) of degree 2 that satisfies f(1) = 2, f (1) = -1 and f' (1) = 2. We now state Taylor's theorem which gives us the connection between a function and its Taylor polynomials at a point.