Advanced High-School Mathematics

SECTION 5.4 Polynomial Approximations 291 = ∑n k=0 f(k)(a) n! (x−a)k. We expect, then, to have a pretty good approximation f(x) ...

292 CHAPTER 5 Series and Differential Equations For most of the functions we’ve considered here, this is true, but the general r ...

SECTION 5.4 Polynomial Approximations 293 Example 2. (A handy trick)If we wish to compute the Maclaurin series for cosx, we coul ...

294 CHAPTER 5 Series and Differential Equations Example 6. (A handy trick)In the above series we may substitute −x^2 forxand get ...

SECTION 5.4 Polynomial Approximations 295 Further valid sums can be obtained by differentiating. Exercises Find the Maclaurin s ...

296 CHAPTER 5 Series and Differential Equations Consider the function defined by settingf(x) = ln(1 + sinx). (a) Determine the ...

SECTION 5.4 Polynomial Approximations 297 (i) By equating the imaginary parts of DeMoivre’s formula cosnθ+isinnθ= (cosθ+isinθ)n= ...

298 CHAPTER 5 Series and Differential Equations Step 4. Use step 1 to write the above as m(2m−1) 3 < (2m+ 1)^2 π^2 ∑m k=1 1 k ...

SECTION 5.4 Polynomial Approximations 299 Having been reminded of theMean Value Theorem, perhaps now Taylor’s Theorem with Remai ...

300 CHAPTER 5 Series and Differential Equations u=f(n)(t) dv= (x−t)(n−1) (n−1)! dt. du=f(n+1)(t)dt v=− (x−t)n n! From the above, ...

SECTION 5.4 Polynomial Approximations 301 The above remainder (i.e., error term) is called theLagrange form of the error. We’ll ...

302 CHAPTER 5 Series and Differential Equations In this case, as long as −^12 ≤ x ≤ 1, then we are guaranteed that ∣∣ ∣∣ ∣ xn+1 ...

SECTION 5.4 Polynomial Approximations 303 Assume that you have a function f satisfying f(0) = 5 and for n≥ 1 f(n)(0) =(n 2 −n1) ...

304 CHAPTER 5 Series and Differential Equations (c) The fourth derivative off satisfies the inequality ∣∣ ∣f(4)(x) ∣∣ ∣≤ 6 for a ...

SECTION 5.5 Differential Equations 305 solutiony =y(x). This results in the initial value problemof the form y′=p(x)y+q(x), y(a) ...

306 CHAPTER 5 Series and Differential Equations From the above slope field it appears that there might be alinear solution of th ...

SECTION 5.5 Differential Equations 307 Show thaty =Ke^2 x−^14 (2x+ 1) is a solution of the linear ODE y′= 2y+xfor any value of ...

308 CHAPTER 5 Series and Differential Equations C(x) =xn+an− 1 xn−^1 +···+a 1 x+a 0 has a real zeroα, i.e., thatC(α) = 0. Show t ...

SECTION 5.5 Differential Equations 309 which describes a rapidly- decreasing exponential func- tion ofx. The slope field, to- ge ...

310 CHAPTER 5 Series and Differential Equations 2 x^2 dy dx = x^2 +y^2 can be reduced to the form (5.2) by dividing both sides b ...