 6.4.1: Let f(x) := cosax for x E JRwhere a :1=O.Find f<n)(x) for n EN, x E...
 6.4.2: Let g(x) := Ix31for x E JR.Find g'(x) and g"(X) for x E JR,and glll...
 6.4.3: Use Induction to prove Leibniz's rule for the nth derivative of a p...
 6.4.4: Show that if x > O,.then 1 + ~x  kx2 ::: .vI + :X::: 1 + ~x.
 6.4.5: Use the preceding exercise to approximate v'f2 and../2. What is the...
 6.4.6: UseTaylor'sTheoremwithn = 2 to obtain more accurate approximations ...
 6.4.7: If x > 0 showthat 1(1+ x)1/3 (1 + 1x  ~x2)1::: (5/81)x3. Use this...
 6.4.8: If f(x) := eX, show that the remainder term in Taylor's Theorem con...
 6.4.9: If g(x) := sinx, show that the remainder term in Taylor's Theorem c...
 6.4.10: Let h(x) := e1/x2 for x :1=0 and h(O) := O.Show that h(n)(0) = 0 f...
 6.4.11: If x E [0, 1] and n E N, show thatI (x2 x3 n)Ixn+1In(1+x) x++2....
 6.4.12: We wish to approximate sin by a polynomial on [1, 1] so that the e...
 6.4.13: Calculate e correct to 7 decimal places.
 6.4.14: Determine whether or not x = 0 is a point of relative extremum of t...
 6.4.15: Let I be continuous on [a, b] and assume the second derivative I" e...
 6.4.16: LetI S;JRbe anopeninterval,let I : I + JRbe differentiableon I, a...
 6.4.17: Suppose that I S; JRis an open interval and that I" (x) ~ 0 for all...
 6.4.18: Let I S; JRbe an interval and let eEl. Suppose that I and g are def...
 6.4.19: Showthat the functionI(x) := x3  2x  5 has a zero r in the interv...
 6.4.20: Approximate the real zeros of g(x) := x4  x  3.
 6.4.21: Approximatethe real zeros of h(x) := x3  xI. ApplyNewton'sMethods...
 6.4.22: The equation lnx =x  2 has two solutions. Approximate them using N...
 6.4.23: The function I(x) = 8x3  8x2 + 1has two zeros in [0, 1].Approximat...
 6.4.24: Approximate the solution of the equation x =cos x, accurate to with...
Solutions for Chapter 6.4: Taylor's Theorem
Full solutions for Introduction to Real Analysis  3rd Edition
ISBN: 9780471321484
Solutions for Chapter 6.4: Taylor's Theorem
Get Full SolutionsSince 24 problems in chapter 6.4: Taylor's Theorem have been answered, more than 8783 students have viewed full stepbystep solutions from this chapter. Chapter 6.4: Taylor's Theorem includes 24 full stepbystep solutions. This textbook survival guide was created for the textbook: Introduction to Real Analysis, edition: 3. This expansive textbook survival guide covers the following chapters and their solutions. Introduction to Real Analysis was written by and is associated to the ISBN: 9780471321484.

Absolute minimum
A value ƒ(c) is an absolute minimum value of ƒ if ƒ(c) ? ƒ(x)for all x in the domain of ƒ.

Addition principle of probability.
P(A or B) = P(A) + P(B)  P(A and B). If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B)

Arcsecant function
See Inverse secant function.

Arithmetic sequence
A sequence {an} in which an = an1 + d for every integer n ? 2 . The number d is the common difference.

Average velocity
The change in position divided by the change in time.

Compound fraction
A fractional expression in which the numerator or denominator may contain fractions

Cubic
A degree 3 polynomial function

Cycloid
The graph of the parametric equations

Dependent variable
Variable representing the range value of a function (usually y)

Division algorithm for polynomials
Given ƒ(x), d(x) ? 0 there are unique polynomials q1x (quotient) and r1x(remainder) ƒ1x2 = d1x2q1x2 + r1x2 with with either r1x2 = 0 or degree of r(x) 6 degree of d1x2

Factoring (a polynomial)
Writing a polynomial as a product of two or more polynomial factors.

Graph of a function ƒ
The set of all points in the coordinate plane corresponding to the pairs (x, ƒ(x)) for x in the domain of ƒ.

Instantaneous velocity
The instantaneous rate of change of a position function with respect to time, p. 737.

Intermediate Value Theorem
If ƒ is a polynomial function and a < b , then ƒ assumes every value between ƒ(a) and ƒ(b).

Line of travel
The path along which an object travels

nset
A set of n objects.

Negative angle
Angle generated by clockwise rotation.

Ordered set
A set is ordered if it is possible to compare any two elements and say that one element is “less than” or “greater than” the other.

Righthand limit of ƒ at x a
The limit of ƒ as x approaches a from the right.

ycoordinate
The directed distance from the xaxis xzplane to a point in a plane (space), or the second number in an ordered pair (triple), pp. 12, 629.