 4.8.1: Let f(x) = x2 x.(a) An interval on which f satisfies the hypotheses...
 4.8.2: Use the accompanying graph of f to find an interval [a,b]on which R...
 4.8.3: Let f(x) = x2 x.(a) Find a point b such that the slope of the secan...
 4.8.4: Use the graph of f in the accompanying figure to estimateall values...
 4.8.5: Find a function f such that the graph of f contains the point(1, 5)...
 4.8.6: 58 Verify that the hypotheses of the MeanValue Theorem aresatisfie...
 4.8.7: 58 Verify that the hypotheses of the MeanValue Theorem aresatisfie...
 4.8.8: 58 Verify that the hypotheses of the MeanValue Theorem aresatisfie...
 4.8.9: (a) Find an interval [a,b] on whichf(x) = x4 + x3 x2 + x 2satisfies...
 4.8.10: Let f(x) = x3 4x.(a) Find the equation of the secant line through t...
 4.8.11: 1114 TrueFalse Determine whether the statement is true orfalse. Exp...
 4.8.12: 1114 TrueFalse Determine whether the statement is true orfalse. Exp...
 4.8.13: 1114 TrueFalse Determine whether the statement is true orfalse. Exp...
 4.8.14: 1114 TrueFalse Determine whether the statement is true orfalse. Exp...
 4.8.15: Let f(x) = tan x.(a) Show that there is no point c in the interval ...
 4.8.16: Let f(x) = x2/3, a = 1, and b = 8.(a) Show that there is no point c...
 4.8.17: (a) Show that if f is differentiable on (, +), and ify = f(x) and y...
 4.8.18: Review Formulas (8) and (9) in Section 2.1 and use theMeanValue Th...
 4.8.19: 1921 Use the result of Exercise 18 in these exercises. An automobil...
 4.8.20: 1921 Use the result of Exercise 18 in these exercises. At 11 a.m. o...
 4.8.21: 1921 Use the result of Exercise 18 in these exercises. Suppose that...
 4.8.22: Use the fact thatddx [x ln(2 x)] = ln(2 x) x2 xto show that the equ...
 4.8.23: (a) Use the Constant Difference Theorem (4.8.3) to showthat if f(x)...
 4.8.24: (a) Use the Constant Difference Theorem (4.8.3) to showthat if f(x)...
 4.8.25: Let g(x) = xex ex . Find f(x) so that f(x) = g(x) andf(1) = 2.
 4.8.26: Let g(x) = tan1 x. Find f(x) so that f(x) = g(x) andf(1) = 2.
 4.8.27: (a) Use the MeanValue Theorem to show that if f isdifferentiable o...
 4.8.28: (a) Use the MeanValue Theorem to show that if fis differentiable o...
 4.8.29: (a) Use the MeanValue Theorem to show thaty x < y x2xif 0 <x<y.(b)...
 4.8.30: Show that if f is differentiable on an open interval andf(x) = 0 on...
 4.8.31: Use the result in Exercise 30 to show the following:(a) The equatio...
 4.8.32: Use the inequality 3 < 1.8 to prove that1.7 < 3 < 1.75[Hint: Let f(...
 4.8.33: Use the MeanValue Theorem to prove that x1 + x2 < tan1 x < x (x > 0)
 4.8.34: (a) Show that if f and g are functions for whichf(x) = g(x) and g(x...
 4.8.35: (a) Show that if f and g are functions for whichf(x) = g(x) and g(x...
 4.8.36: Let f and g be continuous on [a,b] and differentiableon (a, b). Pro...
 4.8.37: Illustrate the result in Exercise 36 by drawing an appropriatepicture.
 4.8.38: (a) Prove that if f (x) > 0 for all x in (a, b), thenf(x) = 0 at mo...
 4.8.39: (a) Prove part (b) of Theorem 4.1.2.(b) Prove part (c) of Theorem 4...
 4.8.40: Use the MeanValue Theorem to prove the following result:Let f be c...
 4.8.41: Let f(x) =3x2, x 1ax + b, x > 1Find the values of a and b so that f...
 4.8.42: (a) Let f(x) =x2, x 0x2 + 1, x> 0Show thatlimx0 f(x) = limx0+ f(x)b...
 4.8.43: Use the MeanValue Theorem to prove the following result:The graph ...
 4.8.44: Writing Suppose that p(x) is a nonconstant polynomialwith zeros at ...
 4.8.45: Writing Find and describe a physical situation that illustratesthe ...
Solutions for Chapter 4.8: ROLLES THEOREM; MEANVALUE THEOREM
Full solutions for Calculus: Early Transcendentals,  10th Edition
ISBN: 9780470647691
Solutions for Chapter 4.8: ROLLES THEOREM; MEANVALUE THEOREM
Get Full SolutionsCalculus: Early Transcendentals, was written by and is associated to the ISBN: 9780470647691. Chapter 4.8: ROLLES THEOREM; MEANVALUE THEOREM includes 45 full stepbystep solutions. Since 45 problems in chapter 4.8: ROLLES THEOREM; MEANVALUE THEOREM have been answered, more than 39729 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Calculus: Early Transcendentals, , edition: 10.

Deductive reasoning
The process of utilizing general information to prove a specific hypothesis

Equivalent systems of equations
Systems of equations that have the same solution.

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

Finite sequence
A function whose domain is the first n positive integers for some fixed integer n.

Infinite limit
A special case of a limit that does not exist.

Initial value of a function
ƒ 0.

Inverse cosecant function
The function y = csc1 x

Law of cosines
a2 = b2 + c2  2bc cos A, b2 = a2 + c2  2ac cos B, c2 = a2 + b2  2ab cos C

Linear programming problem
A method of solving certain problems involving maximizing or minimizing a function of two variables (called an objective function) subject to restrictions (called constraints)

Minor axis
The perpendicular bisector of the major axis of an ellipse with endpoints on the ellipse.

Multiplicity
The multiplicity of a zero c of a polynomial ƒ(x) of degree n > 0 is the number of times the factor (x  c) (x  z 2) Á (x  z n)

Oddeven identity
For a basic trigonometric function f, an identity relating f(x) to f(x).

Onetoone function
A function in which each element of the range corresponds to exactly one element in the domain

Opens upward or downward
A parabola y = ax 2 + bx + c opens upward if a > 0 and opens downward if a < 0.

Plane in Cartesian space
The graph of Ax + By + Cz + D = 0, where A, B, and C are not all zero.

Polar equation
An equation in r and ?.

Power rule of logarithms
logb Rc = c logb R, R 7 0.

Tangent line of ƒ at x = a
The line through (a, ƒ(a)) with slope ƒ'(a) provided ƒ'(a) exists.

Transpose of a matrix
The matrix AT obtained by interchanging the rows and columns of A.

Transverse axis
The line segment whose endpoints are the vertices of a hyperbola.