A partition P = {I0, I1, I2,...,In 1,Tn} of the interval [a, b) is said to be "regular" if the subintervals [2i 1,7] all have the same length given by Az = (b- a)/n. Let P, be a regular partition of (a, b). Show that if a function f is continuous and increasing on (a, b), then U(f, P,) – L(f, P;) = [f(b) – f(a)|Az.

A partition P = {I0, I1, I2,...,In 1,Tn} of the interval [a, b) is said to be "regular" if the subintervals [2i 1,7] all have the same length given by Az = (b- a)/n. Let P, be a regular partition of (a, b). Show that if a function f is continuous and increasing on (a, b), then U(f, P,) – L(f, P;) = [f(b) – f(a)|Az.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Describe the interval(s) on which the function is continuous. f(x) = sec 7EX 8 ..., (-16, -8), (-8,…

Q: Let f (x, y) = 1 5+√x²+y²-1 Then range of f is:

Q: f(x) = 3x² - 2x-2, [-1,2]

A: To determine if the function is continuous over the interval [−1,2], we need to check three…

Q: Suppose f is a function defined and continuous on the open interval (a, b). Suppose g is a function…

A: Given that: f is a function defined and continuous on the open interval (a, b).g is a function…

Q: p) State Cauchy's Schwarz Inequality and prove this for Space of Continuous functions on[-n, n].

Q: *The interval [a, b] for a function x^3 - 4 x- - ...... 9 = 0 is [2,3] O [0, 2] O [0, 1] O [1, 2] O

A: We need to find the interval for a function f(x)=x3-4x-9=0

Q: Let Z∼N(0,1). Find P(-2<Z<1).

A: From the provided information,Z~N (0, 1)The probability P (-2<Z<1) need to be calculated.

Q: التاريخ 27 أمر موح let I = (a,b) be an open interval show that - for every positive sM(IDS.

Q: Let /= (-00, 1] and let f(x) =V1-x for x E I. Let G = (-e, e). One of the following open intervals H…

Q: Show that the interval [a,b] is complete

A: Given

Q: 1) Consider the function f(x) = 2x³- 5x² +8 over the interval [1,5]. a) Approximate the area under…

A: Given: 1) f(x)=2x3-5x2+8 and the interval 1, 5. To find: a) The area under the curve over the…

Q: Let f be a function that is continuous on [1,7] and differentiable on (1,7). Suppose 2 < f'(x) < 3…

Q: 7. has a fixed point. ) Prove that if f is a continuous function from [2, 5] to [2, 5], then f

Q: Determine whether each of these functions from Z to Z is one-to-one. a) f(n)=n−1 b) f(n)…

Q: Let f be continuous on an interval I from a to b with only one critical number c in (a, b). If f'(c)…

A: Step 1: Step 2: Step 3: Step 4:

Q: 5. Let f be defined by the floor function f(x) = [] and let S = {-3, -2, -1, 0, 1, 2, 3}. a)…

Q: (d) Prore that the function f(x) =X is Continous but not unifomly Continous on the intenal

A: Use the absolun delta definition for uniformly continuous

Q: 3. 2- a) Find the average value of f on the interval [1, 5] b) Does the Mean Value for Integrals…

Q: Let f(x) = Vsin x. Compute the midpoint sum for f corresponding to the partition of [0, 2] into 20…

A: Draw the graph of the function .

Q: Find the domain and the range of the functions and describe it as regions in relevant spaces as well…

A: The objective is to evaluate the domain and range of x+116-x2-y22-z2. To find the domain and range…

Q: Let f1, f2, .. fn be n real-valued, differentiable functions.- Prove that e :[fif2 ... fn] = (fif2…

A: We have to prove that the given formula.

Q: let f(x) = 2x²-36x² + 20. Find the intervals where f is increasing. a) (-3,0)√(3, ~) b) (-3,0)0(0,3)…

A: Let To find the intervals where is increasing.

Q: Construct a continuous function that shows that the intervals [0,1] and [2,6] are homeomorphic

A: In the given question we have to find a continuous function from the interval [0,1] to the interval…

Q: What is an example of a function f : [2, 3] -->R which is not continuous at 1 but which is…

A: Here's an example of a function that is not continuous at 1 but is integrable on the interval [2,…

Q: Use the intermediate value theorem to prove that any continuous function with domain [0, 1] and…

Q: Prove that f(2) = z/(z* + 1) is continuous at all points inside and on the unit circle Iz] = 1…

A: we need to find those points where the denominator ( z^4+1) equals to 0

Q: Use a calculator to compute the left, sum, midpoint sum, and right sum for the function f, using a…

Q: Let f be a function defined on [11, 25]. Suppose for every x = (11, 25) d. dx -f(x) = 0. If f(12) =…

A: Given query is to find the value o f(24).

Q: Consider the piecewise linear function with graphs that pass through the following list of points on…

Q: consider the ff. function: f(x, y) = cot" "^ (1+x) - 2x + ³y 3y 2e

A: Given the function f(x, y)=cot-1(1+x)-2ex+3y Where x=s2+t2, and…

Q: Let f(x) = x-*,x > 0. Find the interval on which f is increasing or decreasing and find the local…

Q: 호호 -^ z-vz44 Z

Concept explainers

Riemann Sum

Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.

Riemann Integral

Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.

Question

A partition P = {ro, I1, T2,. .. ,Tn 1, Tn} of the interval [a, b] is said to be "regular" if the subintervals
[2i 1, Z;] all have the same length given by Az = (b – a)/n.
Let P, be a regular partition of (a, b]. Show that if a function f is continuous and increasing on [a, b), then
U(f, P,) – L(f, P,) = [f(b) – f(a))Az.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Definite Integral$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Show that C[0, 1] ⊆ L1[0, 1]. Compare ||f||sup and ||f||1.Justify your answer. Generalize this to the interval [a, b].
Let f be a function defined on [0,2], and f (x) = -x² + 1, Find L (p. f) only where p, is sequence of partition divide the interval into n equal parts.
Consider the function f(x) 1 on the interval [- 4, 4). a. Is f(-4) = f(4)? Yes No b. Is f continuous on [- 4, 4]? Yes No c. Is f differentiable on (- 4, 4)? Yes No d. What do your answers to parts a., b., and c. imply? O Rolle's Theorem can be applied The Mean Value Theorem can be applied O Both Rolle's Theorem and the Mena Value Theorem can be applied O Neither Rolle's Theorem nor the Mean Value Theorem applies to this problem e. If you can use Rolle's Theorem or the Mean Value Theorem, find the value of c satisfying said theorem. If not, enter DNE. Answers should be in exact form (i.e., do not use a calculator to get a decimal approximation). c = 4. 5n 6.
Write down the first order and second order conditions for maximum and minimum of afunction y = f(x1, x2, x3,............ xn).
Let f :[-2,6] R be the function given by * if -2 sx
c) Evaluate f x*(In x)²dx
Graph the function f(x) = x + 5/x and the secant line that passes through the points (1, 6) and (10, 10.5) in the viewing rectangle [0, 12] by [0, 12].
The function flo)=- x° +) satis fies the hypothess of the mean-Value Theorem over the interval To,27 since this a polynomial. Find all valves of c in the interval (0,2) at which the tangent line to the graph of f(x) is parallel to the secant line joining the points (9, 4 (0, f (0)) and (2, f(2) . Do Not Rond. Exact Anauers.