Find the limit of definite integral ∫_1^3 (16-x2), when P is the partition of [1, 3] into n equal subintervals.   Determine expressions for L(P,f) and U(P,f).  Find their values for n = 4 and n = 40, and take the limit of L(P,f) and U(P,f) as n goes to infinity

Q: Let xn -> 0 and (yn) be bounded. Prove xnyn -> 0.

A: A sequence is bounded if it is bounded above and below, that is to say, if there is a number, k,…

Q: Suppose f is an odd continuous function whose mean value (with respect to integral) on the interval…

A: We have to use property of odd function here.

Q: Provide a function f, continuous on (0, 1), such that for all r where f'(x) exists, f'(x) > 0 but f…

A:

Q: Let x, y ∈ R. Suppose x^2 +y^2 = 0. Prove that x = 0 and y = 0.

A:

Q: Calculate, if possible, the area of the region that lies between the curve of the function f(x) = _6…

A: First, we should recognize that the function F(x) is defined for all real numbers x except for x = 2…

Q: Linear operator A = [0 1;1 1]: R^2 to R^2, omega = {x in R^2 | x1<3, x1-2x2>-2, x1+2x2>2}.…

A: To sketch the range A(ω) and the initial domain ω for the linear operator A = [0 1;1 1]: ℝ² → ℝ², we…

Q: If the percent decrease is 25%, what is the common ratio?

A: Common Ratio The common ratio is defined as the amount between each number in a geometric sequence.…

Q: Express the limit as an integral. lim (8(x)²-3(x) ³)Ax over [0, 3] /=1 X ) dx

A: Express the limit as an integral.limn→∞∑i=1n(8(xi∗)2−3(xi∗)3)Δx over [0, 3]

Q: Find an area similar to the example f(x)=x^2 from [0,1] that we couldn't have computed from…

A: Given That: f(x)=x2 To Find : Area using Riemann sums

Q: Let a be a positive constant. Let f(x) = x (Vx +a). Which of the following numbers is\are critical…

A: given function :

Q: d. y= x² + y x + cos y

A:

Q: Let X be f(x) = x^(2/3) , X is a subset of R x R satisfying the given equation. Define a bijective…

A: Given X be the set of all points x,fx, where fx=x23. That is X=x,x23: x∈ℝ. Now define a map g:X→ℝ as…

Q: Is a function Continuous ?why? integrable anod

A:

Q: Approximate the area under f(x)=3x^2 +1 on [-5,0] using RIGHT Riemann sum with 5 approximating…

A: According to question,Given Function,f(x)=3x2+1,In the interval -5, 0

Q: 22. Explain why the Mean Value_Theorem for Integrals applies to the function f(x) = x² on the…

A:

Q: find the maximum and minmum values of f(x,y) = y^2 + 6x^2 on the set R={(x,y) 2x^2+y^2<=9}

A: Given function is: fx,y=y2+6x2 and the given set is R=x,y:2x2+y2≤9

Q: 2. Which function is bounded both above and below? B. 1. f(x) = x' S(x) = x+2 %3D С. D. f(x) = 4e…

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Q: A function h(x, y) is defined by h(x,y)=(x^2 y)/(〖7x〗^6+y^3 ). Verify the limit over h(x, y) exists…

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Q: 5. (Section 4.2) Use 4 equally sized subintervals to approximate the area under the graph of 1. f(x)…

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Q: Find  limit x 0  1/x^2if it exists.

A: We find the limit numerically

Q: let f: R^ -> R be a function given by 2 f(x₁, x₂... x₁) x.x²2. x²n, where = € 2.1 Show that the…

A:

Q: Suppose f" is continuous on (-00, 00). (a) If f'(-5)=0 and f"(-5)= 5, what can you say about f? ○ At…

A: Q(a)At x=-5, f has a local minimum.Q(b)More information is needed to determine if f has a maximum or…

Q: Show that if F: AB and G, H F"" (GUH) = F"" (G) U f" (H) and F" (Gn H) = F²¹ (6) ^ F-1 (H) are…

A:

Q: Let f(x)= X^2+1 on the interval [-1,2]. find the area under the graph as a limit of a reimann sum

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Q: Express as a definite integral the limit as x approaches infinite of 1/n[(1/n)^2 + (2/n)^2 +...+…

A: Step 1:

Q: do 2

A: Given- Let fx = 7x be a function defined on 0, 2.  To find-  Find the simplified form of the…

Q: The function f(x)=4e^x is defined over the interval [-2,3]. Appproximate the area A by partitioning…

A:

Q: Rocky Mountain Chocolate Factory (RMCF) founder and president Frank Crail employs 220 people in 361…

A: for net price  equivalent rate determine first find the complement of the chain discount rate…

Q: d (n, m): = 0 si n = m si n‡m 5-c(n-m)

A: Question seems wrong

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.