It would be nice if average values of integrable functions obeyed the following rules on an interval [a, b] . a. av(ƒ + g) = av(ƒ) + av(g) b. av(kƒ) = k av(ƒ) (any number k) c. av(ƒ) <= av(g) if ƒ(x) <=g(x) on [a, b] . Do these rules ever hold? Give reasons for your answers.

Q: If f(x) is an integrable function defined on the closed interval [a, b] ], ƒ (x) dx represents the…

A:

Q: Which of the following is always true for integrable functions f and g? ® Sirw)+ g(x}]dx = [owax+…

A:

Q: et [a, b] be a real interva b]→X is continuous, the is continuously differentia

A:

Q: Let $c$ be a critical number of a function $f$ that is continuousinterval, except possibly at $c$,…

A:

Q: f(x) is a positive continuous function on R what does the function F(x)= | f(t) dt a represent (for…

A:

Q: Find the average value of f(x) over the interval [−3,2] given that f(x)=−5x+2

A:

Q: For a continuous function f(x) on the set 2 [- 5,2], the net area Tx)dx must be -5 always positive…

A: The given question has been solved in detail

Q: Find the average value of the function on the given interval. а. f(x) 3D seсx tan x on [0, п] 6x2 b.…

A:

Q: Is f(x) = ln(x) uniformly continuous on (0, 1)? Justify your response.

A:

Q: If f(x) and g(x) are integrable on the closed interval [a, b], and k is a constant, which of the…

A: B option is correct.

Q: f(x) = 1+x tan? r is f(x) uniformly continuous on [0, +o0)? Prove your result.

A: TheoremIf f:[0,∞) →R Be continuous on[0,∞) and limx→+∞ f(x)=0, f isuniformly continuous on…

Q: If f(x) and g(x) are integrable on the closed interval [a, b], and k is a constant, which of the…

A: Topic:- application of integration

Q: If f(x) and g(x) are integrable on the closed interval [a, b], and k is a constant, which of the…

A:

Q: If f and g are continuous integrable functions on [a, b] then Lo f (x) + g (x) dx = [" ƒ (x) dx + S"…

A: To prove ∫f(x)+g(x)dx=∫f(x)dx+∫g(x)dx , All are definite integrals with upper limit a and lower…

Q: Find the average value of the function f (x) = x – x² on the interval [0,2]

A:

Q: • Let X be a continuous r.v; with pdf f(x) : 0 < x < 1, x, f(x) = {2 – x, 1< x < 2, 0, elsewhere.…

A: We have given that, Let X be a continuous random variable with pdf: f(X) = {  x,   0&lt;x&lt;1…

Q: Say if the following statement is true or false: Suppose f is continuous at a critical point c. If a…

A:

Q: Find the average value of the function f(x) = 4(x + 1)/x2 , over the interval     [2, 4].

A:

Q: Let a, b, c, d be numbers and f(x,y) be any function. Then . f(x, y) dxdy This statement is --- --…

A:

Q: What value of c makes the following function continuous at (0,0)? f(z, y) = C, x+ y³ +2, (x,y) +…

A:

Q: Suppose that X1,..., Xn are i.i.d. following pdf given by f(x; 0) = že-5, where both x and O are in…

A:

Q: Find the normal line to f(x) = ax - 3ax at x=2. Assume that a is a positive constant.

A: The given function fx=ax2-3ax at x=2.   Formula used:   Formula for finding normal to the function…

Q: Is f(x) = x² uniformly continuous on (0,1)? Why or why not?

A:

Q: Find the center of mass of the thin plate bounded by the graphs of the given functions. g(x) =…

A:

Q: Use the results of Exercises 15 and 16 to find the area of the region Ω shown in Exercise 13.

A: In questions 15 and 16, we can find the area of the bounded region by using the definite integral…

Q: (3) Determine the values of a for which x4 + y* , if(x, y) # (0, 0), (x² + y?)ª f(x, y) = ,if(x, y)…

A: solution.

Q: Let f be an integrable function and g be a differentiable function over R and let ∫02f(3x+5)dx=16.…

A:

Q: Find the point(s) on the interval (0, 1) at which ƒ(x) = 2x(1 - x) equals its average value on  [0,…

A: Given: fx=2x1-x we know that average value of any given functionfx on interval a,b is given by…

Q: Suppose f is continuous on the closed interval [a.b]. If Fis an antiderivative of f for all real…

A:

Q: Let (s n(x-y) |x|+ly\ = 0 f(x, y)={ |x|+|y]| (0, (x, y) = (0,0) Is f continuous at the origin? Why?

A: The continuity of the function can be checked by checking the limits of the function at the point.…

Q: Find the normal line to f(x) = ax - 3ax at x=2. Assume that a is a positive constant. %3D

A: Given: f(x)=ax2-3ax &amp; x=2 To determine: Equation of normal line to given f(x)

Q: At what points (x,y,z) in space are the functions continuous? a. h(x,y,z) = In (5z° - x-y² - 3) 1 b.…

A: In this problem, we have to check condition for having the function continuous.

Q: Find the average value of the function over the interval f (x) = x3 − 2x + 2, [−1, 2]

A:

Q: If  m ≤ f(x) ≤ M  for  a ≤ x ≤ b,  where m is the absolute minimum and M is the absolute maximum…

A:

Q: f m ≤ f(x) ≤ M for a ≤ x ≤ b, where m is the absolute minimum and M is the absolute maximum of f…

A: Given integral is ∫02x3dx Given property  If m ≤ f(x) ≤ M for a ≤ x ≤ b, where m is the absolute…

Q: Let f(x) = e2sinx. Use the local linear approximation of fat x = 0 to approximate e²

A:

Q: If |fn(x) – fn(y)l 0 and all x, y in a compact interval, show that {fn} is uniformly…

A:

Q: If f(x) and g(x) are integrable on the closed interval [a, b], and k is a constant, which of the…

A: Definite integral Property : ∫ab fx±gx dx = ∫ab f(x) dx ± ∫ab g(x) dx. ∫ab k f(x)dx =k∫abf(x)dx ;…

Q: true or false with reason c) If f and g are continuous and f(x) ≥ g(x) for a ≤ x ≤ b, then…

A: Given: If f and g are continuous and, then  integration ∫abf(x)dx&gt;∫bag(x)dx

Q: 2. The function f in |f(x)dx = F(x)+C is called A. integral B. derivative C. antiderivative D.…

A:

Q: If f(x) and g(x) are integrable on the closed interval [a, b], and k is a constant, which of the…

A:

Q: Suppose X has the density f (x) = (2 – x)² for 0 < x < 5.

A: Given that fx=3352-x2 0&lt;x&lt;5

Q: The average revenue is defined as the function R(x) Ř(z) (x > 0) Prove that if a revenue function…

A: Let Rx denotes the revenue function. The average revenue function is defined by Rx=Rxx where x&gt;0.…

Q: Compute the average value of the function f(x) = (0x² + ax + B.n)e* on the interval [-a,0].

A:

Q: It would be nice if average values of integrable functions obeyed the following rules on an interval…

A:

Q: Estimate f(x) dx given that f is a continuous function containing (-2,8), (–1,0), (4, 125) , and…

A: We will first construct the Lagrange interpolation polynomial for the Newton-Cotes quadrature…

Q: give an example where f is not derivable at [0, ∞), but is uniformly continuous

A: give an example where f is not derivable at [0, ∞), but is uniformly continuous

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.