If f is continuous on [-1, 1] and f(x)dx = 0, then there exists some ce [-1,1] for which f(c) = 0.

Q: Suppose f and g are continuous functions on [a, b] and that g(x) > 0 for all x E [a, b]. Prove that…

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Q: Find the value(s) of C in the given interval that satisfies the mean value Theorem for integral.…

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Q: Determine the values of a, b, and c so that the function (sin(a + 1)x + sin x x0 is continuous for…

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Q: Suppose f is continuous everywhere, and | f(x) dx = -1 and 3 Find f(2)dz. f(x) dx 2. Find dx. Cannot…

A: As per our guidelines we are supposed to solve only one question i.e. the first question. For the…

Q: Find the global maximum and minimum of the continuous function f(x) = x²-8 ln x on [1,4]. %3D

A: Given:F(x)=x2-8lnx        1,4

Q: If f is continuous on [a,b], and f(x) > 0 then b [²√f(x) dx = a √5°₁ S f(x) dx

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Q: Let f : [0, 1] → [0, 1] and f be continuous on [0, 1]. Show that there exists x E [0, 1] such that…

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Q: Let f(x) be a continuous function on R-{0} such that (f(x) dx exists for all a > 0.If g(x)= |- dt, t…

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Q: and g are continuous on [a,b] and f(x) > g(x) for all x €[a,b] then f(x)dx> ] g(x)dx a

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Q: If f : [0,2] → R is defined by f(x) = 1 if 0 ≤ x ≤ 1, f(x) = 2 if 1 < x ≤ 2. Show that f is…

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Q: =f and g are continuous on [a,b], then [*[f(x) + g(x)]dx = f(x) [ √° g(x) f(x)dx+ Sº g(x)dx (A) True…

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Q: Find the Wronskian for the set of functions.{x, sin x, cos x}

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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: 2. If f is a continuous function such that S(t)dt = cos(2x)e* - [e s(t)dt for all x, find an…

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Q: 4. If f(x) and g(x) are continuous on [a, b], which one of the following statements is true? [f{x)–…

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Q: "For a continuous function f(x), defined on an interval a , b, we have d f(x) dx = f(x), dx a…

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Q: Suppose that f (x) is a continuous function, f(-1) = 6, f(2) = and f(x)dx: = -3 = T. Evaluate the…

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Q: 1. Suppose that f : [0, 0) → R is continuous and f(x) # 0 for all x > 0. If we have (f(x))? = 2…

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Q: Suppose f and g are continuous functions on [a,b] and that g(x) > 0 for all x E [a, b]. Prove that…

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Q: It is frequently required to solve equations of the form f(x) = 0. When f is a continuous function…

A: To solve an equation fx=0, if f is continuous on an interval a,b such that fa and fb are of opposite…

Q: Find the Wronskian for the set of functions.{x, ex, sin x, cos x}

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Q: Let f and g be continuous on the closed interval [0, 5]. Suppose that J f(x)dx = A, f(x)dx = B, and…

A: Given: ∫05f(x)dx=A∫03f(x)dx=B∫53g(x)dx=C…

Q: Prove that if f is continuous on [a, b] and F(x) = ſx ƒ (t)dt α then F is continuously…

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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: 10. If f(x) and F(x) then f(2x)dx = are continuous functions such that F'(x) = f(x) on R, %3D %3D

A: We have to solve given problem:

Q: Let f be a continuous (and therefore uniformly continuous) function on [a,b). Show that, given e> 0,…

A: Consider the given information.

Q: If f is a continuous function satisfying f(x) = x* - x+(1-x)[,f'f)dt + x[ (t -r°)f(@dt for all x,…

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Q: 2. If f is continuous on [0, 1] such that x* f(x) dx=0, for n=0, 1, 2, 3,.. then show that f(x)3D0…

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Q: Find numbers a and b, and c, so that f is continuous and differentiable at every point. 4) x- + C,,…

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Q: 4. If L is a Continuous function on H, then unique YEH such that L(x)=(x,y). уєн there is a

A: Given that L is continuous on H. To show: y is unique in H.

Q: 6. Let f be a continuous function on [0, 2] whose second derivative exists on (0, 2). Assume f(0) =…

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Q: Find a value of k making h(x) continuous on [0, 5]. kx , 0 < x < 1 h(x) = x +3, 1<x < 5

A: Given: hx=kx,0≤x&lt;1x+3,1≤x≤5 given function will be continuous over 0,5, if it will be continuous…

Q: Suppose that f has continuous derivative and is twice differentiable in R. Suppose that f(a) = f(b)…

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Q: 1. Suppose f(r) is a continuous function on r E [a, b]. Prove f(x)dx < Slf(x)| dx.

A: According to our guidelines we can solve only first question and rest can be reposted.Thank you

Q: If f is continuous on [a,b], then the function g defined by g (x) = S f (t) dt, a < x < b is…

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Q: Find the linearization L(x) of the function at a. f(x) = cos(x), a = 3n/2 %3D

A: We have to find the linearization L(x) of the given function.

Q: Find a value of k making h(x) continuous on [0, 5]. 0 <x < 2 x +1, 2<x < 5 eka, h(x)

A: Givenh(x)=ekx    0≤x&lt;2          x+1  2≤x≤5To find a value k when h is continuous on…

Q: Suppose that f is a continuous function on R and, g and h are differentiable functions on R.…

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Q: Can the Fundamental Theorem of Calculus be used to find x²-x+3 dx ? x²-x+3 O No, f(x) = is not…

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Q: Let g be a continuous function such that g(x) dx = –12. Compute the 1 definite integral x° g(3x*)…

A: Definite integral: ∫03 g(x)=-12We need to find the value of ∫01 x3g(3x4) dxLet3x4=z12x3dx=dz

Q: Let f be a differentiable function with f(0) = 1 and / f'(1) dt = x². Then Of(x) = COS X Of(x) = (x+…

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Q: (3x² -1, x 1, Determine c and d so that f(x) is continuous everywhere.

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Q: 2-3. If f is integrable on [a, b], then ("f(x) dx = lim E f(x) Ax i=1 • In the theorem above, define…

A: We know that integration is equivalent to summation as mentioned above. Here, △x is the small step…

Q: 1. Suppose f(r) is a continuous function on r € [a, b]. Prove f(x)dr < S°S(x)| dx.

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Q: A function f is defined by f (x1) = / Cos(x – t) os(x-t) dt cos(t) f (x) : 01 Simplify f(x) and…

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Q: Let f(x) be continuous on [0, oo) and twice differentiable on (0, co). If f'(x) > 0 and f"(x) 1,…

A: Let f(x) be a continuous function on [0,∞) and differentiable at 0,∞. The given double derivative…

Q: Consider the sequence of functions {fn} defined on the interval (1, 0), where fn (x) 1 Vx E (1, 0),…

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Q: The Fundamental Theorem of Calculus (Part 1) states that if f(x) is a continuous function on an…

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.