Let f be differentiable on [a, b]. Suppose that f'(x) 2 0 for all x E [a, b] and that f' is not identically zero on any subinterval of [a, b]. Prove that f is strictly increasing on [a, b].

Q: If f(t) is a continuous function on the interval a, b), then the integral | f(t) dt is defined to be…

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Q: Show that if f is differentiable and f'(x) > 0 on (a, 6), then f is strictly increasing provided…

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Q: consider the function f defined on [0,∞),f(x)=(x^r)sin(1/x),for x≠0 and f(x)=0 and for x=0, where r…

A: Solve the following

Q: Let the function ƒ : R → R be given by 1- cos(x) if x + 0, f(x) := if x = 0. %3D 1 Prove that f is…

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Q: Use the limit definition of definite integral in the form of f(æ) dæ = lim {(#:)Ax, i=1 6 - a and r;…

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Q: Define f : R2 → R by letting (y² – x)² (y4 + x²) f(x, y) = if (x, y) # (0, 0), and f(0,0) = 1. ||…

A: The given function is f:ℝ2→ℝ defined as: fx,y=y2-x2y4+x2if x,y≠0,01if x,y=0,0 Claim: The given…

Q: Suppose f is integrable on [-b, b] and f is an odd function-that is, f(-t) = -f(t) for all tE(-b,…

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Q: Suppose a function a increases on [a, b}, a < xo sb, a is continuous at xo. Let f be a function…

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Q: Define f : R" → [0, 0) by letting f(x) = (1 – ||æ|| 1. Prove that f is continuous on R™. (To prove…

A: As per the instruction, we will be using the Lemma 3 (which is written in the question) to show that…

Q: Consider the function f(x, y) = V[ry[. Show that Taylor's expansion about the point (x, x) is not…

A: We know the Taylor series expansion of a function f(x, y) is given by, f(x,y)=f(a,b) +(x-a)…

Q: Let f be the function defined by S1+x, when 0<x<}, }<x<1 0, when x-1. f(x)= Show that fis integrable…

A: Recall the Result/Theorem A bounded function f which has only a finite number of points of…

Q: 2. Let f(x) = { I , if r € [0, 1] 1, if r € (1, 2] Using the definition of the definite integral,…

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Q: Define f(0,0) in a way that extends f to be continuous at the origin. 2.2 11x - xy + 11y 2 f(x,y) =…

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Q: let f be a function on [0,1] defined by {x ifxcQn [01] F(x) = -xif x = [0,1] \ \ (Recall that is the…

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Q: If f is continuous everywhere except at x = 2, then f(x +1) dx is not integrable. True False

A: To check whether the given statement is True or False."If f is continuous everywhere except at x=2,…

Q: Let f be continuous on the interval [a,∞). Show that if the improper integral ∫∞a ∣f(x)∣ dx…

A: If f  is a continuous function on the interval [a, ∞). If  ∫a∞f(x)dx converges, Then, we use…

Q: Let f (x) = x² +2x +1. Use the Riemann sum theorem to calculate the following limit. It is an…

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Q: Consider the function F(x) = | s(t) dt , where a is a constant and f is continuous everywhere. By…

A: Find we have to use definition of derivative.

Q: For real number p, let f and g be defined by f(x) =e¯*x³ and g(x) = 1/xP, respectively, on [1,0).…

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Q: Suppose f is continuous ont he closed interval [a, b]. If F(x) is an antiderivative of f(x), for…

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Q: Suppose that |- f(x) is not constant. Il- f(x) is differentiable on 0, o0 II- S f(t)dt = (x +1)f(x)…

A: In the given question we have to choose a function which satisfies the given three condition in the…

Q: Assume f(x) is continuous on [a, b]. Prove that if f(x) > 0 for all x € [a, b], then S. f(x) dx > 0.…

A: Extreme Value theorem states that if a real-valued function f is continuous on the closed inteval…

Q: Let f be differentiable on [a, b]. Suppose that f' (x) > 0 for all a E [a, b] and that f'is not…

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Q: Suppose f is Reimann integrable on [a,b]. If f (x)< 0 for all x E [a, b], then S f(x)dx

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Q: Q2: If f(x) is Riemann-Stieltjes integrable function with respect to g(x) on [a,b] and if…

A: It is given that,  f(x) is Riemann-Stieltjes integrable function with respect to g(x) on a, b such…

Q: Suppose that f is integrable on [a.b] and define the functions f*(x)= [ f(x), f(x)20 f(x) s0 and…

A: Given f is integrable on a,b and define a function f+x=fx, if fx≥00     , if fx&lt;0 and f−x=0, if…

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

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

A: If intergral is define in given interval and contious we can add a constant value

Q: Let f be a positive decreasing function defined on [a, +co] such that f(x) →0 as x→ + o. Let a be…

A: Consider the given positive decreasing function f defined on the interval a, +∞ such that fx→0 as…

Q: Consider the function f defined by V10t – Vt, 0 e Using the conditions for continuity, determine if…

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Q: Suppose f> 0 is integrable on [a, b], continuous at xo € [a, b], and f(xo) > 0. Prove that f>0.

A: Given f≥0 is integral on a,b continuous at x0∈a,b and fx0&gt;0

Q: Use the Cauchy-Riemann Equations to show that f(z) = z is differentiable for 16. all z and find…

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Q: Consider the function defined on the closod interval (a, b) as f1 if I=c h(r) = if Itc where a <c<b.…

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

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Q: Consider the function f defined on [0,∞) f(x) = xrsin(1/x), for x≠0 and f(x) = 0, for x=0 where r >…

A: Given: f(x)=xr sin1x,   x≠00         ;     x=0     a) f is continuous on [0,∞) f is continuous if…

Q: suppose f (x) = Vx (x>0),prove that f is differentiable (0,00) and find f'(x) on

A: Function f(x) = √x (x&gt;0)  We know f(x) =Xn

Q: Prove that each of the following functions is uniformly continuous on (0, 1). sin z (a) f(x) = (b)…

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Q: Suppose a function a increases on [a, b}, a < xo Sb, a is continuous at xo. Let f be a function…

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Q: If f is integrable on [a,b], show by example that F (t) = | f (x) dx is continuous but need not to…

A: We need to show by an example that if f is integrable on [a, b], then F(t) (as defined in the…

Q: Assume that J is continuous on [0, 3 and differentiable on (0, 3) with f (z) > 6 for each z in (0,…

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Q: Assume f and g are functions that are both integrable on the closed interval [a, b]. Prove that if…

A: Given f and g are integrable functions on the closed interval [a,b]. And f(x)≤g(x) for all x∈[a,b]…

Q: Consider the following theorem. Theorem If f is integrable on [a, b], then [ºf(x) dx = dx = lim…

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Q: Consider the following theorem. Theorem If f is integrable on [a, b], then f(x) dx lim f(x;)Ax %3D…

A: This way of evaluating definite integral is known as " Integral using limit of sum " Here we will…

Q: Q2: If f(x) is Riemann-Stieltjes integrable function with respect to g(x) on [a,b]and if…

A: detailed explanation mentioned below.

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: 1. Given that f is continuous on a, b|, compare f(b)(b – a) | f(x) dx, and if f is constant on a,…

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Q: Consider the following theorem. Theorem If f is integrable on [a, b], then FX) dx = lim ° fx;,)ax n-…

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: