Suppose that f:I → R and g:I R are differentiable at ce I. Prove that if k E R, then the function kf is differentiable at c and (kf)'(c) = k f'(c).

Q: Let f € L'(0,1). Assume that f(t)t2"dt = 0 for all n f = 0 a.e. 1,2,3,.... Prove that %3D

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Q: Assume that is differentiable on a ≤ x ≤ b and that f (b) < f (a). Show that f' is negative at some…

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Q: Let f : R → (−1, 1) be given by f(x) = x/(1 + |x|). Show that f is differentiable at x = 0, with…

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Q: Show that a function f : R? → R defined by S (7, y) = ! if (x, y) + (0,0) 0, if x = y = 0 is…

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Q: Let f(x) = e2sinz 1. Use the local linear approximation of f at x = 0 to approximate e2 sin(0.01

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Q: Let F(x) –3 cos(ť²) dt. Use the Fundamental Theorem of Calculus to find F'(x). F'(z) =

A: Given: F(x)=∫x−1(−3cos(t2))dtF'(x)=ddx∫x−1(−3cos(t2))dt=?

Q: (e) If f(x) is reducible over Q then f(x +a) : a E Z is reducible over Q.

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Q: A function (f) is homogeneous of degree • if and only if : ôf + y = Af (x, y). Does the function…

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Q: For the function f(x, y) show that fx(0, 0) and fy(0, 0) both exist but that f is not differentiable…

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Q: Prove that if f :V → W is differentiable at a point a E V it must be continuous there.

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Q: Suppose that z = f(x, y) is differentiable at the point (1024, 256) withf(1024, 256) = 2 and f,…

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Q: Suppose f(r) z2 + sin z and F is an antiderivative of f that sausfies F(T) 0. Then F(2) = ( -1)

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Q: If f is a differentiable function at a, then xf is differentiable at a and (af)'(a) is: Select one:…

A: We need to evaluate the following expression xf'a

Q: iii. is continuous at a provided g(a) # 0.

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Q: Let g(x, y) = x4 + y² – 4x³ + 5x – 6y+3. Show that g(x,y) continuous (actually at least aC')…

A: Recall [ Implicit Function Theorem ]

Q: Let F(x) E C'[0, 1] and F(x) be twice differentiable in (0, 1). Suppose that f(x) = F'(x) E R[0,1]…

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Q: Let h(x) = f(g(x)), where f and g are differentiable on theirdomains. If g(1) = 3 and g′(1) = 5,…

A: Given: Let h(x) = f(g(x)), where f and g are differentiable on theirdomains. If g(1) = 3 and g′(1) =…

Q: Suppose f(z) = r + sin z and F is an antiderivative of f that satisfies F(#) = 0. Then F(z) =

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Q: Suppose f(x) is continuous on R such that F(x) is an antiderivative of f(x). Furthermore i Find F(1)…

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Q: Let f(z) (x-iy)e- ) Determine the points at which exist, and find r(2) at these points.

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Q: If the function of f:[0,4]-R is differentiable, then show that 4 (i) [f(t)dt = 2| af (a²)+Bf (8)* |…

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Q: Let g(x) = 8 In(t) dt. Use the Fundamental Theorem of Calculus to find g'(x). gʻ(x) =||

A: The given function,         gx=∫1x8 ln(t) dt

Q: Show that the subtangent of f (x) = ex is everywhere equal to 1.

A: Formula to find the subtangent of the function helps to find the required subtangent. Derivative of…

Q: If z is a function of x and y implicitly defined by əz əz find and əx ду at (x, y, z) = (0, 2, -1).…

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Q: Let f (x) belong to Zp[x]. Prove that if f (b) = 0, then f (bp) = 0.

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

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Q: Find the linearization L(x) of f(x) = cot x at x = 4 %3D -

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Q: Given f(x)= x' – rx + k where r E r > 0 and r # 1, and k to be any constant. Prove tha if 0 1, fhas…

A: The given function: Where and k to be any constant. To prove that: If 0&lt;r&lt;1, f has a…

Q: If f:R⇒Rf:R⇒R is defined by f(x)=e−cosxsin(x)f(x)=e−cos⁡xsin⁡(x) for every x∈Rx∈R then…

A: The given function is: f(x)=e-cos x sin x To find the derivative first. f'(x)=ddxe-cos x sin x…

Q: Consider the function F whose criterion is = / dt. 1+ t2 F(x) sena When determining dF/dx the result…

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Q: 10.Show that F(x, y) = (1 + xy)e*yi + (ev + x?e*y)j is conservative and find a function "f" such…

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Q: Suppose fn → f and the functions fn all satisfy the Lipschit condition |fn(x) – fn(y)| < Mx – y| for…

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Q: Let g be a continuous function. Using the substitution u = 3x – 1, the integral (g (3x – 1)) dx is…

A: We can use the concept of method of substitution in order to solve the given problem.

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 if m(x) is differentiable on (−∞,∞) and its derivative m'(x) is bounded then m is…

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Q: Let f(x) = V31 – x. Compute f'(6) using the limit definitic f'(6) = %3D Find an equation of the…

A: Given: f(x)=31-x

Q: Suppose that the functions f and g, with linear approximations L, and Lz at x = a as shown below.…

A: Limit problem

Q: Show that the p.d.f. of Z is given by g1(r) = e-1(+A)cosh(vAx), x > 0, where cosh(2) = }(e² +e=*) is…

A: Note: Hi there! Thank you for posting the question. As you have posted multiple questions, as per…

Q: Consider the following. F(x) = %3D t(t8 + 1) dt (a) Integrate to find F as a function of x. F(x) =…

A: Part (a)

Q: Show that there is no differentiable function f on [0, 2] with f (0) = -4, f (2) = 4 and %3D f' (x)…

A: Given: A function f such that f(0)=-4,f(2)=4 and f'(x)≤3, ∀x∈[0,2] According to Mean Value Theorem,…

Q: Suppose that f:I → R and g: I → R are differentiable at c E I. Then, a) If k E R, then the function…

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Q: Let f be a C' function on the line, and let g(x) = S f(xy)y² dy. Prove that g is a C' function and…

A: f ∈ Cn  means f is atmost n times differentiable

Q: Suppose that F(x, y, z) = 0 defines z as a function of x and y and F is differentiable. Prove that…

A: This is a problem of the chain rule and Implicit function theorem.

Q: Assume that ƒ is differentiable on a&lt;=x&lt;=b and that ƒ(b)&lt;ƒ(a). Show that ƒ′ is negative at…

A: Given,