2. Assume that h: [a, b²] → R is a continuous function and let G : [a, b] → R denot the following function, G(x) = [*² h(t)dt. Show that G is differentiable and find its derivative.

Q: Consider the following formula to compute the third derivative of f: ƒ”" (x) = −…

A: The given formula for the third derivative of…

Q: f(x)=210, where is it continuous?

A: Given: fx=210 We know that graph of given function is horizontal line for every value of x

Q: Suppose F(x) is a differentiable function for all real numbers x. Evaluate the following derivative.…

A: We need to find the derivative of the function F(x^4) with respect to x.

Q: The derivative of g() Select one: O a. Positive for all x O b. Negative for all z O c. Negative for…

A: To find- The derivative of gx = 12x is Positive for all x Negative for all x Negative for x > 0…

Q: Suppose that f(a) = f(b) = g(a) = g(b) = 0 and the second derivatives of f and g are continuous on…

A:

Q: 1. Show that a frination fiłamb ]->1R is differentiable at xolamb) is and only if its lower and…

A: Solution: f:[a,b]→ℝ is differentiable at x0(a,b). Let us take upper derivative at x0+. Then,…

Q: 1. Let f(x)=√x-2 / x-4 . Show that f(x) has a removablediscotuiti t x=4   2. Given f(x)= √(x+1).…

A: Here we have to find the derivative of a function at a point.

Q: x²+x' Evaluate the derivative at x = 1 for y = In () %3D

A: Some common derivatives are, ddx(ax)=ax.loga+C ddx(logf(x)=1f(x).f'(x)+C Quotient rule of…

Q: a) Calculate the derivative of f Cx) by first expanding and Simplifyıng the function fCx)=(2-3x) 2

A: Given,           fx=2-3x2

Q: f(x) = 4√x + 2x1/2 – 8x-7/8 + x2 1 23 +2

A: The given function is fx=4x+2x12-8x-78+x2-1x3+2. Find the derivative of the given function. Here we…

Q: 6. Use the definition of the derivative to show that the function f(z) = Z is nowhere…

A:

Q: Find the derivative Of 2x² usino efinition of The derivative -SPF pjest W

A:

Q: Consider the function y=g(x)=-x2+3. Use the limit definition of the derivative to compute a formula…

A:

Q: If f has a derivative at x = 2, then f is continuous at x = = 2. True False

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Q: 3. Express the derivative of f (x) = Vx as a limit of the difference quotient ("h → 0")

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Q: Let ƒ (x) E R[x]. Suppose that f(a) = 0 and f'(a) = 0, where f'(x) is the derivative o f(x). Show…

A: Since, fa=0, fx can be factorized as fx=x-angx where n is a positive integer and g(x) is another…

Q: Use the definition of derivative, f'(x) = lim! the derivative of the following function: to…

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Q: Let f(x) E R[x]. Suppose that f(a) = 0 but f' (a) + 0, where f' (x) is the derivative of f(x). Show…

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Q: 7 x = R¹, ƒ(x):=· + 3√x² 1-x 1+x² +ea² +1 +π².

A: The power rule of differntiation states that d dxxp=pxp-1, where p is any real number. The quotient…

Q: Q9. Provide the following examples: (a) a function g whose domain is R\{7} and whose derivative at 0…

A: To determine: a) A function g whose domain is ℝ\ 7 & whose derivative at 0 equals 2. b) A…

Q: A not uncommon calculus mistake is to believe that the product rule for derivatives says that (fg)′…

A: Derivative: f(x)=ex2We know the product rule,    ddx(u.v)=u.dvdx+v.dudx