1. Let f(x) = x', a = -1, b = 1. Verify that there is no number c for which f (b) – f(a) b - a | f'(c) = Explain why this does not contradict the mean value theorem.

Q: Find the equation of the normal line of f(x) = x⁴ + 2x at x = 1 using the limits definition.

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Q: 2. Find the derivative of a function at a certain number using the formula. f(x) – f(xo) f'(x) = lim…

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Q: The tangent line to y=f(x)y=f(x) at (−3,−2)(−3,−2) passes through the point (−4,−3)(−4,−3). Compute…

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Q: Suppose that a PDF for a continuous variable X is given by if 0 x)=3/4

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Q: : Verify Mean value theorem for the function f(z) = 4x over the interval [0, 2].

A: As given function is: fx=4x3 Given interval is: 0,2 According to mean value theorem for a continuous…

Q: Suppose f(x) = df Find the derivative dx In(In(In(In(x)))). f' (x) :

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Q: 6 Use the Linear Approximation to estimate Af for f(x) = when a = 2 and Ax = 3+x2 (Use decimal…

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Q: 9. Suppose for sume funchion f we have its derivative, f'x) = e*sin(x), on the %3D interval o< x<…

A: We can use application of derivative in order to solve the given problem.

Q: Compute the derivative of f (x) = x 1/3 using the limit definition.                            (x +…

A: Given: fx=x13 To Find: The derivative of the given function using limit definition

Q: Let f(x)=x² – 3x'. For what value(s) of x in [–1, 1] is the slope of the line tangent to the graph…

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Q: 4. Consider the function f(x) = Vx – 1 (a) Use the limit definition of derivative to find f'(x)

A: Ans:- Consider the function f(x)=x-1(a) we have to use the limit definition of derivative to find…

Q: 4.9. Show that if f has a bounded derivative on [a, b], then f is of bounded variation.

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Q: Apply the Mean Value Theorem for the function f(x) = 2 ln(4x + 7) over the interval [1,3]. Show your…

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Q: (a) f(xi), f(xi+1) and f(x+2) are three data points of f(x) and equally spaced at x with the step…

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Q: Use forward and backward difference formula with h = 0.1 to approximate the derivative of f(x) =…

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Q: II. Compute the right-hand and the left-hand derivatives as limits to show that the following…

A: Given,

Q: 4.3. Show that the function f(x) = Ja² sin, if x 0, 0, if x = 0 is not of a bounded variation over…

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Q: 4. Verify the Mean Value Theorem for the function y=ln (2x + 3) in

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Q: n(x)=3x?4k+1 %3D

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Q: Verify Mean value theorem for the function f(x)=4x3+x+2 over t interval [0,1].

A: Prove hypothesis of mean value theorem Differentiate continuous and value of c

Q: 55. By normalizing the function f(x)==-defined in the domain [0, 2], we will get as a result the…

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Q: The tangent line to y = f(x) at (-1,-8) passes through the point (-7,5). Compute the following a.)…

A: The tangent line to y=f(x) at (-1,-8) passes through the point (-7,5)

Q: 2 Find the derivative of f(x) =x² – 6 from limit definition. - Show all steps of working. f(x) :…

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Q: 2. Let f(x) = x² In a + 1. Use the second derivative formula to approximate f"(1) when h = 0.2. %3D…

A: Given that, f(x)=x2ln x +1 To find f''(1) when h=0.2 Using second derivative formula,…

Q: Find the derivative of f(x) = x In(x) 1- In(x) O In(x) O1+ In(x) O In(x) /x

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Q: 10. Show that f (x) = { x² sin(1/x) if x + 0 if x is differentiable, but the derivative is not…

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Q: How to find the median with an integral when f(x)=4(1-2x) on the interval [0,1/2]

A: Consider that a continuous random variable X has a probability density function f(x), over the…

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

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Q: For f(x)=(x^2+x−2)/(x+2) does the Mean Value Theorem apply on the interval [0, 2] and why?

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Q: 1) Verify the Mean Value Theorem for f(x)= x+sin(2x) over the interval [0, 2]

A: given f(x) = x + sin (2x)   defined over interval  [0, 2π]

Q: In the interval 3 < x < 4 of the first derivative test, f(x) is a. Not moving b. Changes direction…

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Q: 15. Apply the Mean Value Theorem to f(x) = Vx – 2x using the interval [0.4]

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Q: 4. Show that the function is continuous on (-00, 0). f(2) = {0 x* sin(1/x) if x 01 if x = 0 S

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Q: In(8z) Siven f(z) the derivative f'(x) is 8. %3D O 8 8x o 1 O 78z 1 8x 1/8

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Q: Compute the derivative of f(1) = In(x In(x² In(r³))).

A: The given function f(x)=lnx lnx2 lnx3

Q: Use backward, forward , and centered difference approximation to estimate the second derivative of…

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Q: 5) Use the tangent line at x = 9 to approximate the value of the function f(x)= Vx at x= 8.8.

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Q: 6. Find f(x) if f(1) = 1 and the tangent line at (x, f(x)) has slope f(x)

A: Slope of tangent is obtained by derivative of the function f(x).

Q: 5. (a) Use the limit definition of the derivative to find the derivative of f(x) = V5x +1. V5yl th)-…

A: Given; f(x)=√(5x+1)

Q: 4. Use the limit definition of derivative to find f'(x) for f(x) = V5x + 1. You will only get credit…

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Q: Prove that d-function is the distribution derivative of the Heaviside function { if x > 0, if x < 0.…

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Q: The tangent line to y=f(x) at (−8,−5) passes through the point (−3,10). Compute the following f(-8)…

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Q: The general anti-derivative of the function f (x) - In(x) is given by x In( x)+ C Inx)-x+ C x In(x)…

A: The integral (anti derivative) of fx=lnx can be evaluated by using integration by parts. The…

Q: Fix a constant r>1 Using the Mean Value Theorem that enx > 1 + x for any fixed x >0 prove

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Q: Find Tn cebtered at x=0 fir akk b for the function f(x) = 1/(1+x) at a=0

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Q: 7. Use the limit definition of the derivative to find f'(x). Then find the equation of the tangent…

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