Let f(x) = 3x + 1, x, = -1, x, = 0, x3 = 1, x4 = 2, and X5 = 3. Find E f(x,).

Q: If f(x) to dt then f'(æ) =

A: We have to solve the problem

Q: Moreover, suppose F(x) = f(g(x)) and G(x) = g(f(x)). Use the chain rule to calculate the following…

A: f(x)=cot x  and  g(x)=5x-2

Q: f (x) e1/x x2

A: Given function is fx=e1xx2

Q: Find the antiderivative F of f(x)=4-3(1+x^2)^-1 that satisfies F(1)=-2.

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Q: Find the first derivative for the following function. Vx4 – 3 (2х — 5)"] f(x) = ln (2x-5)7 Vx4-3 2x3…

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Q: For what value(s) of x is the function f(x) = } 2x + 4 (x² +x +1 if – 2 sxs1 if x>1 NOT continuous?

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Q: 4) Calculate the first derivative of the following functions: 1. f(x) = In 2x2+1° 2. f(x) =…

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Q: 5. Find the derivative of the function. f(x) = x°/1- 8x x(16-136x) 2/1-Sr x'(16+136x) f (x)3D 2/1-8x…

A: The function is given by,

Q: Use the u-substitution to find the anti-derivative of m'(x)= 1/xln(x)

A: solution :- antiderivative is the integration  So put ln(x)  = u Then (1/x )dx  = du And then…

Q: Use the chain rule to show that for any differentiable function f(x), [e/(e) = f(x)e/(=),

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Q: Let g=yf(xy). Show that: - дд у ду дд - Х = g дх -

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Q: Prove that for all a, b e R, the function f(x) = 3x - 4x³ + 6x² + ax + b has exactly one point where…

A: f'(x) is increasing  function. But f'(x)  is polynomial of degree 3. So it has exactly one real root…

Q: Use the Chain rule to differentiate each function (a) f(x) = In(x +sin(x)) b) g(x) = (-2x² +3x + 1)3…

A: Chain rule is used to determine the derivative of the composite functions. Consider a composite…

Q: Prove that for all a, b = R, the function f(x) = 3x² - 4x³ + 6x² + ax + b has exactly one point…

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Q: If f(x) = and g(x) = Vx, then the derivative of f(g(x)) is -VE (A) ( + 1) (B) -(x + 1)- -2x (C) ( +…

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Q: if f'f′ is the derivativeof ff with 2∫1f'(x)dx=10∫12f′(x)dx=10 and f(2)=4,f(2)=4, then f(1)

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Q: Let y= (x+sin(x))^5 Find g(x) and f(x) so that y= (f∘g)(x), and compute the derivative using the…

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Q: Suppose that a continuous function g satisfies: g(x)dx = 42 and g(2x)dx = 3. %3D 2 Find æg(3x²)dx.

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Q: Find the second, third, and fourth order derivatives of: y= f(x )= ax7 + bx3 + c y= f(x) = 3x/1-x…

A: It is simple derivative

Q: Find the linearization L(x) of the function at f(x) = x4 + 2x2, a = 1 L(x) =

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Q: Find all antiderivatives of function:  f (x) = -4x

A: An antiderivative of a function f(x) is the indefinite integral ∫f(x)dx. Denote this antiderivative…

Q: Let 'f be a continuously differentiable real valued function defined bx + a by f(x) ={ 5x x<1 x21…

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Q: Use the Chain Rule to find f'(x). f(x) = (cos x – x²)5

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Q: Let y = (x + sin(x))³. Find g(æ) and f(x) so that y = (f o g)(x), and compute the derivative using…

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Q: If 7х- 5,х <1 { h(x) = 3x2 – x , 1 < x< 3 Is h(x) continuous at (x=1)?

A: Continuity of function at a Point : Suppose a function f:A→B where A,B⊆ℝ .Then f is continuous at…

Q: Compute the derivative of the given function. f(x)=−4x^π+4.1x^1.3+π^1.3

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Q: write the function as a composite f (g(x)) and compute the derivative using the Chain Rule. y =…

A: Given : The function is y = cos ( x3 ) .

Q: The derivative of In x is and the derivative of a constant C is, sO X dx = In |x| +C for x> 0.

A: Given x&gt;0, The differentation of natural log x is 1/x  And differentation of constant are zero.

Q: Let h(x)=f(x)/g(x). If f(x)=−cos(x) and g(x)=4x2+2x−3, what is h′(x)?

A: If f(x)=−cos(x) and g(x)=4x2+2x−3, what is h′(x)?

Q: Find the constant c that makes g continuous on (-0;0) x-2x-8 g(x) = a+1 if x+2 if X=-2

A: Continuity of a function at a point x=c : If function is defined at x=c and limx→c fx=fc then we can…

Q: Suppose f and g are differentiable functions on R such that, (x²+1)f (x) + 3xf(x) = e*2 + 7x and…

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Q: Find the derivative of the function f (x) = /e* – In(x) 12 – 1/x) O (ex12 – 1) O (e2 – 1/x) e? -x)

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Q: Find all critical points of the function f(x)=xlnx.

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Q: Evaluate the derivative of the function: x2 f(x) = ( x²+1 8x (æ²+1)° O 8a(a² –1) (2²+1) O 1 4x…

A: Use the chain rule for finding the derivative

Q: Find a point c satisfying the conclusion of the MVT for the function y(x) = 16/x on the interval [0,…

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Q: If the function f(x) = {cos(2πx−a),x&lt;−1x3+1,x≥−1 is continuous, what is the value of a?

A: fx=cos2πx-a,  x&lt;-1        x3+1,    x≥-1 Since Cosine function is continuous for all real numbers,…

Q: Let y = (x + sin(x))³. Find g(æ) and f(x) so that y = (f o g)(x), and compute the derivative using…

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Q: If f (x) = In (4a + 1), then its third derivative is f(3) (æ) = %3D

A: In this question, concept of derivative is applied. Derivative The derivative expresses the rate of…

Q: Suppose G, (x) and G2(x) are both antiderivatives of a function g(x). Using complete 3. sentences,…

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Q: Suppose that a continuous function g satisfies: | g(x)dx = 12 and g(2x)dx = 27. Find æg(3x²)dx.

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Q: The function f(x, y) -{¹. - x² - y², 0, if if x² + y² 1 , is continuous

A: Given function is fx,y=1−x2−y2, if x2+y2≤10,             if x2+y2&gt;1. We have to check whether the…

Q: Find all the x-intercepts of f(x)=cos(x) on the interval [0,2π)

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Q: Given f(x) In(5x) the derivative 5 f'(x) is 1 5x 1 5x O 5 O x5x 1

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Q: Suppose that a continuous function g satisfies: g(x)dx = 42 and | g(2x)dx = 12. 2 | #g(3x²)dx. Find

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Q: Let –1, -T<¤ < 0, 0, 0 < x < T. f(x) = * = 0 1,

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Q: Given g(x) = ln(11x³ + x), the derivative s! (x)f 3x2 11x° + x 33x2 + 1 33x2 + 1 11x³ + x 33x2 +1 O…

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Q: If f(x)=e^cosh(4x) then f'(x)=

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Q: Let f : (0,∞) → (0,∞) be given by f(x) = ln(2 + x). Prove that f has a fixed point located between 1…

A: The given problem is to prove that the function f(x) has a fixed point in between 1 and 1.2 Given…

Q: Find f'(x) (the derivative of f(x)) if f(x) = 7x'3 - 2x'2 + 4x - %3D 5. O 21x^2 - 4x + 4 + C O 21x^2…

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