Show in detail that div(ϕF) = ϕ div F + F • grad ϕ.

Q: Differentiate f(x) = 4sin (6z – 7) – 4 cos (). Enter your answer as a function of æ in the box…

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Q: Verify Rolle's theorem for f(x) = x* – 4x+8 over the interval [0, 2].

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Q: If 1– cosh x f(x) : 3+ cosh x then f'(x) =

A: The quotient rule of the differentiation uv'=u'⋅v−u⋅v'v2.

Q: What is frz, fyy, and fyz of f(x, y) = In(y²")?

A: Given:           f(x,y)=lnyx2          fx=∂∂xlnyx2         Treating y as constant variable…

Q: Suppose that f(x, y) is continuous at (2, 3) and that f(2, y) = y3 for y ≠ 3. What is the value f(2,…

A: Given that f(x, y) is continuous at (2, 3) and that f(2, y) = y3 for y ≠ 3. We have to find the…

Q: Suppose that | f(z)dz = 2, | f(z)dz = 1, %3D [ 9(2)dz = - 5, | 9(2)dz - 2. Then, | (=) – g(2))dz =

A: Given: ∫25fxdx =2, ∫23fxdx =1, ∫25gxdx =-5, ∫23gxdx =-2 To evaluate:∫35fx-gxdx We know that…

Q: az ) Use Implicit differentiation to find ax and for x2 – y2 +z? – 2z = 4 ду

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Q: What is the derivative of f(x) = %3D (² - (z – 1)")* 02 (2 – (a – 1)°) (-3z² + 8z – 3) O2 (2² - (2 –…

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Q: Show that the function is continuous. (x) = 3x cos 3x

A: Show that the function is continuous. (x) = 3x cos 3x

Q: Find f'(x). f(x) = O O O 6x - sin² (5x) f'(x) = O f'(x) = f'(x) = f'(x) = = 6-10sin(5x)cos(5x) 2√6x…

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Q: Prove that 1 3V2 dx < 3 + x (Hint: Theorem 5.9(ii) and carefully chosen functions should do the…

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Q: Let f(x) = sinh(2x) then f'(In2) = %3D %3D

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Q: Prove: cosh 3x = 4cosh'x- 3 cosh x %3D

A: We will solve the following.

Q: The function f(x) = x^2 is decrea sin g on (- ∞,0)  True or False

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Q: Prove the relationship =<- dt

A: we have to calculate  ddt&lt;px&gt;  .....(1) &lt;px&gt;=-ih∫ψ*∂ψ∂xdx .....(2) substituting equation…

Q: Use the definition of the hyperbolic functions to prove that 2cosh (x)-1=cosh(2x).

A: As per the instruction, we will be answering only the first question. For (a),      2 cosh2(x)-1=2…

Q: use product rule, find derivative of x.y3 O 1.y3 + x. 3y2y' O 1.y° + x. 3y? O x. 3y2y' noe of the…

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Q: Use the Linear Approximation to estimate Af = f(4.8) – f(5) for f(x) = x – 5x². (Use decimal…

A: Given data:   The given expression for the function is f(x)=x-5x2. The given value a=5.   The…

Q: Prove that f (x) = x3 − 12x is differentiable. Compute f'(x) and findan equation of the tangent line…

A: For a function f(x) to be differentiable the limit limh→0f(x+h)-f(x)h must exist. This limit is…

Q: Explain why the fundin f(2)= 526+ z COSZ - z’e? is entire ?

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Q: Find the higher-order deriv f"(x) =6Vx – 4 f(5)(x) =

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Q: The derivative of f(z) - (+10z) is O6z (2z + 10) O6z (2z+10) O 6z (2z + 10) O 6z+60

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Q: Use the Linear Approximation to estimate Af for f(x) = when a = 2 and Ax = - %3D 5+x? (Use decimal…

A: fx=115+x2,a=2;∆x=-12 Now, differentiate the function with respect to x.…

Q: complete solution

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Q: Show that f(x) = x – cos(x) has only 1 real zero on the interval [0, .

A: given fx=x-cosxx∈0,π2

Q: in of (fo f)(x). (Enter your answer using

A: We have to find the following conditions according to the given information

Q: then what is y at x=1?

A: We can find y at x=1 in the following steps.

Q: find fsubxz if (x, y, z) = ye^x + x ln z

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Q: Use the linear approximation to estimate Af = f(3.2) – f(3) for f(x) = 1+x² * (Use decimal notation.…

A: Given: f(x)=81+x2

Q: The derivative of f(x) = 9z3 12 8. is %3D 3z f'(x) O f(r) – 9x?+ 12x 4 – 9r– 12x + 8e 7 O f(r) - 9x…

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Q: 2. Let f(x) = r³, 0, if reQ if r ER\ {Q} Show that f'(0) = 0.

A: We have given a function ,  fx = x3 , x ∈ ℚ0 , x ∈ ℝ \ ℚ We need to show that , f'0 = 0 We know the…

Q: Find the maximum value of f(x)=2sin(x)cos(x).

A: Given: f(x)=2sinxcosx The above function can be written as: f(x)=sin2x…

Q: Find the derivative using the definition fex)- = C fcx) fCx) = そり f ex) = KX" f cx) fx) = gx).hcx) 9…

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Q: Find one of the roots of the function using Regula-Falsi method. £ (x) = x- V+1

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Q: Show that div (r x a) = 0

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Q: 4. If f(x) = sin x, show that f(2.x) = 2f(x)f(-x) %3D

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Q: 13 Use the Linear Approximation to estimate Af for f(x): when a = 2 and Ax = 2+x %3D (Use decimal…

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Q: II) By using the limit definition of derivative, prove the identity J dzf(2)8' (2) = - / da f…

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Q: Use the First Fundamental Theorem of Calculus to find F"(x) if = [*_ sinh√/1 +ť² dt. F(x) =

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Q: EXAMPLE 2 Find af/ay as a function if f(x, y) = y sin xy.

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Q: Find fqix(q|x) for x € {0,1} and all q.

A: Given Information: Q be the continuous random variable with pdf, fQ (q) =6q(1-q)       ; 0≤q≤10…

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 the derivative of the fünction. y = 7(6 - x2)6 %3D

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Q: inh´x + cosh²x = 1 = x²arccsch(x²) then f

A: 1. The function fx=y=coshx is represented as , coshx=ex+e-x2 To check whether the function is odd…

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

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Q: If f(z) is analytic with constant modulus show that f(z) is constant u=3x²y+2x²-y³-2y²

A: To Proof :-  If f  z is analytic with constant modulus , show that f  z is constant.  Given :-  The…

Q: 6. y 45

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.