The graph of y=x'-3x-24x +32 is concave upward on the interval (1, oo) because: O'=y"=0 Oy' is positive on the interval (1, 00) Oy' is negative on the interval (1, 00) y" is positive on the interval (1, 0 O" is negative on the interval (1, 0)

Q: A curve passes through the point (2,0) has gradient dy at point (x, y) that satisfy the dx equation…

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Q: Find the first partial derivatives with respect to x, y, and z. f(x, y, z) = 3x²y – 9xyz + 10yz²…

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Q: Find both first partial derivatives. 18y2 x az %3D ax dz ду

A: The given function is:z=18y2x

Q: az Find if yz – In z = x + y ax

A: To find: The partial derivative ∂z∂x, if yz-ln z=x+y. Calculation: The partial derivative ∂z∂x can…

Q: Find a function w = ƒ(x, y) whose first partial derivatives are ∂w/∂x = 1 + ex cos y and ∂w/∂y = 2y…

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Q: Find both first partial derivatives. %3D az ax az ây

A: Solution is given below:

Q: Find the value of 0z/0x at the point (1, 1, 1) if the equation xy + z3x - 2yz = 0 defines z as a…

A: Given, the equation is xy + z3x – 2yz = 0 and defines z as a function of the two independent…

Q: . Form the differential of the system of circles touching the y-axis at the origin.

A: The solution is given as

Q: Find partial derivatives of  hxx ( x,y) , hxy(x,y), hyy (x,y)  For function  h(x,y)= ln ( x^3 +…

A: Given, h(x,y)=ln(x3+y3)

Q: Find the value of dz at the point (1)1, If the equation 5xy+ z4x -Qy2 =4 %3| defines z as a function…

A: Given 5xy+z4x-2yz=4to determine dzdx  at  (1, 1, 1)

Q: Find the value of ∂z/∂x at the point (1, 1, 1) if the equation xy + z3x - 2yz = 0 defines z as a…

A: Given, the equation xy + z3x – 2yz = 0, defines z as a function of two independent variables x and y…

Q: Find the first partial derivatives with respect to x, y, and z. f(x, y, z) = 4x2y - 7xyz + 8yz2…

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Q: Evaluate (2x-y)dx+(x+3y)dy along the path C: curve on y = 1-x² from (0,1) to (1,0).

A: Given the line integralI=∫C(2x−y)dx+(x+3y)dyCurve y=1−x2

Q: 2+ yety y' : 2у - хеху

A:

Q: Find both first partial derivatives. sinh(5x + 4y) az %3D dz %3D ду

A: Explanation of the answer is as follows

Q: Find partial derivatives of x3sin y + cos2x tan5y with respect to x and y

A: We have to find the partial derivatives of  x3siny+cos2xtan5y with respect to x and y.

Q: Find the value of ∂x/∂z at the point (1, -1, -3) if the equation xz + y ln x - x2 + 4 = 0 defines x…

A: Given equation is xz+ylnx-x2+4=0. Given point is 1,-1,-3. Partial derivative of given equation with…

Q: Find the first partial derivatives of 3x - y f(x, y) = 3x + Y at the point (x, y) = (4, 4). af (4,…

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Q: Compute the partial derivative fxxyz for f(x, y, z) = cos(cos(x3 + z2 )) + x2 ln(y2 + z2 ).

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Q: Find the indicated partial derivatives. y+ Sz az dy ex

A:

Q: Find the first partial derivatives of f(x,y) = 2 ex"y of II

A: Consider the given function. fx,y=2ex10y The objective of the problem is to find the first partial…

Q: Find the first partial derivatives with respect to x, y, and z. Ex, y, 2) = 3x?y - 6xyz + Byz? ,(X,…

A: We have to solve this problem by partial derivative method.

Q: 2.Evaluate the total derivative of u with respect to x for u = 6x²+4y² + 4z² when y = cosx and z =…

A: We have to find total derivative of u with respect to 'x'    where     u = 6 x2 + 4 y2 + 4 z2  Given…

Q: Find both first partial derivatives. z = 4y²/x az ax az ду || ||

A: z=4y2x

Q: Find the partial derivative of f(x,y) with respect to y at the point (3,2) f(x,y) = ln(x^2y-xy^2)

A: A partial derivative of a function of several variables is its derivative with respect to one of…

Q: Find all the second partial derivatives forf (x, y) = x³y5 + 2x*y.

A: All the solutions are given below.

Q: Find the value of at the point (1, 1, 1) if the equation xy+ z°x – 2yz = 0 defines z as a function…

A: Given equation, xy+z3x-2yz=0

Q: derivative

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Q: ihe fcu,v, w) and its partial derivatives with respect tou and vand w and are Continous that u-メ-ツ…

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Q: Find the first partial derivatives with respect to x, y, and z. f(x, y, z) = 3x²y – 9xyz + 10yz²…

A: As we know that f=f(x,y,z)then fx(x,y,z)=∂f(x,y,z)∂x, fy(x,y,z)=∂f(x,y,z)∂y,  fz(x,y,z)=∂f(x,y,z)∂z

Q: From the partial differential eqiuation by eliminating the arbitrary function z=(x2+y2+z2)

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Q: What is the result of the partial derivative 3(z² + y sin (z)) Select one: a. 2a + y * cos(x)

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Q: Show that the Second Derivative Test is inconclusive when applied to the following function at…

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Q: Find the first partial derivatives of the function. f(x, y, z) = x7yz2+ 4yz

A: From the given problem:   fx,y,z=x7yz2+4yz   So,   fxx,y,z=∂f∂x=∂ ∂ xx7yz2+4yz=∂ ∂ xx7yz2+∂ ∂…

Q: The population of a city is associated by the function P(x,y,z)=x² + 2y² + 2z². If the population of…

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Q: 23-4z2-2一4 y = %3D 1+ Va

A:

Q: If z-f (x y): then its total differential can be expressed as * (),中 (). dyds dy + dz O Option 2 O…

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Q: Find both first partial derivatives. z = cosh xy2

A: Given: z=cosh xy2 To determine: ∂z∂x& ∂z∂y

Q: d²x dx 5 dt2 2- 3x = -9t2 – 1 - - dt dx When t = 0, x = 1, = 0, find x when t = 2 dy %3D -

A: Please see the below picture for detailed solution.

Q: Find the value of dx at the point (1,1,1) if the equation 6xy + z"x-2yz = 5 defines z as a function…

A: Given: (x,y,z)=(1,1,1) and z=6xy+z4x-2yz-5=0

Q: Find all the second partial derivative T = e-4r cos(0) -4r, Trr 16e 'cos (0) %3D Tre = 4e -4r,…

A: “Since you have posted a question with multiple sub-parts, we will solve the first three subparts…

Q: v = exy In(x +y)

A:

Q: but its partial derivatives are not continuous at (0,0).

A:

Q: Find the firsst derivative of y = arctan [ (3 sin x) / (4 + 5 cos x) ] and the first partial…

A: Given:       y=arctan3sinx4+5cosx First derivative:     dydx=ddxarctan3sinx4+5cosx…

Q: Find the first partial derivatives with respect to x, y, and z.…

A: The given function is f(x, y, z) = 3x2y − 5xyz + 10yz2

Q: Find the mixed partial derivative at the point (2, 3) for h(x.y) = x sin(y) + y sin(x) + xy

A: The given function is, Differentiate the above expression with respect x partially.

Q: Find the partial derivative. 2 = (r² + 5z – 6y)“ az ||

A: The given function is: z=x2+5x-6y4

Q: Find the first partial derivatives of 3x – 2y f(x, y) = at the point (x, y) = (1,4). 3x + 2y af…

A: I am going to solve the given problem by using some simple calculus to get the required result of…

Q: Find all the first partial derivatives of; glw,2, y, z) = xe2w - wyz?

A: Topic:- partial derivatives

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: