4 Determine the surface area of a funnel that is generated by revolving the graph of 1 y = f(x) = x³+ 12x on the interval from [1, 2] about the x-axis.

Q: Find the maximum of y = xa − xb on [0, 1], where 0 < a < b. In particular, find the maximum of…

A:

Q: 7-The tangent of the curve x^2+y^2- 2xy3D0 at the point (1,1), is * O a. y=2-x. O b. y=x-2. C.…

A: We find equation of tangent to the curve

Q: Find the orthogonal trajectories to the family of curves: x² = C-v'(ln y² –1).

A:

Q: Use Taylor's formula for f(x,y) at the origin to find quadratic and cubic approximations of f(x,y) =…

A: The function fx,y=11-3x-y Then the taylor's formula for fx,y is, fxx,y=31-3x-y2fyx,y=11-3x-y2 Then…

Q: -Find the D.f ofr (x, y) = (x² – y)' at Po(3, 1) with

A:

Q: What is the isogonal trajectories cutting at angle of 0= arctan 4 to the family of curves whose…

A:

Q: 4. Determine the maximum point of the the graph of y = 12 sin(π – 2x), for x = [-T, T).

A: Given:         y=1-2sinπ-2x     To get the maximum value of the function the derivative is equal…

Q: Find the points at which y=f(x)=x9−17xy=f(x)=x9−17x has a global maximum and minimum on the interval…

A: We find the critical points using f'(x)=0 -1.0827 is not in the interval [0,4]

Q: find the absolute extrema of thefunction on the closed interval y=x \ln (x+3),[0,3]

A: A function is usually represented as f(x) where x is the input to the function and f(x) represents…

Q: Find the absolute maximum and absolute minimum values of the function f(x, y) 2x3 + 3xy + 2y %3D…

A: We have to find the absolute maximum and absolute minimum values of the function fx,y=2x3+3xy+2y3…

Q: (6) Find the orthogonal trajectories of the family of curves r= 2C cos 0, where C being parameter.

A: Consider the given curve. r=2Ccosθ          (1) Find the derivative of the given equation.…

Q: Find the local maximum and minimum values and saddle point(s) of the function. (DNE if doesn't…

A:

Q: Find the absolute maximum and minimum of the function f(x,y) = 7y + 64x – 3 over the closed region…

A:

Q: 4. Find the absolute extrema of the function f(x, y) = ry-x-3y on the triangular region with…

A: To find the absolute extrema of the function f(x,y).

Q: Consider a circle of radius 2 centered at the origin. Use the Lagrange multiplier method to find the…

A: It is given that circle is of radius 2 and centered at origin. The function isfx,y=3x-y+5.

Q: 7. Use the Method of Lagrange Multiplier to find the critical points of the function f(x, y, z) =…

A:

Q: Determine the x-coordinate of the local maximum of the curve y = (2x - 1)2 (3x+4) in the interval…

A: Given: y=2x-123x+4

Q: 6. Locate the absolute extrema of the function H(x) = - x* Inx on 1 3 2' 2 Give exact solutions.

A: We have to find out absolute extrema

Q: Find the absolute maximum and minimum of the function on the given domain. f(x.y) = 6x + 5y on the…

A:

Q: 3. Evaluate f 2xdx + (y + 2)dy along the rectangle with vertices (0, 0), (0,3), (3,3) and (0, 3).

A:

Q: 3. Find the interval on which the curve of y= 2 1+3t+t² dt is concave up.

A: Second derivative is positive.

Q: 49.) Find the minimum distance from the point (2, 1, 4) to x + y + z = 4 by using the LaGrange…

A:

Q: 2. Find the absolute maximum and minimum values of f(x, y) = 4x+6y-(r²+y?) on the rectangular region…

A:

Q: Consider the For at function y= In(x+1) . what valves" of x is y undefined ?

A: The given function is y=ln (x+1) We know that, for the function ln x, x>0

Q: Eliminate the arbitrary constants a. x = -In t, y = at + bt?

A:

Q: 2. y = V4-x , x=0 and y=0 about the y- axis

A: Given that y=4-x2, x=0 and y=0 about the y-axis

Q: Find the linear approximation to f(x) = (x+2) at 0 and use it to approximate (2.001)

A: Given :           f(x) = (x + 2)5 at 0 To find :             The linear approximation of f(x)…

Q: Find the absolute maximum and minimum values of f (x, y) = 4xy – x* – by the line y = -2 above by…

A:

Q: The temperature in degrees centigrade (◦C) at each point (x, y) on a curve 2x + y = 3 is given by…

A: For relative maximum or minimum To maximize or minimize  a function f(x) 1st we need to find its…

Q: Q2 (a) A miniature quadcopter flies in a curve line from one point to another. Suppose that the path…

A:

Q: Find the local linear approximation L (x) of the function f(x) = 5 – x² at =2.

A: To find the linear approximation of given function.

Q: 2. Find and classify all the extrema of the function f (x, y) = 4xye e-(x²+y²)/4

A: The given function is:  f(x,y) = 4xy e-(x^2 +y^2)/4

Q: A rectangular plate with an area of 36 cm? is to be produced on a lathe in such a way that the…

A:

Q: Find the extreme values of the function f (x, y) = e¬¤y given that (x, y) must live in the region…

A:

Q: 9) A function y(x) is increasing and has upward concavity on the interval [-2, 3]. Would a…

A:

Q: 27. Find the length of the curve Vcos 2t dt y = from x = 0 to x = "/4.

A: The given curve is We know that by Second Fundamental theorem of calculus ,if F(x) is continuous on…

Q: Find the absolute maximum and minimum values of f(x, y)= 3+xy-x-2y on the closed triangular region…

A:

Q: Find an equation for the level surface of the function f (x, y) = ln (x² + y²+ z²) that passes…

A:

Q: Find the point at which the function f(r,y) = 2 in z+2 In y attains a maximum subject to the con-…

A: Maxima or minima: For any function f(x), 1) Evaluate f'(x)=0 for critical point. 2) Evaluate f''(x)…

Q: Find the absolute maximum and the absolute minimum of f(x,y)= 3xy-3x-6y+1 on the closed triangular…

A: Given:- The function is, f(x,y)=3xy-3x-6y+1on the closed triangular region whose vertices are (0,0),…

Q: Find the extreme values of the function f at the region of x2 + y² = 4 where: F (x, y) =x² + 2y²-2y…

A: The extrema of the function can be calculated by calculating the derivatives of the function. First…

Q: 2 f(x, y) = x² + 3xy

A:

Q: for the function f(Z) z²-2z find the (z+1)° (2 t4) residues at singular points.

A: The residue of f(z) is solved below

Q: 1o9.. 10 11 12 What is the minimum value of y = -3 – 5 sin (4x)?

A:

Q: Find the minimum and maximum values of y = V120 - V6 sec e on the interval [0, ] fmin fmax

A: Solve

Q: Determine the values of constanas a,b,c, and d so that f(x) = ax^3 +bx^2+cx+d has a local maximum at…

A: Given that f(x) = ax3 +bx2 + cx + d has a local maximum at the point (0,0) and a local minimum at…

Q: Find the absolute maximum and absolute minimum values of the function f(x,y)=x2+y3 on the region…

A:

Q: Find the absolute maxima and minima of the function on the given domain. 6) f(x, y) = 6x + 5y on the…

A: f(x,y) = 6x+5yfx(x,y) = 6     fy(x,y) = 5  (Hence No critical points of the function itself)Lets…

Q: Find the maximum value of J (x) f(x) = V1 + x² on the indicated interval. [0,2]

A:

Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.