Consider F and C below. F(x, y, z) = yzexi + ej + xyek, C: r(t) = (2 + 3)i + (t2 - 2)j + (2- 3t)k, 0sts 3 (a) Find a function f such that F = Vf. f(x, y, z) = F. dr along the given curve C. (b) Use part (a) to evaluate

Q: Let f be a function such that Vf(x, y) = (2x, 1) g(x, y) = x² + y². Find all the points on the curve…

A: Let f be a function such that ∇f(x, y)=2x, 1 and let g(x, y)=x2+y2 We have to find all the points on…

Q: Find the decivative of thne function f(xiy)=x²+xy+y? at the point (5,4) Po the direction 3n which…

A:

Q: Let r(t) (t¹ + 4, t² + 2t¹, − 4t³). Find the line (L) tangent to r(t) at the point (20, 36, 32) = L:…

A:

Q: 21 Let F(x, y) = (3x + 2y)i + (2x - y)j. Evaluate F. dr where C is the curve X y from (0, 0) to (1,…

A: Here we calculate the integral along the curve x = y2 from (0,0) to (1,1)

Q: 13. Find an equation for the tangent plane of the graph of f at the point (xo, yo, f(xo, yo)) for:…

A: Note:-Since you asked multiple questions which are not interlinked to each other, according to our…

Q: a) f : R? → R, f (x, y) = Vy sin x 5) f : R2 → R. f (r. u) = GGrul + In (a2 + 1? – 1). /4 – 72 -

A: We will find the domain

Q: How large should n be to guarantee that the Trapezoidal Rule approximation to -1 √(-2¹ - 122³ - 30x²…

A: Since you have uploaded multiple questions, according to our guidelines we have solved first…

Q: a. Find a function (f(x,y,z)) such that (F = Vf(x,y,z))? b. Use part (a) to evaluate (F. dr ) along…

A: We have to evaluate the ∫CF.→dr→.

Q: Q2: Evaluate . f ds where C is the given oriented curve r(t) = (-2 + 6t ,5 + 8t) for (0 <t< 1) and…

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Q: Let f(x, y) = 2xy – 4x² + 3y². %3D Then a standard equation for the tangent plane to the graph of f…

A: The function is given. Find the partial derivative fx and fy at the given point. Given,…

Q: r(t)=(-6–2t)i+tj– 2tk, 0<ts Evaluate where C is the curve given by C

A: Solve the following

Q: Consider the curve r (t) = (t, tº, tº) ; t > 0 Does the point (1, 1, 1) belong to the curve and if…

A: Please refer the attached image for complete solution.

Q: Find F-dX where C is the curve X(r) = r°I+( 2r+ 1J, 0<I55, and Fxy) = rI+yJ.

A: 9

Q: dy for the curve defined by x° – y = 2xy? dx Which of the following is O 2x + 3y? 3x2 – 2y O 3x2 –…

A:

Q: 2. Consider the function 2r2-9r-18 f(r) r2-r-2 (a) Sketch the curve y f(r). (b) Find the area of the…

A: Let's find.

Q: The population of a community is known to increase at a rate proportional to the number of people…

A:

Q: E.g. Let f(x, y) = x² - 2xy + y². The graph and contour diagram of f is given below right. Compute…

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Q: Let f(x, y) = x2 − 2y and g(x, y) = xey . Show that at any point (x0, y0), the level curve of f…

A: To show that at any point the level curve of f and g intersects perpendicular, it is enough to show…

Q: Sketch the following curve

A:

Q: 2. Given f (x, y) = x + 2y, find F(x, y) =Vf and sketch it along with the contour map of f.

A:

Q: Let (t)=(-5t2+4,t³-+4,2+³) 7 Find the line (L) tangent to (t) at the point (-1,0,2) L: (x,y,₁2)= +t

A: The given problem is to find the line tangent to the given parametric curve at the given point…

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.