12.2.7. Suppose that Q := {(x, y) e R : x > 0 and y > 0} and that f is a continuous function on R? whose first-order partial derivative satisfies |fxl < 1. If %3D F (x, y) := x- I f(u, y) – f(v, y)) d(u, v) B, (0,0) for (x, y) E Q, prove that F is bounded on Q. [Hint: You may use polar coordinates to change variables in F.]

Q: Let r(x) = f(g(x)) and s(x) = g(f(x)), where f and g are shown in the figure. Find r'(1) and s'(4).…

A: our objective is to find the required derivative

Q: 2. Let u,v be functions of x. The first order derivative of uv can be derived by the following…

A:

Q: The function f has continuous second derivatives, and a critical point at (6, -9). Suppose fzz…

A:

Q: 2. Given the following function f(x, y) = x3 +15x2 - 20y2 + 10. (i) Find fr. fy. frx fyy. fzy and…

A:

Q: If we have that f: R → Ris a continuous function , in which f'(7) = 6 and f(7) = -2 an equation of…

A: We have to find out tangent line

Q: Q10

A: Given F(u,v)be a function of two variables. let Fu(u,v)=G(u,v) and Fv(u,v)=H(u,v) Here f(x)=F(u,v)…

Q: 4.9. Show that if f has a bounded derivative on [a, b], then f is of bounded variation.

A:

Q: 4. Suppose that the following information about a function f is known. It has continuous partial…

A: To find the gradient of the function  f  at the origin given the directional derivatives, we need to…

Q: 4. Let f be a twice-differentiable decreasing function. If f(0) = 640 and f(20) following must be…

A: Now, we are given that f is a twice differentiable function . So it is differentiable in the…

Q: 6) When some variable z is a function of time (t), the derivative of z with respect to t is denoted…

A:

Q: (3) Determine the values of a for which x4 + y* , if(x, y) # (0, 0), (x² + y?)ª f(x, y) = ,if(x, y)…

A: solution.

Q: 2(x-1) If y = Vx² – 2x then y' 3x2-2x

A: For the first part, we differentiate the given function y with respect to 'x' and use change rule of…

Q: 4. a. Let f(x) = ax²+3x+y be a quadratic function with a, 3, 7 € R. Consider the closed interval [a,…

A:

Q: 2 Sketch the level curve of the function f(x, y) = y - x² for c = -2,0,2.

A:

Q: 6. Let f be a function such that f(1) = 1, ƒ′(1) = 4, ƒ”(1) = −2 and ² ƒ""(t)(2 – t)²dt = 3. Find…

A:

Q: 6. Let f be a continuous function on [0, 2] whose second derivative exists on (0, 2). Assume f(0) =…

A:

Q: 5. Given the function x+y+2, if x = 1 or y = 1 2, if x 1 and y # 1. Show that fe(1, 1) and fy(1, 1)…

A:

Q: What

A: The correct option is (B).

Q: Let a differentiable function z = f(h²(x² + y3), 2łn(x +y)) satisfying the condition f(4,0) 1 be…

A: We need to find the partial derivative of z with respect to x for the given function.

Q: Let f(x, y) = x³ + 2ry – y² be a func the tangent plane to f(x, y) at the poi

A: Here we have f(x,y)=x3+2xy-y2  we have to find the equation of the tangent plane to f(x,y) at point…

Q: 15. Let f(x, y) = 22 if (x, y) ‡ (0,0), and f(0,0) = 0. Is ƒ continuous at (0,0)? x+y

A:

Q: 7. Modify B(3) to construct a C* function C(2) which satisfies C(2) = 0 if z S 0.. a. b. C(z) = 1 if…

A: please see the next step for solution

Q: 1. (y-1) x²+4 Cy-1)² (x,y)# (0,1) f(ox,y)= %3D (x,y) = (0,1)

A: Explained below...

Q: Find all point(s) of inflection, if any, of the graph of the function f(x) =x* – 2x'. с. (-1, 3) а.…

A:

Q: 5. (а) Find the directional derivative of f at P in the direction of ā if f(x,y) = y In(x) ; P(1,4);…

A: To find The directional derivative of f at p in the direction of a.

Q: 18. The graph of the function g on the closed interval [0, 10] consists of four line segments, as…

A: Value of function from graph

Q: 1. Find the gradicnt of the following functions (here, i = V-1, and A is a constant) (a) f (x) = 3x.…

A: The objective is to find the gradient.