A thin observatory probe having the shape of a plate is probing a volcano in an xy −plane at a point P(x, y). The probe can only withstand a certain amount of temperature; thus,  Represent the temperature as the function of x and y when the temperature calculated in degrees Celsius is inversely proportional to the square of its distance from the origin.

Q: What does it mean for a function of two variables, f (x, y),to be differentiable at a point (a, b)?

A: A function differentiable at a point has a derivative that is defined at that point. This is defined…

Q: consider the function f(x, y) = esin(x), Find the maximum rate of change of f at the point (, 1).

A: The direction of maximum increase of f is given by ∇fx,y.Maximum value of Du→fx,y or rate of…

Q: Prove the formula for the derivative of a) (Inverse of cotangent) b) (Hyperbolic cosecant) y = csch…

A: formula for derivative of both function is following:-

Q: Illustrate local linearity for the function f (x) = xsinx at x = l.

A:

Q: 7 Determine the gradient function of y %3D ४

A: Given y = x^(7/3) Gradient of function is slope of function. Gradient = slope = dy/dx…

Q: Find the ABSOLUTE VALUE of the gradients of the tangents drawn to the ellipse x^2/4+y^2/93D2 at the…

A: We have to find absolute value of gradient at x=2 for the given function.

Q: Differentiate the function.   y = x + 1/ex

A: Case (I): If we have the given function as y=x+1ex        ⋯⋯1 We know that, from the quotient rule…

Q: What is the formula for the linearization of a function f at the point x = a? %3D

A: Given :To provide the formula for linearization of a function f at the point x=a

Q: Find the derivative of the function. g(x) = In()x° – x|) %3D g'(x) = %3D

A: Formula used. ddxln(x)=1x;ddx(x)n=nxn-1

Q: general

A:

Q: Find the maximum rate of change of f(x, y) = (2x - y)3/2 (3,2). at %3D

A: Given,          The function is fx,y=2x-y32 at 3,2.We know that,          The maximum rate of change…

Q: Find the direction for which directional derivative of f (x,y) = 3x² – 4xy + 2y² +29 at (-2,3) is…

A: Formula: The maximum directional derivative of f(x, y) at (a, b) is given by ∇f(a, b) and direction…

Q: Q3. Find the coordinates of the point(s) of the function y=3x – 5x +2, where the tangent has a…

A:

Q: How can linear approximation be used to approximate the changein y = f(x) given a change in x?

A: Given;                 y=fx How can linear approximation be used to approximate the changein y=f(x)…

Q: c. y = 2Vx when x = -1 с.

A: We have to find the gradient

Q: What does it mean for a function ƒ(x, y) to be differentiable? What does the Increment Theorem say…

A: Concept:   The calculus helps in understanding the changes between values that are related by a…

Q: Find the gradient of the function at the given point. y Ri (2, 2) x²+ y2' Vf(2, 2) = %3D Find the…

A:

Q: Find the maximum rate of change of the the function 80 f(x, y, z) = 1+g² + 2y? + 3z² %3D at the…

A:

Q: Let f(x, y) = xe*Y. Find the maximum rate of change of f at the point (2, 1). хе

A: We will find the maximum rate of change.

Q: Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x,…

A: Gradient problem....

Q: b. y = 2x2 when x = 6 %3D 3.

A: Consider the curve: yx=2x23 at x=6. Find the gradient of the curve at x=6.

Q: How is the antiderivative of a function related to the function?

A: Given: antiderivative of a function related to the function.

Q: Find the gradient of the function: y = (5x ln (x)/ 6) + x/3 at the point with x-coordinate 1. Give…

A: We can solve this problem as follows:

Q: Find the linearization of f(x, y) = −x² + 2y² at the point (3,-1). Use the linearization to estimate…

A:

Q: Find the point (xo, yo) on the graph of y = x³ + 3x? + 8x – 3, at which the slope of the tangent…

A: To find the slope of the function we need y'y=x3+3x2+8x-3y'=3x2+6x+8

Q: Differentiate the function. y = In(eX + xe )

A: In this question first apply the property of log and then differentiate as shown...

Q: Find the derivatives of the functions in Exercise

A: Find the derivatives of the functions in Exercise y=x3ex

Q: Find the point (x0,y0) on the graph y=x^3 + 6x^2 +3x -5 at which the slope of the tangent line has…

A:

Q: Find the gradient of the function y = x³ + x² – 2x at points where - a. It crosses the x -axis b. It…

A:

Q: Find the gradient of the function at the given point. g(x, y) = 5xev/x, (8, 0)

A: Follow the procedure given below.

Q: Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x,…

A: Given function and the point are

Q: Find the magnitude of the greatest rate of change of f(x, y, z) = (x2 + z2)3 at (1, 3, -2), and…

A: Consider the provided question,

Q: Find the gradient when x = and y = sin (3x). %3D

A:

Q: Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x,…

A: A vector has 2 components, magnitude and direction which is represented by cos and sin angles. A…

Q: Calculate the gradient. 5x g(x, y) = x² + y? (Use symbolic notation and fractions where needed. Give…

A: first calculate the gradient for partial differentiation.

Q: How to find the gradient of f(x,y) = (e^x)cos(sqrt(x^2 + y^2))

A:

Q: The gravitational attraction F between two objects of masses m1 and m2 is jointly proportional to…

A: see the attachment for answer

Q: Let f(x, y) = xe". What is the maximum rate of change of f at the point (2,0)?

A:

Q: Find the directional derivative of the function at the point P in the direction of the point . f(x,…

A:

Q: Can a function have more than one antiderivative? If so, how are the antiderivatives related?…

A: Can a function have more than one antiderivative? If so, how arethe antiderivatives related?…

Q: · y = Vx - x, (1, 0)

A:

Q: Find the maximum rate of change of f at the given point and the direction in which it occurs. f(x,…

A: We have to solve

Q: Find the maximum rate of change of f at the given point and the direction in which it occurs. (x, Y)…

A:

Q: Find a cubic function y = ax3 + bx2 + Cx + d whose graph has horizontal tangents at the points (-2,…

A:

Q: Find the directions in which f(r, y) = (r+y-2)2+(3x- y- 6)?, increase and decrease most rapidly at…

A:

Q: Find the maximum rate of change of f(x, y) : = ye* at the point (0, 2).

A: Introduction: Let f(x,y) is the function of two variables x and y. then the maximum rate of change…

Q: Find the gradient of the function at the given point. Function f(x, y) = x² Point (4,4) Vf(4,4) = x²…

A: In this question we have to find the gradient of the function. Note: As per our policy we can solve…

Q: Find the direction at which the function f (x, у, 2) %3 хIn(е' — хг) increases fastest at (-3,0,0)

A:

Q: Let f(x,y) = x^2 y + e^xy sin y.   At Point (1,0) State the direction where the function…

A: Given: fx,y=x2y+exysiny

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.