Let f(x) = x2 and g(x) = −x3 + x2 + 3x + 2. Then f(−1) = g(−1) and f(2) = g(2). Using Rolle’s Theorem, show that there is at least one value c in the interval (−1, 2) where the tangent line to f at (c, f(c)) is parallel to the tangent line to g at (c, g(c)). Identify c.

Q: f(x) = sin 2x,x e [0,2n]

A:

Q: Let  f(x, y) = sin(15x2 + 3y2) Calculate the quadratic approximation of f at the point (0,0)

A: Given that,  fx, y=sin15x2+3y2 The quadratic approximation of the function fx, y at the point x0, y0…

Q: Find all the points of g(x, y) = x³ − 3x + 3xy² where a relative extrema might occur.

A:

Q: The derivative of f(r, y) at P(1,2) in th in the direction of-2j is -3. Then the of -i - 2j is

A: It is given that the directional derivative of f(x,y)  at P0(1,2) in the direction of (i+j) is = 22…

Q: Calculate the linearization L of f(x) : 5e* at x = 1. x2 Express your answer in terms of e and x.

A:

Q: Given f (x,y) = x³ + y3 – 3y² – 3x² – 9y, find the critical points and use the Second Derivative…

A: solution :-

Q: find the derivative of the function at P0 in the direction of u. ƒ(x, y) = 2x2 + y2, P0(-1, 1), u =…

A:

Q: In what directions is the derivative of ƒ(x, y) = (x2 - y2 )/ (x2 + y2 ) at P(1, 1) equal to zero?

A: Given, , the unit vector is , here directional derivative is given as,

Q: Use the second derivative test to identify any critical point and determine what each critical point…

A: Critical point: The point where the function fx, y is defined and its gradient is a zero vector or…

Q: Find the derivative of y with respect to x using implicit differentiation in the equation *³y =…

A: We can find the derivative as below.

Q: Define a function f : R² → R² by f 2.у 1. Show that f is differentiable on A = E R² |x > 0, y > 0}…

A: To show that the function f defined as below is differentiable on A and compute its derivative

Q: dy + ydx %3D 3x²y?dx,

A: Given differential equation is bernaulli's differential equation.,  and we can solve it by reducing…

Q: Supose z is implicitly defined by the equation qy+ 3x + xyZ + 32=9 in an enmrament of the point…

A:

Q: For the given function 9 = x-yz and 62 -(@C) at C= 3x?yi+ yz²j- xzk, Find 5yöz бубz point (1, – 2, –…

A:

Q: Let ƒ(x) = √2x + 1. Compute ƒ'(2),the derivative of ƒ at x = 2, and use the result to find an…

A: Consider the given function, fx=2x+1 differentiate with respect to x, f'x=ddx2x+1=122x+1·2=12x+1…

Q: find the critical point and classify the function f(x,y)= x3 +y3 - 6xy

A:

Q: Let be a real function of y=f(x) real variable implicitly defined by the equation 2xy−x2y2+ey=0.…

A:

Q: Prove that f (x) = x3 − 12x is differentiable. Compute f'(x) and findan equation of the tangent line…

A: For a function f(x) to be differentiable the limit limh→0f(x+h)-f(x)h must exist. This limit is…

Q: Find given that z is a differentiable function of x and y, and xz + yz3 = 2 +x²ev

A:

Q: Find the local linear approximation of f(x)= at 2+ x xo = 2, where Ax = x – 2. NOTE: Enter the exact…

A: while finding out the local linear approximation of any function f(x), what we do is to determine…

Q: Find the derivative of the function. h(x) = In[(2x + 8)(3x +6)] a. (2x + 8) (3x +6) 1 1 Ob. (2x + 8)…

A: The objective is to find the derivative of the function h(x) .

Q: Find the derivative of the function at Po in the direction of u. f(x, y) = In( 6 x - 3 y), Po(-7, -4…

A:

Q: Find the derivative of y with respect to x, t, or 0, as appropriate. 1-x y = In (x + 2)5 4x - 7 (x +…

A:

Q: Find the derivative of f (x y z) = x^3 − xy^2 − z at (1, 1, 0) P0 in the direction of A = 2i - 3j +…

A: Given,           f(xyz)=x3−xy2−z  and point Po =(1,1,0) and the direction vector A=2i-3j+6k.

Q: For f(x, y) = 2y3 + x² – 4x – 4xy – 9, find the critical points and use the - Second derivative test…

A:

Q: Write the equation of the tangent line to f(x) = + In(a²) - 4 at x = 1. O y = 3x - 6 Oy = 4x - 7 Oy…

A:

Q: Let y be a function of x and suppose y=In(x²+y²). What is the correct result for the derivative of y…

A:

Q: Find of the function f(x,y) = sin(x²y³). əx3

A: The product rule of differentiation : (PQ)'=PQ'+QP' sinx'=cosxcosx'=-sinxxn'=nxn-1 Use the given…

Q: Shows the derivative function of: u³ + 2u, where u = x² + 3

A: Let's find.

Q: Find the linearization L(x) of the function at a. f(x) = sin(x), a 6

A:

Q: find a linearization at a suitably chosen integer near a at which the given function and its…

A: (1)Given:

Q: Find a positive value, c, for x, thot satisfies the Mean valve Theorem for derivatives for…

A:

Q: a.) If     f(x) dx   = 4x9 − 8x6 + C, find f(x). f(x) =

A: Given: ∫fxdx=4x9-8x6+c                                          ..1 for finding fx, we differentiate…

Q: Find the derivative of the function at the given point in the direction of A. f(x.y) = 6x+10y,…

A:

Q: The derivative of the function y = tan(In(vx² – 6x + 1)) is: %3D

A: Formula:ddx(tan x) = sec2xddx(ln x) =1xddx(xn) =n xn-1

Q: Find the linear approximation L(x) of the function g(x)=  ∛(2+x ) at a = 0 and use it to approximate…

A:

Q: The finite difference approximation of the first derivative in the simple implicit method used to…

A: given  ∂2u∂x2=∂u∂t to find the finite difference approximation of the first derivative in the…

Q: Supposed that z is defined implicitly as a function of x and y by the equation x2 + y2 +z2 = 4. Find…

A: The given data is: x2+y2+z2=4 To find the value of dzdx

Q: Obtain the first and second derivative of the following real-valued functions 1. ƒ : R → R, such…

A: Solution :1.fx=23x3-2xfirst derivative ,f'xf'x=29x2-2Second derivative…

Q: Assuming that x> 0, use differentiation to justify the formula X dx = In |x| + C. (=| d dx = In |x|+…

A:

Q: Find the differential dy of each function: a) f(x) = 2x^1/3 - 3x^1/2 b) f(x) = ( x^2 +1) / (x+1)

A: we will find the derivative of given function using basic differentiation rules.

Q: Find the derivative using the theorems on derivatives. dx x2-2 1 y = 2.y = (x? – 1)³(x² + 3)? x3+4x…

A:

Q: Evaluate the 3rd derivative of M with respect to (x) M= x6 y3 Cos (x6 y3) if x =6 and y= 3

A: The function is M=x6y3cosx6y3. Using implicit differentiation, find dydx.…

Q: find the derivative of y with respect to x, t, or θ, as appropriate.

A: On simplification, we get

Q: find the derivative of the function at P0 in thedirection of v. ƒ(x, y) = 2x2 + y2, P0(-1, 1), v =…

A: Given: fx,y=2x2+y2  We know that, The direction of v is the unit vector obtained by dividing v by…

Q: 4) Let g(x,y) = 2x² + y² – x²y. a. Find the critical point(s) of g. b. Use the second derivative…

A:

Q: Find the standard (slope-intercept form) eq at the specified points. 1. f(x) = 3x² – 12x +1 at…

A: 1) Tangent line of the function f(x) at x1,y1 is given by y=f'x1x-x1+y1  3x2-12x+1 at P(0,1) x1=0,…

Q: Find ý of the function x2 - e2x+3y - In(xy) = 0. %3D

A: Consider x2-e2x+3y-ln(xy3)=0We differentiate it implicity, we…

Q: Find the critical points for the function: f(x,y)=x3+y3-9x2-48y-3 and use the Second Derivative Test…

A: To determine the critical points for the function and classify the critical points.