For the function f(x) = 6x² + 14x + 31, find f' (x) using the definition f (x + h) – f(x) lim h→0 h O 12x + 14 O 6x O 28x + 31 O 6x + 14 O 12x

Q: Find the linearization of the function f (x) = x' at x = 1. L (x) =

A:

Q: Find the linearization of f(x) = (1+ x)20 at x = 0. Use it to approximate (1.001)20.

A:

Q: Suppose F(r, y) is a C' function such that F(0, 0) = 0. What conditions on F will guarantee that the…

A: Let us consider, F^(x,y)=F(F(x,y),y)  On applying the implicit function theorem , can get a solution…

Q: Let f (x) = x² – 4. a. Find f'(x) using the formal limit definition of derivative. -

A: The formula of formal limit definition of derivative is  : lim h →0  [{f(x+h) - f(x)} / h ]

Q: Prove that = va² – x* if y =(f)sin~+ () + (9)v&* - x²

A: Differentiation wrt x

Q: Assume that f(x) is a differentiable and invertible function so that f(3) = 4 and f'(3) = .…

A:

Q: Find linearization L(x) of f (x) = Vx at a =4 and use to approximate 7.8.

A:

Q: dy Prove that dx of x2 +2x is f'(x)=2x + 2

A:

Q: Use the limit definition of the derivative to show that the derivative does not exist at x = a for…

A:

Q: Use the quotient rule to find the derivative. 2.1 +6 f(x) = x2.0 + 1 2.6 3.7 +2.1x x1.1 - 15.6x 1.6…

A: You need to just apply quotient rule.

Q: Find the local linear approximation L (x)of the function f (x)= x² +3 at x = 1

A: Given that: f(x)=x2+3

Q: Suppose F(x,y)=sin²(xy +e"), where x=u² +v and y = In(u² +v). Find the . ƏF ¸ ÔF value of when u =1…

A: The given function is: F(x,y)=sin2xy+eywhere:x=u2+v         and         y=lnu2+v

Q: Find the values of k that make the function continuous. Jk² + 18x, x > -3 f (x) = kx, x < -3

A:

Q: Find the local linearization of f(x)=x^(2) at 2

A:

Q: for x 2 - X Is differentiable at x = 2 but not at x = 1.

A:

Q: Find the value of k that makes function continuous at z = 4 | 2x + 8 x> 4 4x k x < 4 f(x) =

A: If the function is continuous at x=4 then the left and right-hand limit at x=4 should match. That…

Q: of For the function f(x.y) = In (x + 6y), find- of and dy

A: It is partial derivative so when we differentiate with respect to one variable consider other…

Q: Find linearizing ( L(x) ) f(x) at x=a . where f(x) = tanx and a = π a) π b)  x - π c) π - x d)…

A: SOLUTION-

Q: Use the e - S definition to show that the function x- 1 f(x) = x +2 %3D is continuous at x = 3.

A:

Q: 4. Find the differentiation dy. using definition with limit of the following: y = V2x + 1.

A:

Q: Find the linearization L(x)of f(x)at x = a where f(x) = Vx² + 9 at a = –4

A:

Q: Find for the following function: 1- y = etan'x 2-x'y+ x*y 2 at x = 2

A:

Q: Find the global maximum and global minimum of the function f(x)=sinx−x in the interval [−π,π].

A:

Q: Find the value k that makes the function continuous. kx + 14, х > 2 f (x) = 11x + 2, x < 2

A:

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

A:

Q: Find the linearization of the function f (x) = x³ at x = 1. L(x) = Using this linearlization, ƒ…

A: Answer is in next step.

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: Find the linearization of the function f (x) = x° at z = 1.

A: Given, f(x)=x6

Q: Find the linearization L(x) of the function at a. f(x) = sin(x), a =. 3 L(x) =

A:

Q: 2. Find whether the function F-(* - y' )i + (v² - 3x)j + (:² - xy is irrotational.

A:

Q: lim, f(x)= 0, then | f(x)dx diverges

A: Statement:  limx→∞ f(x)≠0,  then ∫1∞ f(x) dx   diverges.

Q: (B) Let f(x) = 3x – 1. Use the limit definition of the derivative to find f'(x). %3D

A: Explanation of the answer is as follows

Q: Find the value of K that make the function f continuous at x=2 -4 スー3x+2 sin(nx) + kx + k if x 2

A:

Q: Suppose that f(x, y) = e¯4z² - ly + 7z + 2y then find the discriminant of f.

A:

Q: Use the first derivatives test to find the local exterma of f(x) =x3-12x on R

A:

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: Find the linear approximation L(x)of f(x)=x^4 at the point a = −5 L(x)=

A: Given that f(x) = x4 We have to find the linear approximation L(x) of f(x)=x4 at the point a= -5.

Q: Find the linearization L(x) of the function at a. f(x) = sin(x), a = 3 L(x)

A:

Q: - Given f(x, y, z) = x In(yz³), find fzy and fyz.

A:

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

A:

Q: Show that the linearization of f(x) = (1 + x)k at x = 0 is L(x) = 1+ kx.

A: We need a linear approximation formula for f(x).

Q: For f(x) = 2x2 +1, use the definition of the derivative f'(x) = lim (e+h)-}(=) to find f'(x). h0 h O…

A: f(x) is differentiable then  h→0limf(x+h)-f(x)h Here f(x)=2x2+1

Q: Integrate from f(x) = x4 from 0 to 1 by (Trapezoidal Rule) with h = 1, h = 0.5, h = 0.25 and…

A: Here, f(x)=x⁴ So, I=  Integrate from f(x) = x⁴ from 0 to 1. Now we can solve by Trapezoidal rule,…

Q: If ƒf (x) =¸ (A) (B) Sº sec²x 1+elanx In (secx) √✓ 1+tan²x tanx 1+e sec ³x tanx 1+e 1+1² tanx…

A: Using Leibnitz integral rule.

Q: Find and sketch the domain of the function. f(x, y, z) = ln(64 − 4x2 − 16y2 − z2)

A: Given function is      f(x,y,z) = ln(64-4x2-16y2-z2) We have to find the domain of this function.

Q: If F(X) is an antiderivative of *-2 + x-', then what is the value of F(1) - F(-1)?

A:

Q: Find constants c1 and c2 such that F(x) = c1xe -x + c2e -x is an antiderivative of f(x) = 3xe -x .

A: Given: Fx=C1xe-x+C2e-x  and fx=3xe-x

Q: Integrate f(x) = 10 ( 1 – x)9 where 1 ≤ x ≤ 3, Using Boole’s rule

A: The function f(x)=101-x9 where 1≤x≤3. We have to integrate this integral using the Boole's Rule.