2. Which of the following functions satisfy the hypothesis of Rolle's Theorem on the interval [0, 2]? 1 (a) f(x) = |x-1| (b) f(x) = |x – 1| (c) f(x) = x² – 2x (d) f(x) = 2x + 3 2*+3 %3D n→-0 3x-1 3. lim (а) — оо (b) +00

Q: 1. Show (in terms of e- 6) that a function f : R* → R defined by f(x, y, 2) = (2x + 3y + 42) is…

A:

Q: 3. Find the maximum and minimum values of a 1 āz - on |z| ≤ 1 (where a with la| < 1 is fixed)

A:

Q: The equations of the vertical asymptotes of the function with rule f (0) = are: 1+ cos 0 e = 3rk…

A: Take denominator of f is equal to zero.

Q: If f (x) is an invertible, differentiable function with tangent line y = 4 x + 7 at the point (0, 7)…

A: we have to swap variable for finding the inverse function

Q: 6. Let u (r, t) be the solution of the initial-boundary value problem ut = 9uzz 0 0, %3D uz (0, t)…

A:

Q: Let y : R →R be the real-valued function defined on the real line, which is the solution of the…

A: Ans:-  Initial value problem: y'=-xy+x,  y=0=2          solution : y'+xy=x, 40=2          is of…

Q: 1. Show (in terms of e – 8) that a function f : R³ → R defined by f (x, y, z) = (2x + 3y + 4z) is…

A:

Q: Sketch a graph of each of the following functions that captures the limiting behaviour as t → ∞ and…

A: Note:- As per our guidelines, we'll answer the first three sub parts since exact three were not…

Q: 2. The graph of y = cosh a is (a) concave up on its entire domain because y" = coshr > 0 for all r €…

A:

Q: 15. If g(x) and h(x) are both antiderivatives of some func- tion f(x), then what can you say about…

A: Consider that gx and hx are both antiderivative of function fx. Find gx-hx.   Since gx and hx are…

Q: 2Use divectional devivative delinihion of f(Hiy)=x y'at the divecionof ū =i jatthe Point (le2>

A:

Q: 6. Consider the function f(x, y) = cot ¹(1+x) — 2e¹+³y If x = s²-t² and y = 3t+tan(1-s), use the…

A: Let's evaluate given partial derivative.

Q: We have found the following Wronskians. W = -e-³x; W₁ =-e-2x(4+ex)-¹; W₂ = e¯x(4+ e*)-¹ We want to…

A:

Q: 3. Suppose o1 and o2 are solution to y" + ay +by = f, where a and b are real constants and f is a…

A:

Q: (a) Express the initial value problem: 1 s'(x) = -f(x), for x +1 f(0) = 1, as an integral equation…

A: The Initial value problem is given as, f'x=-1x+1fx, for  -13≤x≤12                         (1) with…

Q: Find solution of an equation x X f(x) 1 1.0000 1.25 0.8000 1.50 0.6667 0.5714 1.75 2.0 0.5000 Using…

A:

Q: 1 1 Select all the antiderivatives of f(x)=-+ %3D x² +1

A:

Q: b) If y(t) is the solution satisfying the initial condition y(0) 1, find the value of lim y(t).

A:

Q: let f(x) = { ax2+x+1 if x is less than or equal to 1                 { bx if x >1   find a and b…

A:

Q: 1. Consider the function z = x In(1+ y²), if x 2-t and y= Vf, use an appropriate form %3D dz at t…

A:

Q: f(x) = 3 x4-4 x3 -12 x2+1, has local minimum at x = OA. OB 2- OC. OD.

A:

Q: Differentiate the following

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: Xn Xn=3-(-1)" xn = cos (¹₂) √1+n² Xn= 27-5 Xn=¹ where (yn) is bounded 4 lim inf(x) lim sup(x₂)

A: We know the fct that for every  bounded  sequence  limsup(xn) and  liminf(xn) exists finitely. since…

Q: 1. Evaluate the limits. e' cos (t – 4) dt - 4 (а) lim x2 x2 – 4 e2 + 2 (b) lim x00 e2x + 100

A: Solve the following

Q: If z is a function of x and y implicitly defined by əz əz find and əx ду at (x, y, z) = (0, 2, -1).…

A:

Q: 2. Give an example of a regulated function f on [0, 1] such that f(x) > 0 for all æ e (0, 1), So…

A:

Q: 1 Let H(s) Using convolution to find h(t) s2 (s-a)

A: Inverse Laplace transferom

Q: If y(t) solves the IVP dy = y2021 y2019 y(2021)= -2019, dt then lim y(t) is: t- 00 Oa. 1 O b. 00 O.O…

A: Given differential equation is dydt=y2021-y2019.       ........1 It is an autonomous differential…

Q: 2. Suppose that z = f (x, y) and both x and y are functions of a single variable t such that dy t2 –…

A:

Q: (3) (a) Find the local linear approximation of f(x) = /1+x at xo = 0. Use your approxima- tion…

A:

Q: -1 Evaluate: lim a. 00 O b. 1 O c. 0 d. - 00 QUESTION 3 Using differentials, if determine the…

A:

Q: Chapter 7, Section 7.7, Question 012 Use the inequalities JEM| = ƒ (x) dx 24n2 (b - a) K2 |Er| = f…

A: Given that the integral is ∫0414x+1dx. a) It is required to find the upper bounds on error obtained…

Q: 23. lim Vx2 + y² In(x² + y²) (xy)→(0,0)

A: This question is related to limit in two variables.

Q: 4.2) Suppose you wanted to find the absolute extrema of f(x, y) = ax² + bxy+cy? + dx + ey on the…

A: The given function is, fx,y=ax2+bxy+cy2+dx+ey We have one critical point on the rectangle,…

Q: Given the function y"=-6X(x²-ax+4) (x+a)³ (x -1)³ 3. with (X+2)(x-) - analysiS Enclude all maximums…

A: given that y=-x3x+2x-2find the maxima and minima for the given function

Q: and that there is a deleted neighborhood U of 0 su and n 2 0. Define f(x) = x -4g(x)(1 – cos x) for…

A:

Q: 3.2 Let f(x) = Approximate the integral x4 x L f(x) dx using the Trapezoidal rule with n = 4. -2

A:

Q: 1 3. Integrate the function f (x, y) = over the region bounded by x-axis, line x = 7, and the curve…

A:

Q: 4. If R(T) is the autocorrelation function of a stationary process {X(t)} with no periodic…

A:

Q: 2. Evaluate f(x) = e3x + 2 at x = -1. Round to the nearest thousandth. %3D %3D f(-1) = ...

A:

Q: 1. Use local linear approximation to estimate In (0.99). 2. Find the absolute extremum values of…

A:

Q: 2t – 2 t2- 1, Let r(t) = - î - tan (t) ĵ+ In(2t) k. %3D (a) Determine the domain of r. (b) Compute…

A:

Q: Suppose that r1 and r2 are roots of ar2 + br + c = 0 and that r1 ≠ r2; then exp(r1t) and exp(r2t)…

A:

Q: If f(x) > g(x) >0 for X in [1,0) and g(x) dx diverges to +0, then | F(x)dx also diverges to +. True…

A: It is TRUE, that 0&lt;g(x)≤f(x) for x in [1,∞)  and                               ∫1∞g(x)dx diverges…

Q: F.M: 15 1. Use second derivative test to classify each of the given critical numbers as a relative…

A: 1. A) f(x) = e-x(x² - 3x + 1)  So, f'(x) = e-x(2x - 3) - e-x(x² - 3x + 1) So, f'(x) = e-x(2x - 3 -…

Q: A clamped cubic spline for a function f on 0,3 is given by [1+ax² -2x', S(x) = . [1-2(x-1)+b(x-1)² +…

A:

Q: none 1 point The y-intercept of the tangent line to the aurve of y = Vx +3 at the point (1, ) is…

A:

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.