If functions f(x) and g(x) are continuous for 0 sxs 1, could fx)/g(x) possibly be discontinuous at a point of [0, 1]? Give reasons for your answer.

Q: 1. If f(x) cos(t) dt, then f'(x) =

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Q: 'he function f: E → R is uniformly continuous if

A: We have to find the answer using the definition of uniform continuity.

Q: 12. Consider x²y f(x, y) = g4 + y? for (x, y) (0,0) and zero otherwise. Is f continuous at (x, y) =…

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Q: Find the value of c that satisfies the MVT of f(x) = 1+x + x2 on (0,2).

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Q: Find the linearization of x+1 f(x) = 2 – dt 1 + t“ at x = 1.

A: Given function is

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

A: The given function is

Q: Sx, ifx 1 f(x) = Show that f is R- integrable.

A: R- integrable function

Q: A:- Find the orthogonal trajectories for functions In -1) = - COS X +a y

A: The given family of curves is: lny-1y=-cos x+a ,            where a is an arbitrary constant.

Q: Differentiate f(x) with respect to g(x) of yx=xsiny

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Q: Find the linearization of f(x) = 9 – dt at x = 1. 2+t

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Q: Locate and classify all the critical points of the function. (If an answer does not exist, enter…

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Q: Determine two functions, defined on the interval  ( − ∞ , ∞ ) , whose Wronskian is given by  W ( f 1…

A: To determine two functions having the prescribed Wronskian and decide on their linear independence

Q: Find the local maximum and minimum values and saddle point(s) of the function. (DNE if doesn't…

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Q: Find the linear approximation L (x) to ƒ (x) = xcosx at a = T.

A: Given that, f(x) = x cosx and we have to find linear approximation L(x) at a = π

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 linearization L(x) of f(x) = cot(x) at x = π/4.

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Q: Let g(x, y) = x4 + y² – 4x³ + 5x – 6y+3. Show that g(x,y) continuous (actually at least aC')…

A: Recall [ Implicit Function Theorem ]

Q: If A(x)= f(t)dt is increasing on [0,10], then f(x) must be increasing on [0,10].

A: Given,A(x)=∫0x f(t) dt

Q: 3. Find the interval on which the curve of y= 2 1+3t+t² dt is concave up.

A: Second derivative is positive.

Q: Use a linear approximation to estimate f(0,1), given that y=f(X) is a differentiable  with f(0)=1…

A: Consider the given information

Q: Find the linearization L(x)of the function at a .

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Q: For the function f(x,y)=x^2+x−3xy+y^3−5, does the critical point (1/4,1/2) correspond to a local…

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Q: Consider the For at function y= In(x+1) . what valves" of x is y undefined ?

A: The given function is y=ln (x+1) We know that, for the function ln x, x>0

Q: Find the best linear approximation to f on th e interval [0, 1].                            f (x)…

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Q: Find the linearization L(x) of f(x) = cot x at x = 4

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Q: Q2 (A) At what points f(x, y, z) in which the function f(x, y, z) = continuous? is

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Q: Let f (x, y, z) = ln(x2 + z) + y2 exz. Find fzyx

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Q: x + y + 1 f(x, y) = , x ln (y² х — 1 -

A: A function f=f1, f2, ⋯, fn is said to be continuous at a if and only if the all the components f1,…

Q: + A(- for 0<x< 1, then u attains its maximum in the interval 0 sxs1 either at x=0 or at x= 1.

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

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Q: Show  that the function Z =1/2(eysinx - e-ySin(x) satisfies the La Place’s equation  =0

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Q: Find He valve of fe2) when fg functions such that are dilferentrable %3D g(x) >o 1 for all x, while…

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

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

Q: Verify Young's Theorem f(x,y) = e(2x+y) In(xy)

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Q: Q: Which c.d.f. F is of X is continuous at x=0?

A: If F(0) = F(0-) then F(x) is said to be continuous at x = 0

Q: S1 xe [0,1] f(x) 0 x¢ [0, 1]

A: The given piecewise function can also be represented as, fx=10≤x≤10otherwise Solve for the mean as,…

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

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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: Find the linearization of the function f(x)=x5 at x=1. L(x)= Using this linearlization, f(1.04).is…

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Q: For the function f(x,y)=x^2+2y^2−(x^2)y, does the critical point (−2,1) correspond to a local…

A: Given,  Critical Point: (-2, 1)

Q: Find and classify all critical points of the function defined by h(x,y)=x^3−3x+3xy^2.

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Q: 1. Show that the function x2 – y2 2xy f(x,y) ={ x2 + y? if (x,y) # 0 if (x, y) = 0 is continuous at…

A: If for every ε&gt;0, there exists a δ&gt;0, such that if |x-a|&lt;δ, |y-a|&lt;δ, then the function…

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

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Q: 3. If f(x) = (1–- t²)e*dt, on what interval is the curve y =f(x) upward? concave

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Q: Find the value of k that makes function continuous at z=4 |2z + 6 z> 4 5z – k z< 4 f(z) = ualuo…

A: Continuity.

Q: 13. Prove that the function 2x 1 dt t F(x) is a constant on the interval (0, ∞ )

A: The student has specifically asked for the solution to Question-13. Hence, that question is solved…

Q: Consider the function f(x,y) = -x³ + 4xy – 2y² + 1. Classify these critical points into local cal…

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Q: 以く2 g(x) = X = 2 *一2 h continuous at MArDl (0,2]

A: The given function is: g(x)=x2+1, x&lt;25        ,x=21x-2,x&gt;2

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.