(a) Can [sin (x+y)]/(x+ y) be made continuous by suitably defining it at (0, 0)? (b) Can xy/(x² +y²) be made continuous by suitably defining it at (0, 0)? (c) Prove that f: R? → R, (x, y) → ye* + sin x + (xy)4 is continuous.

Q: Let f (x) = f₁²² sin² (t)dt. Find ƒ' (π). ○ 2π sin² (π) π sin² (7²) ○ sin² (²) ○ 2π sin² (²)

A: The given function is

Q: Ql) check the continuity of fX) at X=-1, 0,1.23 -1<x<1 x-4 2x-1, 3x-2 2, x=3 f(x) = 1sxs2 25x53 Q2)…

A: 1)

Q: t f(x+y)=f(x)+f(y) + 2xy-1 for all real x and y and f(x) be a differentiable function. [f f^ prime…

A:

Q: Let f (x) and g(x) be differentiable functions in (a, continuous at a and b, and g(x) #0 in [a, b].…

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Q: fo) is continuous on (- co,oo). Use the given information to sketch the graph of f f(0)-0, f (20) 0…

A: f(0) = 20Graphs in option B and D show f(0) to be negative. Hence, these two options are ruled out.

Q: f(x)=(-x,             x0) Which is correct? a) x=0 is a vertical asymptote of f(x) b) f(x) is…

A: The given piecewise function is:  fx=-x, x≤0sinxx, x&gt;0 To find the vertical asymptote of f(x),…

Q: Stan"( 1 -tan m+2 -tan m+1 m+1 1 m+3 Evaluate A B Ⓒ D tan" (x) sec¹x dx. (m + -2)(x) + ¹) (x) - 1)…

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Q: let it be 2= f(x,y) and Y=tan' () and X = === In (r²+5²) Prove that fxx+fxy= (y² + S) (fr+fso)

A: We have to prove

Q: If f is a differentiable function at a, then xf is differentiable at a and (af)'(a) is: Select one:…

A: We need to evaluate the following expression xf'a

Q: Let f(u) = u* and g(x) =u = 2x* + 6. Find (fo g)'(- 1). (fo g)'(-1)= (Туре an exact answer.)

A: Given:- Let f(u) = u4 and g(x) = u = 2x4 + 6   To Find (fog)'(-1).

Q: b) If the function f(x), g(x) and h(x) are continuous on [a, b] and derivable on (a, b), show that…

A: The given functions f,g and h are continous on [a,b] and differntiable on (a,b). So intermediate…

Q: What is the A B (C) D af - of f (x, y) = tan (x²y³ + 2xy)? dy [ sec 2 (x2y3 + 2xy (2 xy + 2y) 2 [sec…

A:

Q: for If f (2) is a regular function of z, prove that f (2) 2 = 4 | f'(2) 1?. %3D ve I f12) 1? =4 |…

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Q: Let f(x) — sin(nx) n³ n=1 Show that f is differentiable on R and that f' is continuous on R.

A: Within its interval of convergence, the derivative of a power series is the sum of derivatives of…

Q: Evaluate tan" (x) sec4x dx. 1 1 A -tan 2) (x). + (m+2 -tan" (x) + C m+2 m (m+1) ¹) (x) (m- m+1 1…

A: Evaluate: I=∫tanmxsec4xdx

Q: 2. a) Give an example of a function f: R →R that is differentiable everywhere and 1- to-1, but f'(x)…

A: Let, f(x) = x3 .      Then, f(x), being a polynomial, is differentiable for all x ∈ R, i.e.,…

Q: Let F = (3t + 2) i + sin(2t) j+ 3ť²k Find F'(t), F"(t) and F"(t) 1) F'(t) = (2) F" (t) 3) F" (t) =

A: Solve for the equation for the derivative

Q: Let f(x) be a twice differentiable function on (-o, 0) such that f"(x) is continuous on (-0, 00).…

A: Solution is in next step

Q: Assume that the functions ?,?,ℎ:ℝ→ℝf,g,h:R→R are differentiable such that  f′(a)=b, g′(b)=c,…

A: given: ℎ:ℝ→ℝf,g,h:R→R are differentiable such that  f′(a)=b, g′(b)=c, h′(c)=d, f(a)=e, g(e)=ℓ,…

Q: 2. At what points of R² is the function continuous? (sin(x² + y²) x² + y² f(x, y) = 0, -, if (x, y)…

A:

Q: Prove that f(x)= sin(5x) < 5x on [0,x]

A: Take the help of derivative and try to prove the given statement in two cases.

Q: Suppose f(z) is defined on all of C, is differentiable at z = a, and satisfies f(a) = a. Show that…

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Q: Let f and h be real-valued functions continuous on [a, b], differentiable on (a, b), and h(a) not…

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Q: If f', and y are continuous on [a, b] and derivable on Ja, b[, then show that there is a value c…

A: