Prove the following using the Mean Value Theorem: (a) For any real numbers x and y with x > y, we have: sin(x) — sin(y) ≤ x − y (Hint: try dividing both sides by (x − y).) (b) Let n be an even positive integer. The equation x + ax+b=0 has at most two real roots. (Hint: show that between any two roots

Q: Let f be the function defined by f(x) = sin 2x + cos 7x and let g be the function defined by g(u) =…

A:

Q: 1. Let f(r1, 22, r3) = x1 sin(r23). Compute the following: %3D a) Dif(x1,x2, x3) b) D3f(x1, x2, x3)…

A:

Q: Suppose we define a function f(x) by f(x) = a1 sin(x) + a2 sin(2x) +· + an sin(nx), .. for constants…

A:

Q: Find the smallest interval in x over which the functions sin(x/xo) and cos(Tx/xo) are orthogonal. On…

A:

Q: Given α+ßi = √¹x³¹_x=¹¹dx, Inx 0 α, ß ER, find the analytic expressions for a and

A: Recall the following formula  ln(a+bi)= ln √(a²+b²) + i tan-1(b/a) .

Q: Rewrite sin (x + in terms of sin x and cos x. Enclose arguments of functions in parentheses. For…

A:

Q: 2. If f(g(x)) = sin x + 3sinx - 1, determine possible expressions for f(x) and g(x), where g(x) # x.

A: (1st method/way): Let f(x)=x²+3x-1 and g(x)=sinx Then we can find f(g(x)) as follows…

Q: 45. Which of the following expressions are equal to 1 ? for all real values of x such that 0 < x < 2…

A: Consider the expression sin2xcosec2x. Here, cosec2x=1sin2x implies, sin(2x)×1sin2x=1 Therefore, the…

Q: Show that the extreme values of f(x) = a sin x + bcos x tVa2 + b2. are

A:

Q: 9. Two zeros of f(x) = x* – 7x³ +x² – 63x – 72 are 3i and 8. In what form are the other roots? A. 1…

A: Topic = Polynomial

Q: 61. Prove Theorem 4.13(c): For any real numbers a and b, Si x² dx = (b³ – a³). Use the proof of…

A: To prove : for any real numbers a and b,∫abx2dx=13(b3-a3). Solution: We know, ∫xndx=xn+1n+1+C…

Q: Let y = sin(2nx) + b cos x, where b: constant and n: even integer. TU 2 Evaluate y" when x = (a)…

A: Here is the solution:

Q: A box contains 5 black balls and 10 red balls from which 3 balls were drawn with repetition, what is…

A:

Q: Answer by 'yes' or 'no' each of the following 8 question. x and  y are real numbers

A: Answer and explanation is given below..

Q: Solve x21 ≡ 6 mod 7

A: If p is a prime number then xp-1≡1(mod p) for all x∈ℤ~pk  where pk denotes multiples of p . Suppose…

Q: b) Show that P1(cos(0)) and P2(cos(0)) are orthogonal to each other, P, and P2 being Legendre…

A:

Q: By applying Rolle's theorem, check whether it is possible that the function f(x) = a° + x – 6 has…

A:

Q: 9. The value of f 1-x 1+x A. sin ¹¹x + √1 + B. sin¹x - √1 + dx x² + C x² + C C. sin ¹¹x + √1- x² + C…

A:

Q: Show that the equation sin x - 7x + 6 = 0 has exactly one real root. Let f(x) = sin x - 7x + 6. f(0)…

A: Rolle's theorem.

Q: Show that the function f(x)= x^3 − 15 cos(pi x) + 1 has three zeros in the interval [−4, 4]. Do not…

A:

Q: Given that f(x) = x² sinx, use Leibniz's theorem to find Ⓒd ²³ f dx²³

A: Here the given function is f(x)=x4sin(x). We have to find the third order differentiation of f(x) by…

Q: For the constant numbers a and b, use the substitution z = a cos² u + b sin² u, for 0 <u<, to show…

A: This question can be solved using the concept of integration by substitution.

Q: Express (text in image) for some constants A, B, and C.

A: NOTE: Refresh your page if you can't see any equations. . compare this equation with  so here we…