4. Use the inequality | sin r|< |x| for x E R to show that the sine function is continuous at r = 0. Use this fact and the identity sin x sin u = 2 sin u) cos %3D to prove that the sine function is continuous at any point of u E R.

Q: If sin' 0+y cos e sin e cos e and x sin e = y cos 6, prove that x + y* = 1.

A: Solution

Q: 1. Determine the absolute extrema of f(x) = (sin r – 1) cos r on 2 2 if they exist, and identify the…

A:

Q: Find all values of x in the interval [0, 2π] that satisfy (cos x)2 − 3 sin x − 3 = 0

A: x=3π2 is the only values that satisfy the given condition see the calculation below

Q: Determine the exact x– value(s) of the intersection points of the functions f (x) = 4 sin² +7 sin 4…

A:

Q: Find the values of x in the interval [0, 2π] that satisfy: sin² x-1=0 O x = 0 and x = πT x = TT/4…

A:

Q: 1. Show by defintion that f: R R, f(x) = cos(a), is continuous on R. Hint: You may try to mimick the…

A:

Q: Complete the following argument that sinx#x for any x in the interval (0,1): • For any x in (0, 1),…

A: 1) the mean value theorem . 2) c . 3) is greater than . 4) is greater than .

Q: Prove that x2 cos x dx = T(2 – n) (+ sin æ)?

A:

Q: If f( x ) = sin (x) – e“ , what is f'(x) ? O f'(x)=- cos(x)-e² — сos(2) —е* O f'(x)=sin (x) –e² O f…

A:

Q: 3. Calculate the following integral: à tanʼxdx 1 4 а. -tan“ x + C 4 1 b. tan x - In |cosx|+C 2 1 с.…

A: integration of tan3 x

Q: Find f(x) given that: f'(x) = 2 cos x -3 sin x and f (주) %3D %3D

A:

Q: #2. Verify Clairaut's Theorem for the given function f (s, t)= In (st)-s* sin (6t) +st.

A:

Q: What is the limit of sin when x approaches 0? Select one: O a. 1 O b. 0 O c. Does not exist

A:

Q: Prove that sin x can be approximated by x − (x^3/6) within 0.01 on the interval [-1,1].

A:

Q: 31. Use the identity x + y y sin x + sin y = 2 sin cos 2 to prove that sin(h/2) cos h sin(a + h) –…

A: Given information: sinx+siny=2sinx+y2cosx-y2 To prove: 1: sina+h-sina=hsinh/2h/2cosa+h2

Q: if y = a sin x Prove a +b CosX a cos + b %3D 2 (atb cosx) 11

A:

Q: Evaluate the given function tan²ydy = sin3 xdx is equal to.. 3(tany – y) 30 + cosx А. Зх - cos³x…

A:

Q: For what values of x does g (x)= x – sin x have a horizontal tangent line? Consider values of x in…

A:

Q: If f cos xdx = A + B sin x + Csin²x+ Dsin³x+ E sin*x+ Fsin5x, what is the value of A + B + CD + E +…

A: Consider ∫cos5x dx =∫cos4x cos x dx =∫(cos2x)2 cos x dx=∫(1-sin2x)2cos x dx…

Q: - Find the extreme value/s of f(x) = 4 sin² x cos 2x on (37 3π 5π 22

A: This is a problem related to maxima and minima of the function. Based on the general formula and…

Q: Show that COS x dx a² – x² sin a 2а where a is some positive number.

A:

Q: 7 Use the Intermediate Value Theorem and Rolle's Theorem to show that the equation 2 = sin z+1 has a…

A: use the intermediate value theorem and Roll's theorem to show that the eqautionx2=sinx+1has a…

Q: 3. If v = sin- (4) + tan-' (±), find the value of r + y. 4. If z= ear+by. f(ax – by), prove that b…

A:

Q: 2. Verify that the Mean Value Theorem (MVT) applies to the function f(z) = sin² z- on the interval ,…

A:

Q: Evaluate si (z* - tan(z) cos(z) ) dx 2-1

A: To find the integral: ∫11x2-tanxcosxx3-1dx

Q: The integral S (A S- (в x dx x² + √√√x²-1- sino do 1 + cos sec²0 do 1+tano " cose de 1-sino sece…

A: Use trigonometric substitution

Q: The Fourier integral of a function f(x) is given by 2w sin w cos wrdw. If lim f(x) = 1, what would…

A:

Q: Show that where n is a positive integer.

A: To show

Q: (5) Use the Intermediate Value Theorem to prove that xe* = cos(x) has a solution.

A:

Q: S sin x In(cos x) dx

A:

Q: to the I. sinx(e+ 1)x' - cosx = 1, y(=) = 0 %3D %3D (1 + e) (1 + cos x) = 0 (1 + e) (1 + sin x) = 2…

A:

Q: See image below....

A:

Q: 1. Determine the absolute extrema of f(z) = (sin a – 1) cos z on -. 2 if they exist, and identify…

A:

Q: Use the indicated substitution to evaluate the integral. 2x cos(x) dx , u = x' sin (x²) + C 2 sin…

A:

Q: 4. Prove that the identity is true. 1- cost e = sin² 0 1+ cos e

A:

Q: a Show that (sin(m + n)x + sin(m – n)x) b Hence, without using the product rule, differentiate sin…

A: We will find out the required values.

Q: Find the intervals on which f(x) = 5 x + 5 cos(r) increases for 0 < a < 2n.

A: f(x) is increasing if f'(x)&gt;0 in the interval   Note that  -1 less than equal to (sinx) less than…

Q: Compute Cos (x + k ) cos a dx by using substitution technique where k is a constant. [Hint: cos(A±…

A: The given integral is cos(x+k)/cosx.

Q: х If f(x) = cos(1? + 1) dt, then find f'(x). %3D Select one: O A. None of the others O B. sin(x + 1)…

A:

Q: 9. Prove that | sin n0/sin 0| <n, limiting values being used when sin e = 0.

A:

Q: Use the Intermediate Value Theorem to show that sin x x2 – x has a solution on the interval (1, 2).

A:

Q: The f sin x cos x dx is not equal to {A} -cos 2x + C {B }sin'x + C {c}-cos'x + C {D} sin 2x + C

A: Given query is to find the incorrect option for the integral.

Q: Find all values of in the interval 0, 2 such that sin x = sin 2x

A: To find all values in the interval for the function

Q: 2. Consider finding the root of f(x) = e*(3.2 sin(x) - 0.5 cos(x)) on the interval [3, 4], this time…

A: 2 Given function is fx=e−x3.2sinx−0.5cosx. We have to find the roots of the function fx on the…

Q: csch(3 tan x) + sin² x cos² x dx

A:

Q: Is there a value of c that will make sin² 3x f(x) = C, x = 0 continuous at x = 0? Give reasons for…

A:

Q: Suppose v is in the interval (G, T ), with tan v = –3 Find the exact expression for the indicated…

A: