Find the constants a, b e R so that the given functions are continuous on R. SO -22 + a, a sin ! +1 01 -x3 + a, x <0 x +b, Va + 4, 2 > 1 (c) h: R R, h (x) = 0

Q: - i) By using L'Hopital Rule, find sinx lim x-0 e3x 1 ii) Given that a = sin V3 2, find, csca -1

A: In L Hopital's rule take the derivative of numerator and denominator separately and than put the…

Q: Where is the function f(x) is continuous, such that f (x) : In(1-2x)+sin (x 4x+3 %3D [-1, ] – (-) O…

A: We know, ln(x) is defined for x&gt;0. sin-1(x) is defined for x∈-1, 1. f(x)g(x) is defined if…

Q: Compute the Riemann sum L4 for the function f(x)=2 cos x on the interval [0, 7]. Enter your answer…

A: To find the left riemann sum.

Q: Consider the function f(x) = cos x − 3x + 1. Since ƒ (0)ƒ () < 0, f(x) has a root in [0]. If we use…

A: For finding a zero of a function f, the secant method is defined by the recurrence relation.…

Q: Use integration by parts to show that, for n > 2, 1 cos a sin"-1 n 1 sin"-2 xdx. n - sin" xdx = x +…

A:

Q: Express the limit as a definite integral on the given interval. n lim x; In(1 + x²) Ax, [0, 1] i = 1

A:

Q: 5a) When p is a real number greater than 1, it can be shown that the improper 1 1 integral |dx…

A: Convergence of Improper Integrals at left-end point: Let us suppose that x=a be the point of…

Q: Show that f is continuous on (−∞, ∞). f(x) =    1 − x2     if x ≤ 1 ln(x)     if x > 1 On the…

A: Given function is : fx=1-x2    , x≤1lnx       ,x&gt;1 To show the given function is continuous in…

Q: Express the limit as a definite integral               n lim         sigma  e^(x_i)/(1+x_i)*deltax…

A:

Q: Consider the function f(x) = cos x − 3x + 1. Since ƒ (0)ƒ (=) < 0, ƒ (x) has a root in [o,]. If we…

A:

Q: Determine whether it converges or diverges. You must provide justification for your answer. sin(æ')…

A:

Q: Determine the points in the interval (0, 2n) at which the graph of the function below has a…

A:

Q: 10. Determine whether sin (x)dr converges or diverges. If this improper integral converges,…

A:

Q: Find the intervals on which ƒ is increasing and the interval on which it is decreasing.

A: Given function is fx=9-x2+sin-1x3.

Q: Express the limit as a definite integral on the given interval. lim n → ∞  n [2 − 5(xi*)2 + 6(xi*)4]…

A: Given query is to find the integral form of limit.

Q: Determine whether it converges or diverges. You must provide justification for your answer. sin(x)…

A:

Q: Consider the function f(x) = cos x - 3x + 1. Since f (0)f() <0, f(x) has a root in [0, 1]. If we use…

A: The given function is                    f(x)=cos x-3x+1 It is known that f(x) has a root in 0, π2.…

Q: hich the extrema occur. g(x)=x^2+cos⁡(x^2 ) on [-√π,√π]

A: The function given is gx=x2+cosx2 on the interval [-√π,√π]. To find the extrema of the function…

Q: Consider the function f(x) = cos x − 3x + 1. Since ƒ (0)ƒ (²-) < 0, ƒ (x) has a root in [0]. If we…

A:

Q: Show that tan(x) > x for 0 0 for 0 1 for 0 x for on 0 < x < -

A:

Q: Select all statements that are true. If f is continuous everywhere except at x = 0, then lim t→0- |…

A: This is a question related series calculations. If it leads to the particular value then it is…

Q: Perform a test to determine if it converge or diverge -1 x 2sin 1 dx 1– x -1

A: The function is given by: ∫-112sin-1x1-xdx If the function fx=2sin-1x1-x is positive, continuous and…

Q: sin (nx +3) S,(x) =- is convergent point wise to zero function. %3D п - 1

A: First one is true second is false.

Q: . Approximate f (x) by a Taylor polynomial with degree n at the number a f(x) = x sinx, a = 0, n =…

A:

Q: Find the intervals on which ƒ is increasing and the interval on which it is decreasing. ƒ(x) = -2…

A: Differentiate the function f(x)=-2cos x-x with respect to x to obtain the value of f'(x) and equate…

Q: Let f(x) = 1 - 2x where 1 0 ≤ x ≤ Expand f(x) to a 4 Fourier sine series converging to 1 f(x) on the…

A:

Q: Consider the sequence of functions n+ cos(x) 3n + sin(x) defined on the R. 1. Find the function f to…

A:

Q: At the point of discontinuity on(0, 00), the Fourier Sine integral converges to: OA None f(x+) -…

A: Here  At the point of discontinuity on  0,∞ the Fourier sine integral converges to

Q: Define the function f(x) by 1 if x = 0, f(x) = sin x/x if 0 < x (Note: We define f(0) = 1 so that f…

A:

Q: Consider the function f(x) = cos x - 3x + 1. Since f (0)ƒ () <0. f(x) has a root [] If we use secant…

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: 3. Prove that: cosh x - sinh x = e¯*

A: Proved Coshx - Sinhx = e-x

Q: b) Determine whether the improper integral 1+ sin x 1+ x2 converges or diverges.

A:

Q: Consider the function f(x) = cos x − 3x + 1. Since f (0)f () <0. f(x) has a root in [0]. If we use…

A:

Q: Consider the function f(x) = cos x − 3x + 1. Since ƒ (0)ƒ () < 0, f(x) has a root in [0]. If we use…

A:

Q: Use the Intermediate Value Theorem to show that 2x – 3 - if – 3 0 has a zero on the interval [-2,…

A: First compute the function if they are continuous then apply the intermediate value theorem.

Q: Determine whether it converges or diverges. You must provide justification for your answer. sin(x)…

A:

Q: f (x) = 4 sin(3x + 7) is continuous at every point x = x0; that is, lim 4 sin(3x + 1) = 4 sin(3xo +…

A:

Q: At which points does f(x) = 2 sin x - cos 2x have an absolute maximum value and what is the maximum…

A:

Q: Consider the function f(x) = cos x − 3x + 1. Since ƒ (0)ƒf () < 0. f(x) has a root in [0]. If we use…

A:

Q: It is given that a sequence of Newton iterates converge to a root r of the function f(x). Further,…

A: We have been given that the sequence of Newton iterates converge to a root r and that the root has a…

Q: Express the limit as a definite integral on the given interval. 5 V 3x; + (x;) Ax, [2, 3] * lim n →…

A:

Q: Find the limit of sin2x/sin3x as x approaches to 0.   a. 1 b. 1/2 c. 0 d. 2/3

A: here we must know about  L'Hopital's rule to solve the given problem which is used whenever we…

Q: The approximation of 1 = f*cos(x³ - dx using composite Simpson's rule with n= 3 is:

A:

Q: Consider the function f(x) = cos x − 3x + 1. Since ƒ (0)ƒ () <0, f(x) has a root in [0]. If we use…

A:

Q: Consider the function f(x) = cos x − 3x + 1. Since ƒ (0)ƒ (=) < 0, ƒ (x) has a root in [o,]. If we…

A:

Q: Only piecewise continuous functionsf (x) on every finite interval (0, 0) such that 00 | S(x)| dxis…

A: According to the condition of convergence for Fourier integral,  Let f and f' be piecewise…

Q: Prove that for all x € [1, 00), – sinh (- |cosh'(x)|) = V -1. %3D Ouest!

A: Proving of function  sinh(cos-1(x))

Q: Expand the function f(x) = x, 0 < x < 2, in half range sine series. 4 O (A) (#) = - sin()- sin() -…

A:

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.