2. Let f(x) = (x + 1)( arctan(1 – 2x) + /. f(t) dt x2 Which of the following statements are correct? I. f'(0) + f'(1) = -3 – 7 T – 4f(x) II. lim — 4 — п sinh x III. (f'(x))² + f(x)f" (x)) dæ (9л + 20) 16 0, Së f(t)dt IV. lim x→1 1– 2ln x 2 (a) I, II (b) I, III (с) I, II, I (d) I, II, IV (e) II, III, IV

Q: 10. (a) State the Fundamental Theorem of Calculus. -x sin t %3D (b) Let J(x) = S*. Evaluate: (i)…

A: Fundamental theorem of calculus

Q: Verify the following properties for ƒ(x) = sin x and p3(x) = x - x3/6:                    ƒ(0) =…

A: We first find the derivatives

Q: 2. Find the derivative of the following functions and write the down the inner and outer functions…

A: Solution: 2.(a) fx=2x5-4ex

Q: TLP iadade mnts & smt Findf F'(t) =sec(t) (seclE)+ tan (ED), -st < f- FCt) = -<ts f (--7 2.

A: We simplify f'(t) by expanding  To find f(t) we integrate f'(t) C=integrating constant

Q: 13. Find ƒ. (1)ƒ'(x)=x² + 3√x (2) f(t)=2cost+sec²t,-/2 <t</2,ƒ(π/3)=4; (3)ƒ*(x) = 6x+sinx.

A:

Q: 3. If f(x) = sinx ex find f'(0).

A: fx=sinxex

Q: Let h(t) = tan(3t + 8). Then h'(5) is and h"(5) is

A: As we know that the chain rule: fgx'=f'gx·g'x The Power law: ddxxn=nxn-1   Given: ht=tan3t+8 Now…

Q: 1. Let f(x)=√x-2 / x-4 . Show that f(x) has a removablediscotuiti t x=4   2. Given f(x)= √(x+1).…

A: Here we have to find the derivative of a function at a point.

Q: Let h(x) = tan (2x + 6). Then h'(2) is and h'(2) is

A: topic- functions and derivatives

Q: true or false prove your answer If (an) is Cauchy and p is a number such that for infinitely many…

A: Given that              an is a Cauchy sequence and p is a number such that for infinitely many…

Q: Find f(a)  a=2

A: We know that the definition of derivative is: We write the given equation in this format and find…

Q: Let f(4) (t) denote the 4th derivative of f (t). Assume limt →00 f(3) (t) e-st = 0. Then L{f(4) (t)}…

A:

Q: Given f(x) = 3 x² – 2 vx , which of the following expressions will represent the first step when…

A: Given,          fx=3x2-2xWe know that,          The derivative of a function, fx using the…

Q: d Suppose that f(x) and g(x) are differentiable functions such that f(5) = 1, f'(5) = 7, g(5) = 2,…

A: The objective is to simplify the given differential equation.

Q: Let f(x) = x°e*. Find f '(x).

A: Product rule:d(uv)dx=udvdx+vdudx