(1) Let f : [0, 1] → R be defined by f(x) = Compute U(Pn, f) and L(Pn, f). x-1 x Qn [0, 1] - X 2 x & Qn [0,1]] n and Pn = }· {0, 1/1, 2/1. n' n' " n

Q: Let r(x) = f(g(x)) and s(x) = g(f(x)), where f and g are shown in the figure. Find r'(1) and s'(4).…

A: our objective is to find the required derivative

Q: Compute r''(t) and r'''(t) for the following function. r(t) = (9t2 + 3,t+ 1,1) = (9r?. Find r'(t).…

A: Find the first, second, and third derivative of the following vector-valued function. r(t) = <9t2…

Q: Let F(x,y,z) = (x)i + (y)j + (z2)k. Compute the defined operations and if it is not defined, write…

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Q: Let X ~ N (0, 1). Demonstrate that E (X)=0 and V (X)=1.

A: To demonstrate that E(X)=0 and V(X)=1 for a standard normal distribution X∼N(0,1), we need to…

Q: 1) Let f(z,y)- . Compute DS(z,y) where u- (4-9)

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Q: Let f(=) = In[=*(z +8)*(z² + 8)°] f'(z) %3D

A: Given that fx=lnx3x+85x2+82 To find f'x

Q: Let z = e**y* , x = tan¬²(st5) and y = əz Evaluate as (2s+t)²* Select one: O a. 2xy³e{x³y?} t5 t –…

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Q: Let T: R³ R² be defined by T (C) X2 - [3²1-22] = -3x3 -3 7 -- {~ - [ ] - - [E] - - [ ]} -- ² - {~ -…

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Q: Theorem: The Chain Rule provided If m(x) = EI(x) is a composite function, then that E'[I(x)) and…

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Q: 2) Let f be a (1:1) Pinction such that differenhable f(3) = 7 P(7) = 8, f'(3)=2, P(1) -3 If glad = f…

A: topic - derivative of inverse function if f(a)=bg(x)=f-1(x)then g'(b)=1f'(a)

Q: 1) x(t) = 1(t- 47T)

A: Solution 1: Given function is x(t)=1(t-4T)           (1) Let the z-transform of x(t) is X(z)…

Q: 1 1 T+ 2 1 B T+ - T(2)"-1 3(T, T)

A: To prove  βT+12,T+12=πT24T-11βT,T It is known that  βm,n=ΓmΓnΓm+n Now, for m=n=T+12,…

Q: A. T(f(t)) = |B. T(ƒ(t)) = f(−t) from Pg to Pg = tª ƒ' (t) from P₂ to P5 = f(t)ƒ' (t) from P7 to P13…

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Q: 1 1 LGt-5- At²est + Ceisinh Dt = [(s – 1)3 ' s² + 2s – 5]

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Q: 15) Suppose that L {f(t)} = F(s) Then L{-te* f(t)} Corresponds to: a) F'(s – 5) b) –F'(s – 5) c)…

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Q: Solve the given initial value problem using the method of Laplace transforms. 5, Ost≤7, y" + 2y' +…

A: One has to solve using Laplace Formula:

Q: Differentiate the function L(k) = cosh¯' (/T+ k² ). O (k² + 1)-1/2 O 1+ k2 1+k

A: ddkcosh-11+k2

Q: f (x, y) ~ bo + b1,0(x − 1)+b0,1(y−2)+ b2,0 (x - 1)² + b₁,1(x − 1)(y-2) + bo,2(y − 2)² +…...

A: We find the Taylor series of the given function

Q: Let f(x) = 1+ VI-1, then f 1'(x) = %3D Select one: O a. (r +1)2 +1 of O b. (z - 1) -1 O c (x – 1)² +…

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Q: K=Zp(t) ,then f(x)=x^p-t is irreducible over K[x]

A: GIVEN: K = Zp(t)

Q: (x + 3)³ 12 16 11 f(x) Consider the function 4(х — 1)2 4 1 (x – 1)2 4. Note that - (x – 9)(x + 3)²…

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Q: Let f(æ) = In(z') f'(2) =

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Q: Suppose that f: [1,9] →→ R² such that f(1) =(2,-2) and f' (t) Find f(9) (v₁, 71) √t, √t

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Q: Oind Let g(2) = u(2)+iv(2) be ol complete function hiC →C, h(2) = e¯( v(a)+iu(2). Show that if h is…

A: gz=uz+ivz is a complete function and h:ℂ→ℂ by hz=e-ziuz+vz is a entire functionTo prove: g is a…

Q: 2(x-1) If y = Vx² – 2x then y' 3x2-2x

A: For the first part, we differentiate the given function y with respect to 'x' and use change rule of…

Q: The linear functional F: R^2[x] -> R is defined by the formula F: p -> ∫(from 1 to 0) p(t)dt.…

A: Step 1: Step 2: Step 3: Step 4:

Q: (1,1) S[3x*+5y+i(x°-y°)] dz along y=x Evaluate (0,0)

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Q: Show that the function f(x) = x’ + 1 is invertible from R to R.

A: First prove that the  function is one -one  let f(x)=fyx7+1=y7+1x7=y7x=y therefore, the function is…

Q: Consider the function F whose criterion is = / dt. 1+ t2 F(x) sena When determining dF/dx the result…

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Q: F(x+y=z₁ x² + y²) where 2= z(x,y) Prove that x2² - y2/²/² = x-y dy if it was

A: We have to show that x∂z∂y-y∂z∂x=x-y for the given integral surface of the partial differential…

Q: (c) Evaluate the Riemann integral tion Pn ={1," i=1 n+1 n+2 n' n 1 dx by using definition with the…

A: Hint: Use the formula ∫abf(x)dx=limn→∞b-an∑i=nfa,ib-an to obtain the required solution.