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6. (a) If f(x a) for -al R, show that i-0 fO(a) fori 0,1,2, (where f0 (a) f(a), and fori 1 f(a) d...
b) i. Using e-8 definition show that f is continuous at (0,0), where f(x,y) = {aš...
b) i. Using e-8 definition show that f is continuous at (0,0), where f(x,y) = {aš sin () + yś sin () if xy + 0 242ADES if xy = 0 ii. Prove that every linear transformation T:R" - R" is continuous on R". iii. Let f:R" → R and a ER" Define Dis (a), the i-th partial derivative of f at a, 1 sisn. Determine whether the partial derivatives of f exist at (0,0) for the following function. In...
Question l: Consider the function f(x) = sin(parcsinx),-1 < x < 1 and p E R (a) Calculate f(0) in terms of p. Simplify your answer completely fX) sin(p arcsinx) f(o) P The function fand it...
Question l: Consider the function f(x) = sin(parcsinx),-1 < x < 1 and p E R (a) Calculate f(0) in terms of p. Simplify your answer completely fX) sin(p arcsinx) f(o) P The function fand its derivatives satisfy the equation where f(x) denotes the rth derivative of f(x) and f (b) Show thatf0(n2p2)f(m)(o) (x) is f(x). (nt2) (nti) (I-x) (nt 2 e 0 (c) For p E R-仕1, ±3), find the MacLaurin Series for f(x), up to and including the...
2 er Let I be an interval of R, and define the function f :I→ R by f(x) 1 +e2z or every z EZ. (a) Find the largest...
2 er Let I be an interval of R, and define the function f :I→ R by f(x) 1 +e2z or every z EZ. (a) Find the largest interval T where f is strictly increasing. (b) For this interval Z, determine the range f(T) (c) Let T- f(I). Show that the function f : I -» T is injective and surjective. (d) Determine the inverse function f-i : T → 1. (e) Verify that (fo f-1)()-y for every y E...
A tinctlon of series y I Taylor The 6. Taylor's Remainder Theorem. fn)(0) where fw) is the n-th d...
a tinctlon of series y I Taylor The 6. Taylor's Remainder Theorem. fn)(0) where fw) is the n-th derivative of f, and the remainder term Ry is given by NN+1 for some point c between 0 and z. (Note. You do not need to prove Taylor's Remainder Theorem.) Problems (a) (5%) write this series for the function ez for a general N (b) (10%) Apply Taylor's Remainder Theorem to show that the Taylor series of function f = ez converges...
3. Let f: RP-R (a) If f(x)-Ax + b, x E R A є Mq.p and b є R9, show that f is p. where differentia...
3. Let f: RP-R (a) If f(x)-Ax + b, x E R A є Mq.p and b є R9, show that f is p. where differentiable everywhere and calculate its total derivative (b) If f is differentiable everywhere and Df (x)A, for some A E Mp and all q.p x E Rp, show that there exists b E R, such that f(x) = Ax + b for all x E Rp
3. Let f: RP-R (a) If f(x)-Ax + b,...
1) Let f:R-->R be defined by f(x) = |x+2|. Prove or Disprove: f is differentiable at...
1) Let f:R-->R be defined by f(x) = |x+2|. Prove or Disprove: f is differentiable at -2 f is differentiable at 1 2) Prove the product rule. Hint: Use f(x)g(x)− f(c)g(c) = f(x)g(x)−g(c))+f(x)− f(c))g(c). 3) Prove the quotient rule. Hint: You can do this directly, but it may be easier to find the derivative of 1/x and then use the chain rule and the product rule. 4) For n∈Z, prove that xn is differentiable and find the derivative, unless, of course, n...
A random variable X has a distribution with probability function f(x) = K(nx)2x for x = 0,1,2,...,n where n is a positive integer.
A random variable X has a distribution with probability function f(x) = K(nx)2x for x = 0,1,2,...,n where n is a positive integer. a. Find the constant k. b. Find the expected value M(S) = E(esX) as a function of the real numbers s. Compare the values of the derivative of this function M'(0) at 0 and the expected value of a random variable having the probability function above. c. What distribution has probability function f(x)? Let X1, X2 be independent random variables both...
I. Let f : R → R be a continuous function. Show that ER sup is a Fo set
I. Let f : R → R be a continuous function. Show that ER sup is a Fo set
I. Let f : R → R be a continuous function. Show that ER sup is a Fo set
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to f...
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to find .后f(x)dr Problem 2. Consider the function -2, zEQ 2, O f(r) = Show that f is not Riemann integrable on 0,1 but s Reemann integrable on this interval. Problem 3. (a) Let f be a real-valued function on a, b] such thatf()0 for all c, where c E [a, b Prove that f is Riemann integrable on a, b...
(i) Find a non-zero polynomial in Z3 x| which induces a zero function on Z3. f(x), g(x) R have degree n and let co, c1,...
(i) Find a non-zero polynomial in Z3 x| which induces a zero function on Z3. f(x), g(x) R have degree n and let co, c1,... , cn be distinct elements in R. Furthermore, let (ii) Let f(c)g(c) for all i = 0,1,2,...n. g(x) Prove that f(x - where r, s E Z, 8 ± 0 and gcd(r, s) =1. Prove that if x is a root of (iii) Let f(x) . an^" E Z[x], then s divides an. aoa1
(i)...
b) i. Using e-8 definition show that f is continuous at (0,0), where f(x,y) = {aš sin () + yś sin () if xy + 0 242ADES if xy = 0 ii. Prove that every linear transformation T:R" - R" is continuous on R". iii. Let f:R" → R and a ER" Define Dis (a), the i-th partial derivative of f at a, 1 sisn. Determine whether the partial derivatives of f exist at (0,0) for the following function. In...
Question l: Consider the function f(x) = sin(parcsinx),-1 < x < 1 and p E R (a) Calculate f(0) in terms of p. Simplify your answer completely fX) sin(p arcsinx) f(o) P The function fand its derivatives satisfy the equation where f(x) denotes the rth derivative of f(x) and f (b) Show thatf0(n2p2)f(m)(o) (x) is f(x). (nt2) (nti) (I-x) (nt 2 e 0 (c) For p E R-仕1, ±3), find the MacLaurin Series for f(x), up to and including the...
2 er Let I be an interval of R, and define the function f :I→ R by f(x) 1 +e2z or every z EZ. (a) Find the largest interval T where f is strictly increasing. (b) For this interval Z, determine the range f(T) (c) Let T- f(I). Show that the function f : I -» T is injective and surjective. (d) Determine the inverse function f-i : T → 1. (e) Verify that (fo f-1)()-y for every y E...
a tinctlon of series y I Taylor The 6. Taylor's Remainder Theorem. fn)(0) where fw) is the n-th derivative of f, and the remainder term Ry is given by NN+1 for some point c between 0 and z. (Note. You do not need to prove Taylor's Remainder Theorem.) Problems (a) (5%) write this series for the function ez for a general N (b) (10%) Apply Taylor's Remainder Theorem to show that the Taylor series of function f = ez converges...
3. Let f: RP-R (a) If f(x)-Ax + b, x E R A є Mq.p and b є R9, show that f is p. where differentiable everywhere and calculate its total derivative (b) If f is differentiable everywhere and Df (x)A, for some A E Mp and all q.p x E Rp, show that there exists b E R, such that f(x) = Ax + b for all x E Rp
3. Let f: RP-R (a) If f(x)-Ax + b,...
I. Let f : R → R be a continuous function. Show that ER sup is a Fo set
I. Let f : R → R be a continuous function. Show that ER sup is a Fo set
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to find .后f(x)dr Problem 2. Consider the function -2, zEQ 2, O f(r) = Show that f is not Riemann integrable on 0,1 but s Reemann integrable on this interval. Problem 3. (a) Let f be a real-valued function on a, b] such thatf()0 for all c, where c E [a, b Prove that f is Riemann integrable on a, b...
(i) Find a non-zero polynomial in Z3 x| which induces a zero function on Z3. f(x), g(x) R have degree n and let co, c1,... , cn be distinct elements in R. Furthermore, let (ii) Let f(c)g(c) for all i = 0,1,2,...n. g(x) Prove that f(x - where r, s E Z, 8 ± 0 and gcd(r, s) =1. Prove that if x is a root of (iii) Let f(x) . an^" E Z[x], then s divides an. aoa1
(i)...
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