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Let f be a differentiable function on R. Assume f' is continuous and always positive. You...
3. (25 pts) Suppose f(x) is twice continuously differentiable for all r, and f"(x) > 0...
3. (25 pts) Suppose f(x) is twice continuously differentiable for all r, and f"(x) > 0 for all , and f(x) has a root at p satisfying f'(p) < 0. Let p, be Newton's method's sequence of approximations for initial guess po < p. Prove pi > po and pı < p Remember, Newton's method is Pn+1 = pn - f(pn)/f'(P/) and 1 f"(En P+1 P2 f(pP-p)2. between pn and p for some
3. (25 pts) Suppose f(x) is twice...
5. Let f : R -R be a differentiable function, and suppose that there is a constant A < 1 such tha...
5. Let f : R -R be a differentiable function, and suppose that there is a constant A < 1 such that If,(t)| < A for all real t. Let xo E R, and define a sequence fan] by 2Znt31(za),n=0,1,2 Prove that the sequence {xn) is convergent, and that its limit is the unique fixed point of f.
5. Let f : R -R be a differentiable function, and suppose that there is a constant A
Can you help with this? Thank you always. Suppose that the function f : R-+ R is continuous at the point xo and that f(...
Can you help with this? Thank you always.
Suppose that the function f : R-+ R is continuous at the point xo and that f(xo) > 0. Prove that there is an interval 1 (x,-1/n, xo + 1 /n), where n is a natural number, such that f (x) >0 for all x in I. (Hint: Argue by contradiction.)
Suppose that the function f : R-+ R is continuous at the point xo and that f(xo) > 0. Prove that...
4. (a) Assume a function h is differentiable at some point to. Is it true that...
4. (a) Assume a function h is differentiable at some point to. Is it true that h is continuous on some open-neighbourhood of xo? Provide either a proof or a counterexample. (b) Let f be twice differentiable on R and assume that f" is continuous. Show that for all x ER S(x) = S(0) + s°C)x + [ (x - 1))"(dt. (C) Deduce that for any twice continuously differentiable function f on R and any positive x > 0, x...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint:...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem)
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
Let f and g be differentiable on R such that f(1) = g(1), and f'(x) <...
Let f and g be differentiable on R such that f(1) = g(1), and f'(x) < '() for all r ER. Prove that f(x) = g(2) for 3 >1.
Q3 (Prove that P∞ k=1 1/kr < ∞ if r > 1) . Let f : (0,∞) → R be a twice differentiable function with f ''(x) ≥ 0 for all x ∈ (0,∞). (a) Show that f '(k) ≤ f(k + 1) − f(k) ≤ f '(...
Q3 (Prove that P∞ k=1 1/kr < ∞ if r > 1) . Let
f : (0,∞) → R be a twice differentiable function with f ''(x) ≥ 0
for all x ∈ (0,∞).
(a) Show that f '(k) ≤ f(k + 1) − f(k) ≤ f '(k + 1) for all k ∈
N.
(b) Use (a), show that Xn−1 k=1 f '(k) ≤ f(n) − f(1) ≤ Xn k=2 f
'(k).
(c) Let r > 1. By finding...
Exercise 5.3.4: Let f: [a,b] → R be a continuous function. Let ce [a,b] be arbitrary....
Exercise 5.3.4: Let f: [a,b] → R be a continuous function. Let ce [a,b] be arbitrary. Define po the Prove that F is differentiable and that F'(x) = f(x) for all x € [a,b]. series on the
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some z...
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists an M R such that f(x) < f(xM) for al E R.
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists...
- Let V be the vector space of continuous functions defined f : [0,1] → R...
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
3. (25 pts) Suppose f(x) is twice continuously differentiable for all r, and f"(x) > 0 for all , and f(x) has a root at p satisfying f'(p) < 0. Let p, be Newton's method's sequence of approximations for initial guess po < p. Prove pi > po and pı < p Remember, Newton's method is Pn+1 = pn - f(pn)/f'(P/) and 1 f"(En P+1 P2 f(pP-p)2. between pn and p for some
3. (25 pts) Suppose f(x) is twice...
5. Let f : R -R be a differentiable function, and suppose that there is a constant A < 1 such that If,(t)| < A for all real t. Let xo E R, and define a sequence fan] by 2Znt31(za),n=0,1,2 Prove that the sequence {xn) is convergent, and that its limit is the unique fixed point of f.
5. Let f : R -R be a differentiable function, and suppose that there is a constant A
Can you help with this? Thank you always.
Suppose that the function f : R-+ R is continuous at the point xo and that f(xo) > 0. Prove that there is an interval 1 (x,-1/n, xo + 1 /n), where n is a natural number, such that f (x) >0 for all x in I. (Hint: Argue by contradiction.)
Suppose that the function f : R-+ R is continuous at the point xo and that f(xo) > 0. Prove that...
4. (a) Assume a function h is differentiable at some point to. Is it true that h is continuous on some open-neighbourhood of xo? Provide either a proof or a counterexample. (b) Let f be twice differentiable on R and assume that f" is continuous. Show that for all x ER S(x) = S(0) + s°C)x + [ (x - 1))"(dt. (C) Deduce that for any twice continuously differentiable function f on R and any positive x > 0, x...
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) = L. Prove that f is differentiable at 1-0 with f,(0) = L. (Hint: Use the definition of derivative and then use mean value theorem)
2. Let f: R R be a continuous function. Suppose that f is differentiable on R\{0} and that there exists an L e R such that lim,of,(z) =...
Let f and g be differentiable on R such that f(1) = g(1), and f'(x) < '() for all r ER. Prove that f(x) = g(2) for 3 >1.
Q3 (Prove that P∞ k=1 1/kr < ∞ if r > 1) . Let
f : (0,∞) → R be a twice differentiable function with f ''(x) ≥ 0
for all x ∈ (0,∞).
(a) Show that f '(k) ≤ f(k + 1) − f(k) ≤ f '(k + 1) for all k ∈
N.
(b) Use (a), show that Xn−1 k=1 f '(k) ≤ f(n) − f(1) ≤ Xn k=2 f
'(k).
(c) Let r > 1. By finding...
Exercise 5.3.4: Let f: [a,b] → R be a continuous function. Let ce [a,b] be arbitrary. Define po the Prove that F is differentiable and that F'(x) = f(x) for all x € [a,b]. series on the
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists an M R such that f(x) < f(xM) for al E R.
Let f be a real-valued continuous function on R with f (-o0 0. Prove that if f(xo) > 0 for some zo R, then f has the maximum on R, that is, there exists...
- Let V be the vector space of continuous functions defined f : [0,1] → R and a : [0, 1] →R a positive continuous function. Let < f, g >a= Soa(x)f(x)g(x)dx. a) Prove that <, >a defines an inner product in V. b) For f,gE V let < f,g >= So f(x)g(x)dx. Prove that {xn} is a Cauchy sequence in the metric defined by <, >a if and only if it a Cauchy sequence in the metric defined by...
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