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3. (18pt) Consider the following problems. (31) Suppose that f is a real-valued continuous function on...
I need to answer 1b 2.5. Let f be a real valued function continuous on a...
I need to answer 1b
2.5. Let f be a real valued function continuous on a closed, bounded Theorem set S. Then there exist x1,X2 S such that f(x1) S f(x) s f(x2) for all x e S. Proor. We recall that if T E' is bounded and closed, then y, - inf T and sup T are points of T (Example 4, Section 1.4). Let T- fIS. By Theorem 2.4, T is closed and bounded. Take x, such that...
3. Let the function f be a real valued bounded continuous function on R. Prove that...
3. Let the function f be a real valued bounded continuous function on R. Prove that there is a solution of the equation f(x) = x, xER. Now choose a number a with f(a) > a and define the sequence (an) recursively by defining al = a and a叶1 = f(an), where n E N. If f is strictly increasing on R, show that (an) converges to a solution of the equation (0.1). This method for approximating the solution is...
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...
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...
0, oo) which converges to a certain real Let f be a real-valued continuous function over o0, i.e., lim f(x) = A. Answer...
0, oo) which converges to a certain real Let f be a real-valued continuous function over o0, i.e., lim f(x) = A. Answer the following questions value A as Find the following limit lim aoo a2 f (x)dx
0, oo) which converges to a certain real Let f be a real-valued continuous function over o0, i.e., lim f(x) = A. Answer the following questions value A as Find the following limit lim aoo a2 f (x)dx
Number 6 please S. Let ) be a sequence of continuous real-valued functions that converges uniformly...
Number 6 please
S. Let ) be a sequence of continuous real-valued functions that converges uniformly to a function fon a set ECR. Prove that lim S.(z) =S(x) for every sequence (x.) C Esuch that ,E E 6. Let ECRand let D be a dense subset of E. If .) is a sequence of continuous real-valued functions on E. and if () converges unifomly on D. prove that (.) converges uniformly on E. (Recall that D is dense in E...
PROBLEM # 5 Let S CRd. A function f S is sometimes called a vector-valued function...
PROBLEM # 5 Let S CRd. A function f S is sometimes called a vector-valued function of k. In this case a) Show that a vector-valued function is f = (fi, ,:W : S → Rk continuous iff each component function note that one can write f(x) (fi()..,fk(z)) where each fi S Ris a real valued function. fi: S-R is continuous. b) P Sd C Rd+1-{x e Rd+1 rove that the uit sphere 1 111-1) is always a compact and...
Please answer c d e 3. This problem shows that the metric space of continuous real-valued...
Please answer c d e
3. This problem shows that the metric space of continuous real-valued functions C([0, 1]) on the interval [0, 1is complete. Recall that we use the sup metric on C([0,1), so that df, 9) = sup{f (2) - 9(2): € (0,1]} (a) Suppose that {n} is a Cauchy sequence in C([0,1]). Show that for each a in 0,1], {Sn(a)} is a Cauchy sequence of real numbers. (b) Show that the sequence {fn(a)} converges. We define f(a)...
Consider the real valued function of two variablesz-f(x,y), which satisfies Ar2 +4y2+ 224 (a) Des...
Consider the real valued function of two variablesz-f(x,y), which satisfies Ar2 +4y2+ 224 (a) Describe all points (ro. yo. f(ro. yo)), such that the tangent plane to f(x, y) at (o.yof(ro. yo)) the point (2,0,0). Include scription. (b) Describe all points (roo,f(o, yo)), such that the tangent plane to f(x, y) at (zo. yo. f(o. yo) passes through the point (2,0,27), where >0 is an arbitrary constant. With the aid of a computer or by hand, provide a sketch of...
3. Numeracy: In each case below, the function f is real-valued function with domain R. (a)...
3. Numeracy: In each case below, the function f is real-valued function with domain R. (a) Suppose you know that f(x + y) = f(x) + f(y) for all ry ER. What is (0)? (b) Suppose you know that f(x + y) = f(x)f(y) for all x, y E R. What is f(0) if f is not the zero function?
I need to answer 1b
2.5. Let f be a real valued function continuous on a closed, bounded Theorem set S. Then there exist x1,X2 S such that f(x1) S f(x) s f(x2) for all x e S. Proor. We recall that if T E' is bounded and closed, then y, - inf T and sup T are points of T (Example 4, Section 1.4). Let T- fIS. By Theorem 2.4, T is closed and bounded. Take x, such that...
3. Let the function f be a real valued bounded continuous function on R. Prove that there is a solution of the equation f(x) = x, xER. Now choose a number a with f(a) > a and define the sequence (an) recursively by defining al = a and a叶1 = f(an), where n E N. If f is strictly increasing on R, show that (an) converges to a solution of the equation (0.1). This method for approximating the solution is...
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...
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...
0, oo) which converges to a certain real Let f be a real-valued continuous function over o0, i.e., lim f(x) = A. Answer the following questions value A as Find the following limit lim aoo a2 f (x)dx
0, oo) which converges to a certain real Let f be a real-valued continuous function over o0, i.e., lim f(x) = A. Answer the following questions value A as Find the following limit lim aoo a2 f (x)dx
Number 6 please
S. Let ) be a sequence of continuous real-valued functions that converges uniformly to a function fon a set ECR. Prove that lim S.(z) =S(x) for every sequence (x.) C Esuch that ,E E 6. Let ECRand let D be a dense subset of E. If .) is a sequence of continuous real-valued functions on E. and if () converges unifomly on D. prove that (.) converges uniformly on E. (Recall that D is dense in E...
PROBLEM # 5 Let S CRd. A function f S is sometimes called a vector-valued function of k. In this case a) Show that a vector-valued function is f = (fi, ,:W : S → Rk continuous iff each component function note that one can write f(x) (fi()..,fk(z)) where each fi S Ris a real valued function. fi: S-R is continuous. b) P Sd C Rd+1-{x e Rd+1 rove that the uit sphere 1 111-1) is always a compact and...
Please answer c d e
3. This problem shows that the metric space of continuous real-valued functions C([0, 1]) on the interval [0, 1is complete. Recall that we use the sup metric on C([0,1), so that df, 9) = sup{f (2) - 9(2): € (0,1]} (a) Suppose that {n} is a Cauchy sequence in C([0,1]). Show that for each a in 0,1], {Sn(a)} is a Cauchy sequence of real numbers. (b) Show that the sequence {fn(a)} converges. We define f(a)...
Consider the real valued function of two variablesz-f(x,y), which satisfies Ar2 +4y2+ 224 (a) Describe all points (ro. yo. f(ro. yo)), such that the tangent plane to f(x, y) at (o.yof(ro. yo)) the point (2,0,0). Include scription. (b) Describe all points (roo,f(o, yo)), such that the tangent plane to f(x, y) at (zo. yo. f(o. yo) passes through the point (2,0,27), where >0 is an arbitrary constant. With the aid of a computer or by hand, provide a sketch of...
3. Numeracy: In each case below, the function f is real-valued function with domain R. (a) Suppose you know that f(x + y) = f(x) + f(y) for all ry ER. What is (0)? (b) Suppose you know that f(x + y) = f(x)f(y) for all x, y E R. What is f(0) if f is not the zero function?
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