41 The derivative 43 Example 49 Define f R → R by f(x) = x2 sin(1/x) if x ̸= 0, 0 if x = 0 Then f is differentiable on R (See Figure 1) It follows from the product and chain rules proved below that f is differentiable at x ̸= 0 with derivative f′(x) = 2xsin 1 x −cos 1 x Moreover, f is differentiable at 0 with f′(0) = 0, since limClick here👆to get an answer to your question ️ If y = f (2x 1/x^2 1 ) and f^'(x) = sin x^2 , then dy/dx = (IIT JEE, 19)Essentially correct, but more complicated than required You make some confusion about k and j, though Suppose x and y are rational solutions of x^2y^2=3^k, with k an odd integer
If F X X X 1 1 Y Then The Value Of F Y Is
If y=f(x)=x+2/x-1 then x is equal to
If y=f(x)=x+2/x-1 then x is equal to-I=1 A i for some integer n If n = 2, then X = A 1 ∪A 2, A 1 and A 2 are closed in X, fA 1 A 1 → Y and fA 2 A 2 → Y are continuous and fA 1(x) = fA 2(x) for every x ∈ A 1∩A 2 Hence, by the pasting lemma, we can construct continuous f0 X → Y such that f0(x) = fA 1(x) if x ∈ A 1 and f0(x) = fA 2(x) if x ∈ A 2 It is clearIf x = 0, y =0 Stepbystep explanation Hi, to obtain the values of the variable y in each case, we have to replace the variable x
9 If Y F X Frac X 2 X 1 Then F Y Is Equal To A X B Frac Y 2 Y 1 C Frac X 1 X 2 Quad Left D Frac Y 1 Y 2 Right
If f0 is a homeomorphism of X with Z, then f X → Y is a topological imbedding (or simply imbedding) of X in Y Example 5 Consider F (−1,1) → R defined by F(x) = x/(1 − x2) Then F is continuous and one to one (since F0(x) = (1x2)/(1−x2)2 ≥ 0) and is continuous on R So F is a homeomorphism (where R has the standard topologyClick here👆to get an answer to your question ️ Let f N→ X f(x) = 4x^2 12x 15 Then, f^1(y) = ?(b) f(x,y) = 2x2 xy2 −2, D= {(x,y) ∈ R2;x2 y2 64} Solution (a) Calculating the partial derivatives gives f x = y 2 √ x − 1, f y = √ x−2y 6 Finding critical points f x = 0, f y = 0 =⇒ (x,y) = (4,4) Note that (4,4) ∈ Dand that f(4,4) = 12 The boundary of Dconsists of
If xy=2 and x−y=1, then what is xy?Solutionput (xy)=v then differentiate both sides with respect to 'x' we get 1dy/dx=dv/dx or, dy/dx=dv/dx—1 this value put in to equation (I), first arranging equation (I) dy/dx= (xy1) (xy—2)/ (xy2) (xy—1) or, dv/dx—1= (v1) (v—2)/ (v2) (v—1) or,dv/dx= (v^2—2vv—2)/ (v^22v—v—2) 1X 1 6=x 2 Then f(x;y) = x1 2Fhas di erent values on P 1 and P 2, so Fseparates points Since Ris compact, the StoneWeierstrass theorem implies that Fis dense in C(R) Alternatively, note that Fincludes all polynomials in (x;y), and the polynomials are dense in C(R), so Fis dense in C(R) 7
ProofLet fK g 2A be a family of convex sets, and let K = \ 2AK Then, for any x;y2 K by de nition of the intersection of a family of sets, x;y2 K for all 2 nd each of these sets is convex Hence for any 2 A;and 2 0;1;(1 )x y2 KDivide \frac{f1}{f}, the coefficient of the x term, by 2 to get \frac{1}{2}\frac{1}{2f} Then add the square of \frac{1}{2}\frac{1}{2f} to both sides of the equationThis is the Solution of Question From RD SHARMA book of CLASS 11 CHAPTER RELATIONS AND FUNCTIONS This Question is also available in R S AGGARWAL book of CLAS
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R X f X(x) dx= 1 Alternately, X may be described by its cumulative distribution function (CDF) The CDF of Xis the function F X(x) that gives, for any specified number x∈X, the probability that theIf x = −1, y =1 ;X y = 2 (equation 1) x y = 1 (equation 2) Add the equations for 2x = 3, meaning that x = 3/2 Using x = 3/2 in equation 1 gives y = 1/2 Using x = 3/2 in equation 2 also gives y = 1/2 (3/2)*(1/2) = 3/4 Answer xy = 3/4
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Given function f(x) =x/(x1) Then f(1/x) = (1/x)/(1/x) 1 f(1/x) = (1/x)/(1 x)/x f(1/x) = (x/x)/(1 x) f(1/x) = 1/(1 x) Hence, f(x) =x/(x1) then f(1/x) = 12 (a) Define uniform continuity on R for a function f R → R (b) Suppose that f,g R → R are uniformly continuous on R (i) Prove that f g is uniformly continuous on R (ii) Give an example to show that fg need not be uniformly continuous on R Solution • (a) A function f R → R is uniformly continuous if for every ϵ > 0 there exists δ > 0 such that f(x)−f(y) < ϵ for all xThe total area under f X(x) is 1;
If F X X X 1 1 Y Then The Value Of F Y Is
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94 7 Metric Spaces Then d is a metric on R Nearly all the concepts we discuss for metric spaces are natural generalizations of the corresponding concepts for R with this absolutevalue metric Example 74 Define d R2 ×R2 → R by d(x,y) = √ (x1 −y1)2 (x2 −y2)2 x = (x1,x2), y = (y1,y2)Then d is a metric on R2, called the Euclidean, or ℓ2, metricIt corresponds toBASIC STATISTICS 3 We can then write 9 as i=1 (xi − a)2 = i=1 (xi −x¯)2 i=1 (¯x− a)2 (11) Equation 11 is clearly minimized when a =¯xNow consider part b of theorem 1 Expand the second expression in part b and simplifyY1 n 1 f X(y 1 n) where, f X() is the pdf of X which is given Here are some more examples Example 1 Suppose Xfollows the exponential distribution with = 1 If Y = p X nd the pdf of Y Example 2 Let X ˘N(0;1) If Y = eX nd the pdf of Y Note Y it is said to have a lognormal distribution Example 3 Let Xbe a continuous random variable with
If F X 1 2x 1 Then Show That F F X 2 X 1 2x 3 2s 1 Provided 1 That 2 1 Y Maths Relations And Functions Meritnation Com
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The partition theorem says that if Bn is a partition of the sample space then EX = X n EXjBnP(Bn) Now suppose that X and Y are discrete RV's If y is in the range of Y then Y = y is a event with nonzero probability, so we can use it as the B in the aboveM 2, and M 3 be metric spaces Let gbe a uniformly continuous function from M 1 into M 2, and let fbe a uniformlyAnswer to If f(x)= g(h(x)), then f'(4) = 5 9 8 4 7 6 3 5 yvalues yvalues 4 h(x) 2 3 8(x) 2 1 1 1 4 5 1 4 5 2 3 xvalues
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