(a) Let f: [a, b] - Show that fE R([a, b)). R be abounded monotonic deczeasing function.

Q: There is no function f(x) such that f(x) = f(x) + C. %3D

A: (1) We have the given statement as If fx and gx is continuous function then, ∫fxdx=fx+C. The above…

Q: Consider the function f(x) = xn , where n is odd. Does f −1 exist? Explain.

A:

Q: Let f(x) = V6 – x and g(x) = a? – æ. Then the domain of f og is equal to

A:

Q: Let f be a continuous function such that f(x) dx: = -1 and f(x) dx = = 9. %3D 2 2 | f(x) dæ? What is…

A:

Q: For what values of a is Jaʼx – 2a, x2 2 f(x) ( 12, x< 2 continuous at every x?

A: Given : The function is f(x) =a2x - 2a        , x ≥212,                     x&lt;2 .

Q: 1. Why is f not a function from R to R if a) f(x) = 1/r? b) f(x) = x? %3D

A:

Q: 3. Let f(x) = x, given the interval [0,1]

A: fx=x, x∈0,1 f'x=1&gt;0 for all x∈0,1 So, fx is strictly increasing on the interval 0,1.

Q: Let f(x) = +.Compute the difference quotient for f(x) at x = 4 with h = 0.3 %3D difference quotient…

A: Given,

Q: (b) Consider the functions g : R → R defined by g(x) = r² + 4, x € R and h : R → R defined by h(x) =…

A:

Q: Where is f(x) = x-5 x² –6x+5 discontinuous? f (x)is continuous everywhere o 5 o 1 о 5, 1

A:

Q: Let f(z) = 3 Find (f-)'(x) using the formula %3D I+9 1 ((e) = and check your answer by…

A: I am going to solve the given problem by using some simple calculus to get the required result.

Q: Give an example where | f(x)dg(x) exist even through f is unbounded on [a, b].

A:

Q: Let f(x) sin x g(x) = x2 + 3 %3D %3D Evaluate the composite function (f og)(x)

A: Given:       To determine: (fog)(x)

Q: 18. Suppose that f is a differentiable function such that f(g(x)) = x, and f'(x) = 1 1+(f(x))². Show…

A:

Q: Suppose f:R-IR is a diffeetiable function which satısfies flo) = 0 We wart to show f': IR → IR is a…

A: Given that f:ℝ→ℝ is a differentiable function which satisfies f0=0. It is required to show that f' :…

Q: 3.Show, by means of the example f(t) = sin t, that f ∗ f is not necessarily nonnegative.

A:

Q: What does it mean for f to be differentiable at a ?

A: we have to define that what we mean by the f is differentiable at a.

Q: 6. Use an appropriate substitution to prove f(x)dr = | f(a+b- x)dx.

A: To use an appropriate substitute to prove  ∫abfxdx=∫abfa+b-xdx

Q: Consider the function g :ℕ→ℝ  where  g(n) = 1 if n is odd and g(n) = 2 if n is even. Is g continuous…

A: Given function is g :ℕ→ℝ  where  g(n) = 1 if n is odd and g(n) = 2 if n is even.

Q: Let f(æ) = – 3x* In æ %3D f'(x) = f'(e²) =

A:

Q: Suppose that f is a function such that f' (c) = 1 for some number c in the interval [a, b]. Is it…

A:

Q: Let f(z) be the principal branch of (z)^⅓, find f(-i)

A: What is Principal Branch: A function that chooses one branch of a multi-valued function is known as…

Q: . Suppose that f(r) is a function given by for 0<r<7, f(z) = for -n <I<0.

A:

Q: Let f(x) = Vx. Compute the difference quotient for f at a = 7 with h = 4. %3D %3D

A:

Q: (a) Let x2 if x is rational f (x) = 0 otherwise. Prove that f is differentiable at 0.

A:

Q: Let f (z) =- (!- 2) (I+2) (!E – 2) The function f (z) is not analytic at z=

A: A function is analytic at a point if it is differentiable and all the partial derivatives are…

Q: A function f: Z x Z is defined as f(m, n) = 3n - 4m.  Verify whether this function is injective and…

A: f: Z x Z is defined as f(m, n) = 3n - 4m So, f(m, n) = f(a, b) implies, 3n - 4m = 3b - 4a i.e. 3(n…

Q: If f(3) = 15, f' is continuous, and I f'(t) dt = 24, what is the value of f(4)? %3D

A: In this question, f is a primitive function and we have to find f(4). So I am providing the…

Q: 1. Consider the function f : Z × Z → Z × Z defined by f(a, b) = (a+ b, a – b). Is f injective? Is f…

A: Ans= f is surjective but not injective.............I have explained all the reasons with…

Q: ind the values of c that make f continuous everywhere: { Sa² – c² if x 3

A:

Q: Use the Second Fundamental Theorem of Calculus to find F'(x). F(x) -TVea+3 dt F '(x) =

A:

Q: O No, because f is not continuous on the closed interval [a, b].

A: Given that: The function f ( x) = x2 - 81x - 4     ,  [ -9, 9  ]  .

Q: Let f be differentiable on an interval A. If f'(x) not equal to  0 on A,show that f is one-to-one on…

A: Given that, Let f be differentiable on an interval A. If f'(x) not equal to  0 on A. We have to show…

Q: Let f (x) et . Find a so that f'( a ) = 0. Hint: use the quotient Rule.

A:

Q: Find a function f: Q → Q that is both surjective and injective but not the identity function.

A:

Q: 1. Why is f not a function from R to R if a) f(x)= 1/x? b) f(x) = T? c) f(x) =±V(x² + 1)?

A: We'll answer the first question since the exact one was not specified. Please submit a new question…

Q: Where is f(a) not continuous?

A: Given:

Q: Suppose f(r) = (In z)e. What is f'(x)?

A: d/dx (f(x)g(x))= f(x)g'(x)+g(x)f'(x)

Q: Suppose f(x) = 4nx?e-Tx -e-n for all x € R. Show that f = -f. %3D

A:

Q: Consider the function y = 4ex + 7. What is the exact value of C in the interval [0, 5] such that…

A: We need to find value of C.

Q: If f is increasing on the interval [a, b), then f' does exist. Select one: a. True b. False

A: Let f is increasing on the interval [a, b]. Since, there is not given that the function is…

Q: Suppose f is a bounded real function on [a, b), and f' eR on [a, b). Does it follow that fe R? Does…

A:

Q: Suppose that f(x) is continuous and differentiable, f(2) = -2, f(6) = -6, and f(x)dx =-10. Compute ,…

A: Topic:- Functions

Q: Use this definition to find f'(a) at the given number a. f(x) = V6x + 7,

A:

Q: Consider the function f(x) = 12x' + 60x 100x + 6. f(z) has inflection points at (reading from left…

A: Here, f(x)=12x5+60x4-100x3+6

Q: + The f(x) = Undefined at x = -3 %3D TO F O

A: Given: f(x)= x-13-x A function is undefined at a point x=a, if the denominator of f(x) is 0 at x=a

Q: Give an example of a differentiable function g: R→ R such that f is strictly increasing on R but g'…

A:

Q: / Let f: R → R be a differentiable function with the property that f(-x) = f(x) for all x E R. Prove…

A: Let f:ℝ→ℝ be a differentiable function with the property that f-x=fx for all x∈ℝ.   To Prove:…

Q: for f (z) = z – Z where does f (z) exist?

A:

Q: 3. Let f(r) = 8 – 4x4/5. Carefully explain why f is continuous on [-1, 1), f(-1) = f(1), but there…

A:

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.