Q4(a)- If fE R(a)on [a, b, a < x < b and F(x) = S* f(t) dt] then F(x)is continuous on [a, b].

Q: Suppose f and g are continuous functions on [a, b] and that g(x) > 0 for all x E [a, b]. Prove that…

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Q: Let f be a function such that f' is continuous on [a, b] Show that a

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Q: Define f : R2 → R by letting (y² – x)² (y4 + x²) f(x, y) = if (x, y) # (0, 0), and f(0,0) = 1. ||…

A: The given function is f:ℝ2→ℝ defined as: fx,y=y2-x2y4+x2if x,y≠0,01if x,y=0,0 Claim: The given…

Q: Let f:A-B and X,Y be subsets of B. Then f-¹(XnY) = f'(X) nf-¹(Y). True False

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Q: f:R-R is differentiable on R. There exist monotonic increasing functions g(x) and h(x) such that…

A: Definition continuity of a function f(x). A function f is continuous at x=a when: (i)  f(a) is…

Q: Let ƒ : [a; b] → R be a continuous function and let F(x) = f(t)dt where x > a. (a) Show that if f is…

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Q: Show that the function f defined by (x, y) = (1, –1) f(x, y) = r? + y (x, y) # (1, –1) x + y is not…

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Q: Show that A Function with a Single Point of Discontinuity 2xy x² +1,²² 0, is continuous at every…

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Q: (6) Are all continuous functions on ID differentiable on D?

A: we have to identify the given statement is true or not

Q: sume that f(t) is continuous on [a, b| and that u = g(t) is cont ferentiable on [a, b], then f(u)…

A: Note- Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: Is there a continuous function such that - f: [1,2] U [3,4] → R - f(1) * f(4) < 0 - f(x) not egual 0…

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Q: Exercise 4: Suppose f is differentiable on (2,6) and continuous on [2,6. Suppose further that 1 < f'…

A: Given:f is differentiable on (2, 6) and continuous on [2, 6].

Q: { 2xy? 9x2 + y4 (x, y) # (0,0) (x, y) = (0,0) f (x, y) ·

A: Topic = Limits and  Continuity

Q: Prove that if f is continuous, then f(t)(x-t) dt = f(u) du ) dt

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Q: Suppose that f is a continuous function satisfying stæ) = a f(t)dt + * So f(t)dt + a %3D for all a,…

A: Solve the function

Q: What value of c makes the following function continuous at (0,0)? f(z, y) = C, x+ y³ +2, (x,y) +…

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Q: b) If the function f(x), g(x) and h(x) are continuous on [a, b] and derivable on (a, b), show that…

A: The given functions f,g and h are continous on [a,b] and differntiable on (a,b). So intermediate…

Q: Let f(x) — sin(nx) n³ n=1 Show that f is differentiable on R and that f' is continuous on R.

A: Within its interval of convergence, the derivative of a power series is the sum of derivatives of…

Q: (3) Determine the values of a for which x4 + y* , if(x, y) # (0, 0), (x² + y?)ª f(x, y) = ,if(x, y)…

A: solution.

Q: The linear functional F: R^2[x] -&gt; R is defined by the formula F: p -&gt; ∫(from 1 to 0) p(t)dt.…

A: Step 1: Step 2: Step 3: Step 4:

Q: 10 If f is a continuous function and F(x)=f(x) for all x € R, then x)dx=? A © FS) - F)] B ©2[F(5) –…

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Q: If f(x) is continuous on a, b] thenF(x) = So f(t)dt is continuous. OTrue False

A: We need to check whether the given statement is true or false

Q: Consider the function F whose criterion is = / dt. 1+ t2 F(x) sena When determining dF/dx the result…

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Q: Find all continuous functions f(x) defined on x > 0 with f(x) = 0 such that f(x)³ = f(t) dt a

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Q: 10 1 If f is a continuous function and F'(x)=f(x) for all x € R, then fGx)dx=? LF(10) – F(2)] [FIS)-…

A: We have to find the integral

Q: { 2xy? 9x2 + y4 (x, y) # (0,0) f (x, y) = (x, y) = (0,0)

A: We will discuss limit by using different paths.

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.