a) State the definition that a function f(x) is continuous at x = a. b) Let...

a) State the definition that a function f(x) is continuous at x = a. b) Let f(x) = ax^2 + b if 0 < x ≤ 2

18/x+1 if x > 2

If f(1) = 3, determine the values of a and b for which f(x) is continuous for all x > 0.

Expert Solution

milcah answered 2 years ago

Let f be a continuous function on [a, b] which is differentiable on (a,b). Then f...

Let f be a continuous function on [a, b] which is differentiable on (a,b). Then f is non-decreasing on [a,b] if and only if f′(x) ≥ 0 for all x ∈ (a,b), while if f is non-increasing on [a,b] if and only if f′(x) ≤ 0 for all x ∈ (a, b). can you please prove this theorem? thank you!

9. Let f be continuous on [a, b]. Prove that F(x) := sup f([x, b]) is...

9. Let f be continuous on [a, b]. Prove that F(x) := sup f([x, b]) is continuous on [a, b]

Rolle's Theorem, "Let f be a continuous function on [a,b] that is differentiable on (a,b) and...

Rolle's Theorem, "Let f be a continuous function on [a,b] that is differentiable on (a,b) and such that f(a)=f(b). Then there exists at least one point c on (a,b) such that f'(c)=0." Rolle's Theorem requires three conditions be satisified. (a) What are these three conditions? (b) Find three functions that satisfy exactly two of these three conditions, but for which the conclusion of Rolle's theorem does not follow, i.e., there is no point c in (a,b) such that f'(c)=0. Each...

Let the continuous random variable X have probability density function f(x) and cumulative distribution function F(x)....

Let the continuous random variable X have probability density function f(x) and cumulative distribution function F(x). Explain the following issues using diagram (Graphs) a) Relationship between f(x) and F(x) for a continuous variable, b) explaining how a uniform random variable can be used to simulate X via the cumulative distribution function of X, or c) explaining the effect of transformation on a discrete and/or continuous random variable

1) Let f(x) be a continuous, everywhere differentiable function and g(x) be its derivative. If f(c)...

1) Let f(x) be a continuous, everywhere differentiable function and g(x) be its derivative. If f(c) = n and g(c) = d, write the equation of the tangent line at x = c using only the variables y, x, c, n, and d. You may use point-slope or slope-intercept but do not introduce more variables. 2) Let f(x) be a continuous, everywhere differentiable function. What kind information does f'(x) provide regarding f(x)? 3) Let f(x) be a continuous, everywhere differentiable...

a) use the sequential definition of continuity to prove that f(x)=|x| is continuous. b) theorem 17.3...

a) use the sequential definition of continuity to prove that f(x)=|x| is continuous. b) theorem 17.3 states that if f is continuous at x0, then |f| is continuous at x0. is the converse true? if so, prove it. if not find a counterexample. hint: use counterexample

Let f : R → R be a function. (a) Prove that f is continuous on...

Let f : R → R be a function. (a) Prove that f is continuous on R if and only if, for every open set U ⊆ R, the preimage f −1 (U) = {x ∈ R : f(x) ∈ U} is open. (b) Use part (a) to prove that if f is continuous on R, its zero set Z(f) = {x ∈ R : f(x) = 0} is closed.

(a) State the definition of the derivative of f. (b) Using (a), prove the following:d/dx(f(x) +g(x))...

(a) State the definition of the derivative of f. (b) Using (a), prove the following:d/dx(f(x) +g(x)) =d/dx(f(x)) +d/dx(g(x))

Part A. If a function f has a derivative at x not. then f is continuous...

Part A. If a function f has a derivative at x not. then f is continuous at x not. (How do you get the converse?) Part B. 1) There exist an arbitrary x for all y (x+y=0). Is false but why? 2) For all x there exists a unique y (y=x^2) Is true but why? 3) For all x there exist a unique y (y^2=x) Is true but why?

a) Let S ⊂ R, assuming that f : S → R is a continuous function,...

a) Let S ⊂ R, assuming that f : S → R is a continuous function, if the image set {f(x); x ∈ S} is unbounded prove that S is unbounded. b) Let f : [0, 100] → R be a continuous function such that f(0) = f(2), f(98) = f(100) and the function g(x) := f(x+ 1)−f(x) is equal to zero in at most two points of the interval [0, 100]. Prove that (f(50) − f(49))(f(25) − f(24)) >...

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