Question

Let f : R → R be differentiable with derivative f'. Prove that f(x + h)...

Let f : R → R be differentiable with derivative f'. Prove that f(x + h) = f(x) + f'(x)h + o(h), as h → 0.

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Use the formal definition of derivative to prove f(x) = 10|x| is not differentiable at a=0.
Use the formal definition of derivative to prove f(x) = 10|x| is not differentiable at a=0.
Let f : R → R be a bounded differentiable function. Prove that for all ε...
Let f : R → R be a bounded differentiable function. Prove that for all ε > 0 there exists c ∈ R such that |f′(c)| < ε.
Prove or give a counter example: If f is continuous on R and differentiable on R...
Prove or give a counter example: If f is continuous on R and differentiable on R ∖ { 0 } with lim x → 0 f ′ ( x ) = L , then f is differentiable on R .
Prove or give a counterexample: If f is continuous on R and differentiable on R∖{0} with...
Prove or give a counterexample: If f is continuous on R and differentiable on R∖{0} with limx→0 f′(x) = L, then f is differentiable on R.
Let a < b, a, b, ∈ R, and let f : [a, b] → R...
Let a < b, a, b, ∈ R, and let f : [a, b] → R be continuous such that f is twice differentiable on (a, b), meaning f is differentiable on (a, b), and f' is also differentiable on (a, b). Suppose further that there exists c ∈ (a, b) such that f(a) > f(c) and f(c) < f(b). prove that there exists x ∈ (a, b) such that f'(x)=0. then prove there exists z ∈ (a, b) such...
Let f: R -> R and g: R -> R be differentiable, with g(x) ≠ 0...
Let f: R -> R and g: R -> R be differentiable, with g(x) ≠ 0 for all x. Assume that g(x) f'(x) = f(x) g'(x) for all x. Show that there is a real number c such that f(x) = cg(x) for all x. (Hint: Look at f/g.) Let g: [0, ∞) -> R, with g(x) = x2 for all x ≥ 0. Let L be the line tangent to the graph of g that passes through the point...
For each polynomial f(x) ∈ Z[x], let f ' (x) denote its derivative, which is also...
For each polynomial f(x) ∈ Z[x], let f ' (x) denote its derivative, which is also a polynomial in Z[x]. Let R be the following subset of Z[x]: R = {f(x) ∈ Z[x] | f ' (0) = 0}. (a) Prove that R is a subring of Z[x]. (b) Prove that R is not an ideal of Z[x].
a) Let f : [a, b] −→ R and g : [a, b] −→ R be...
a) Let f : [a, b] −→ R and g : [a, b] −→ R be differentiable. Then f and g differ by a constant if and only if f ' (x) = g ' (x) for all x ∈ [a, b]. b) For c > 0, prove that the following equation does not have two solutions. x3− 3x + c = 0, 0 < x < 1 c) Let f : [a, b] → R be a differentiable function...
Let f : R → R be defined by f(x) = x^3 + 3x, for all...
Let f : R → R be defined by f(x) = x^3 + 3x, for all x. (i) Prove that if y > 0, then there is a solution x to the equation f(x) = y, for some x > 0. Conclude that f(R) = R. (ii) Prove that the function f : R → R is strictly monotone. (iii) By (i)–(ii), denote the inverse function (f ^−1)' : R → R. Explain why the derivative of the inverse function,...
4a). Let g be continuous at x = 0. Show that f(x) = xg(x) is differentiable...
4a). Let g be continuous at x = 0. Show that f(x) = xg(x) is differentiable at x = 0 and f'(0) = g(0). 4b). Let f : (a,b) to R and p in (a,b). You may assume that f is differentiable on (a,b) and f ' is continuous at p. Show that f'(p) > 0 then there is delta > 0, such that f is strictly increasing on D(p,delta). Conclude that on D(p,delta) the function f has a differentiable...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT