Some hints: use the definition: f is a function iff a = b implies f(a) =...

Some hints: use the definition: f is a function iff a = b implies f(a) = f(b) and recall that in informal proofs we show an implication by assuming the if part of the implication, and then deducing the then part of the implication.

The base case will show that a = b implies f(a) = f(b) when f(x) = c₀ (a constant function). The inductive case will assume a = b implies f(a) = f(b) for degree k, and will deduce it is also true for degree k+1.

Show that every polynomial of degree n:
y = f(x) = c_nxⁿ + c_n-1x^n-1 + . . . + c₂x² + c₁x + c₀
is a function by mathematical induction on degree n.
Assume n is a nonnegative integer, all c_is are real, c_n ≠ 0, and x and y are also real.

Expert Solution

This is the desired proof.I hope the answer will help you.Expecting a thumbs up if you are satisfied with the work,it will help me a lot.Thank you.

Colby Messinger answered 2 years ago

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.

Prove a (Dedekind) set is inﬁnite iff there exists an injective function f : N→ A....

Prove a (Dedekind) set is inﬁnite iff there exists an injective function f : N→ A. please help prove this clearly and i will rate the best answer thank you

1. Use the definition of convexity to prove that the function f(x) = x2 - 4x...

1. Use the definition of convexity to prove that the function f(x) = x2 - 4x + 8 is convex. Is this function strictly convex? 2. Use the definition of convexity to prove that the function f(x)= ax + b is both convex and concave for any a•b ≠ 0.

For the function f(x) = x(x+1) find the exact formula for f′(x). Use only the definition...

For the function f(x) = x(x+1) find the exact formula for f′(x). Use only the definition of the derivative.

1. For the function f (x) = 2x² + 8 use the limit definition (four-step process)...

1. For the function f (x) = 2x² + 8 use the limit definition (four-step process) to find f′(x) . Students must show all steps (more or less four of them) to compute the derivative. Simply giving the derivative of the function will not receive much credit. 2. For the function f (x) = x² + 2x a. Find f′(x) . Students may use derivative rules to find this derivative. Show all work and clearly label the answer below. b....

For the following exercises, use the definition of derivative to calculate the derivative of each function. f(x) = 5π

For the following exercises, use the definition of derivative to calculate the derivative of each function.f(x) = 5π

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

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Implement the \void enqueue(int val)" function in the QueueUsingStack.java/QueueUsingStack.h file (ATTACHED BELOW). Here are some hints...

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Explain the difference between the actual definition of a Riemann Integral of function f on the...

Explain the difference between the actual definition of a Riemann Integral of function f on the interval [a,b] and the conclusion of the FTOC Part 2.(Fundamental Theorem of Calculus Part 2)

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