Proof Prove that for any real number y there exists x in (-π/ 2, π/ 2) such that tan x=y .

Name: Problem 124, Chapter 1: Calculus of a Single Variable
Uploaded: 2020-03-30T22:16:09-08:00
Duration: 2 min
Channel: Robert Daugherty
Description: Proof Prove that for any real number y there exists x in (-π/ 2, π/ 2) such that tan x=y .

step 1 Section 10 .4, number 24, we're here to prove that for any real number Y, there exists an X in t

Proof Prove that for any real number y there exists x in...

Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) states that if a function is continuous on a closed interval [a, b] and takes different values at a and b, then it must take every value between those two values at some point within the interval. In the context of the tangent function on (–?/2, ?/2), the IVT guarantees that all real numbers are covered in the range because the limits as x approaches –?/2 and ?/2 are –? and ? respectively.

Strict Monotonicity

Strict monotonicity means that the function is either entirely non-decreasing or non-increasing on a given interval, with every increasing step producing a higher output. This guarantee for the tangent function on (–?/2, ?/2) means that it will hit every value exactly once, ensuring the existence of an inverse function, and by extension, that for every real y, there is a unique x such that tan x equals y.

Continuity

Continuity is a fundamental concept in analysis that means a function has no breaks, jumps, or gaps in its domain. For the given problem, the continuity of the tangent function on (–?/2, ?/2) ensures that small changes in the input x result in small changes in the output tan x, which is essential when applying the Intermediate Value Theorem.

Tangent Function Properties

The tangent function is defined as the ratio of the sine and cosine functions. On its principal domain of (–?/2, ?/2), it is continuous and strictly increasing, making it well-behaved in terms of having no sudden jumps or oscillations. These properties are critical for proving that every real number is achieved as an output of the tangent function.

Transcript

Please give Ace some feedback