Chapter 1: Problem 7
Solve for \(x\). $$ 4^{3}=(x+2)^{3} $$
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Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of \(c\) guaranteed by the theorem. $$ f(x)=\frac{x^{2}+x}{x-1}, \quad\left[\frac{5}{2}, 4\right], \quad f(c)=6 $$
Determine all polynomials \(P(x)\) such that $$ P\left(x^{2}+1\right)=(P(x))^{2}+1 \text { and } P(0)=0 . $$
Show that the function \(f(x)=\left\\{\begin{array}{ll}0, & \text { if } x \text { is rational } \\ k x, & \text { if } x \text { is irrational }\end{array}\right.\) is continuous only at \(x=0\). (Assume that \(k\) is any nonzero real number.)
Rate of Change A 25 -foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, the top will move down the wall at a rate \(r\) of \(r=\frac{2 x}{\sqrt{625-x^{2}}} \mathrm{ft} / \mathrm{sec}\) where \(x\) is the distance between the ladder base and the house. (a) Find \(r\) when \(x\) is 7 feet. (b) Find \(r\) when \(x\) is 15 feet. (c) Find the limit of \(r\) as \(x \rightarrow 25^{-}\).
Prove that if \(\lim _{x \rightarrow c} f(x)\) exists and \(\lim _{x \rightarrow c}[f(x)+g(x)]\) does not exist, then \(\lim _{x \rightarrow c} g(x)\) does not exist.
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