Chapter 1

Graphical Reasoning Consider \(f(x)=\frac{\sec x-1}{x^{2}}\) (a) Find the domain of \(f\) (b) Use a graphing utility to graph \(f\) . Is the domain of \(f\) obvious from the graph? If not, explain. (c) Use the graph of \(f\) to approximate \(\lim _{x \rightarrow 0} f(x)\) (d) Confirm your answer to part (c) analytically.

Proof (a) Let \(f_{1}(x)\) and \(f_{2}(x)\) be continuous on the closed interval \([a, b] .\) If \(f_{1}(a)f_{2}(b),\) prove that there exists \(c\) between \(a\) and \(b\) such that \(f_{1}(c)=f_{2}(c)\) (b) Show that there exists \(c\) in \(\left[0, \frac{\pi}{2}\right]\) such that \(\cos x=x\) . Use a graphing utility to approximate \(c\) to three decimal places.

Approximation (a) Find \(\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}\) (b) Use your answer to part (a) to derive the approximation \(\cos x \approx 1-\frac{1}{2} x^{2}\) for \(x\) near 0 . (c) Use your answer to part (b) to approximate \(\cos (0.1)\) . (d) Use a calculator to approximate \(\cos (0.1)\) to four decimal places. Compare the result with part (c).

Prove or disprove: If \(x\) and \(y\) are real numbers with \(y \geq 0\) and \(y(y+1) \leq(x+1)^{2},\) then \(y(y-1) \leq x^{2}\)

Determine all polynomials \(P(x)\) such that $$ P\left(x^{2}+1\right)=(P(x))^{2}+1 \text { and } P(0)=0 $$