91Ó°ÊÓ

Use the Intermediate Value Theorem \(1.11\) and Rolle's Theorem \(1.7\) to show that the graph of \(f(x)=x^{3}+2 x+k\) crosses the \(x\)-axis exactly once, regardless of the value of the constant \(k\).

Find the rates of convergence of the following sequences as \(n \rightarrow \infty\). a. \(\quad \lim _{n \rightarrow \infty} \sin \frac{1}{n}=0\) b. \(\quad \lim _{n \rightarrow \infty} \sin \frac{1}{n^{2}}=0\) c. \(\lim _{n \rightarrow \infty}\left(\sin \frac{1}{n}\right)^{2}=0\) d. \(\quad \lim _{n \rightarrow \infty}[\ln (n+1)-\ln (n)]=0\)

Find the rates of convergence of the following functions as \(h \rightarrow 0\). a. \(\quad \lim _{h \rightarrow 0} \frac{\sin h}{h}=1\) b. \(\quad \lim _{h \rightarrow 0} \frac{1-\cos h}{h}=0\) c. \(\lim _{h \rightarrow 0} \frac{\sin h-h \cos h}{h}=0\) d. \(\quad \lim _{h \rightarrow 0} \frac{1-e^{h}}{h}=-1\)

a. How many multiplications and additions are required to determine a sum of the form $$ \sum_{i=1}^{n} \sum_{j=1}^{i} a_{i} b_{j} ? $$ b. Modify the sum in part (a) to an equivalent form that reduces the number of computations.

Find the third Taylor polynomial \(P_{3}(x)\) for the function \(f(x)=\sqrt{x+1}\) about \(x_{0}=0\). Approximate \(\sqrt{0.5}, \sqrt{0.75}, \sqrt{1.25}\), and \(\sqrt{1.5}\) using \(P_{3}(x)\), and find the actual errors.

The first three nonzero terms of the Maclaurin series for the arctangent function are \(x-(1 / 3) x^{3}+\) \((1 / 5) x^{5}\). Compute the absolute error and relative error in the following approximations of \(\pi\) using the polynomial in place of the arctangent: a. \(\quad 4\left[\arctan \left(\frac{1}{2}\right)+\arctan \left(\frac{1}{3}\right)\right]\) b. \(\quad 16 \arctan \left(\frac{1}{5}\right)-4 \arctan \left(\frac{1}{239}\right)\)

Find the second Taylor polynomial \(P_{2}(x)\) for the function \(f(x)=e^{x} \cos x\) about \(x_{0}=0\). a. Use \(P_{2}(0.5)\) to approximate \(f(0.5)\). Find an upper bound for error \(\left|f(0.5)-P_{2}(0.5)\right|\) using the error formula, and compare it to the actual error. b. Find a bound for the error \(\left|f(x)-P_{2}(x)\right|\) in using \(P_{2}(x)\) to approximate \(f(x)\) on the interval \([0,1]\). c. Approximate \(\int_{0}^{1} f(x) d x\) using \(\int_{0}^{1} P_{2}(x) d x\). d. Find an upper bound for the error in (c) using \(\int_{0}^{1}\left|R_{2}(x) d x\right|\), and compare the bound to the actual error.

Let $$f(x)=\frac{x \cos x-\sin x}{x-\sin x}$$ a. Find \(\lim _{x \rightarrow 0} f(x)\). b. Use four-digit rounding arithmetic to evaluate \(f(0.1)\). c. Replace each trigonometric function with its third Maclaurin polynomial, and repeat part (b). d. The actual value is \(f(0.1)=-1.99899998\). Find the relative error for the values obtained in parts (b) and (c).

Let $$f(x)=\frac{e^{x}-e^{-x}}{x}$$ a. Find \(\lim _{x \rightarrow 0}\left(e^{x}-e^{-x}\right) / x\) b. Use three-digit rounding arithmetic to evaluate \(f(0.1)\). c. Replace each exponential function with its third Maclaurin polynomial, and repeat part (b). d. The actual value is \(f(0.1)=2.003335000\). Find the relative error for the values obtained in parts (b) and (c).