Chapter 12

Find a bound on the error of the Taylor's polynomial approximation anchored at \(a=0\) and of indicated degree to a. \(f(x)=\cos x,\) of degree \(2,\) on \([0, \pi / 4]\). b. \(f(x)=e^{x},\) of degree 4 on [0,1] . c. \(f(x)=\cos x,\) of degree 4 on \([0, \pi / 2]\). d. \(f(x)=\sin x,\) of degree \(5,\) on \([0, \pi / 2]\). e. \(f(x)=\ln (1+x)\) of degree 5 on \([0,1 / 2]\). f. \(f(x)=\sin x\) of degree 4 on \([0, \pi / 4\).

a. Compute the fourth degree polynomial, $$ p(x)=\pi_{0}+\pi_{1}(x-1)^{2}+\pi_{2}(x-1)^{2}+\pi_{3}(x-1)^{3}+\pi_{4}(x-1)^{4} $$ that matches \(f(x)=\ln x\) at the anchor point \(a=1\). b. From the pattern of coefficients in \(p\), guess the fifth degree term of a fifth degree polynomial that matches \(f(x)=\ln x\) at the anchor point \(a=1\)

Draw the graphs of $$ f(x)=\sin x \quad \text { and } \quad p(x)=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !} \quad \text { for } \quad-7 \frac{\pi}{10} \leq x \leq 7 \frac{\pi}{10} $$ The graphs should be indistinguishable on \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\). The largest separation on \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\) occurs at \(x=\frac{\pi}{2}\left(\right.\) and \(\left.x=-\frac{\pi}{2}\right)\). Compute \(f\left(\frac{\pi}{2}\right), p\left(\frac{\pi}{2}\right)\), and the relative error in the approximation \(f\left(\frac{\pi}{2}\right) \doteq p\left(\frac{\pi}{2}\right)\).

Show that the following functions are invertible. a. \(\quad f(x)=x^{2} \quad 0 \leq x\) b. \(f(x)=\frac{1}{x} \quad 0

For each of the functions, \(F\), draw a graph of \(F\) and the secant through \((a, F(a))\) and \((b, F(b))\) and a tangent to the graph of \(F\) that is parallel to the secant. a. \(\quad F(x)=\frac{1}{x}, \quad[a, b]=[1 / 2,2]\) b. \(\quad F(x)=\frac{1}{x^{2}+1}, \quad[a, b]=[-1,1]\) c. \(\quad F(x)=e^{x}, \quad[a, b]=[0,2]\) d. \(\quad F(x)=\sin x, \quad[a, b]=[0, \pi / 2]\)

Suppose \(y(t)\) solves $$ y^{\prime}=k y(t)(1-y(t)) \quad \text { and } \quad z(t)=M y(k t) $$ Show that \(z(t)\) solves $$ z^{\prime}(t)=k z(t)\left(1-\frac{z(t)}{M}\right) $$

Find a cubic polynomial, \(p(x)=p_{0}+p_{1} x+p_{2} x^{2}+p_{3} x^{3}\) that matches \(f(x)=e^{x}\) at \(a=0 .\) To do so, you should complete the following table. You should conclude from the table that $$ p(x)=1+x+\frac{x^{2}}{2}+\frac{x^{3}}{3 !} $$ is the cubic polynomial that matches \(f(x)=e^{x}\) at \(a=0\). Draw the graphs of \(f(x)=e^{x}\) and \(p(x)=1+x+x^{2} / 2+x^{3} / 6\) on \(-1 \leq x \leq 1\). You should find a pretty good match. The maximum separation occurs at \(x=1\) with a maximum $$ =\frac{\left|1+1+1 / 2+1 / 6-e^{1}\right|}{e^{1}} \doteq \frac{|2.6666-2.71828|}{e^{1}} \doteq 0.02 $$ There is about a \(2 \%\) relative error at \(x=1\) Compute the relative error in the approximation, \(e^{-1} \doteq p(-1)\).

Suppose you are measuring the growth of a corn plant, and observe that a. The plant is \(14 \mathrm{~cm}\) tall at 10: 00 am Monday and \(15.5 \mathrm{~cm}\) tall at \(10: 00 \mathrm{am}\) on Tuesday. What is your best guess for the height of the plant at \(10: 00 \mathrm{am}\) on Thursday? b. The plant is \(14 \mathrm{~cm}\) tall at 10: 00 am on Monday, \(15.5 \mathrm{~cm}\) tall at \(10: 00 \mathrm{am}\) on Tuesday, and \(17.5 \mathrm{~cm}\) tall at \(10: 00 \mathrm{am}\) on Wednesday. What is your best guess for the height of the plant at 10:00 am on Thursday?

Find the intervals on which \(f\) is increasing. Identify the local minima and local maxima. Plot the local minima and local maxima on a graph, and sketch a candidate graph of the function. a. \(f(x)=x^{2}-3 x+7\) b. \(f(x)=-x^{2}+5 x+16\) c. \(f(x)=3 x^{4}+8 x^{3}-6 x^{2}-24 x+17\) d. \(f(x)=x^{5}-5 x+2\) e. \(f(x)=x^{4}+4 x+8\) f. \(f(x)=x \ln x\) g. \(f(x)=x e^{-x}\) h. \(f(x)=x^{2} e^{-x}\)

Rearrange the formula, Equation 12.14, \(f(b)=f(a)+f^{\prime}\left(c_{0}\right)(b-a)\) and show that it is the a statement of the Mean Value Theorem.

Find a number \(c\) between \(a\) and \(b\) for which \(f^{\prime}(c)=f(b)-f(a) /(b-a)\). a. \(f(t)=t^{4} \quad[a, b]=[0,1]\) b. \(f(t)=t^{3}-3 t^{2}+3 t \quad[a, b]=[1,2]\) c. \(f(t)=t^{3}-3 t^{2}+3 t \quad[a, b]=[1,2]\) d. \(f(t)=\ln t \quad[a, b]=[1, e]\)

What is your best estimate of the adult avian population for January 1,2012 based on the following data? Which of the estimates for the four cases seems unlikely. $$ \begin{array}{|c|r|r|r|} \hline {\text { Adult Avian Census }} \\ \hline & \text { Jan 1, 2009 } & \text { Jan 1, 2010 } & \text { Jan 1, 2011 } \\\ \hline \text { Case 1. } & 1000 & 1050 & 1110 \\ \text { Case 2. } & 1000 & 1050 & 1100 \\ \text { Case 3. } & 1000 & 950 & 920 \\ \text { Case 4. } & 1000 & 980 & 1010 \\ \hline \end{array} $$

Find the quadratic polynomial, \(p(x)=p_{0}+p_{1} x+p_{2} x^{2},\) that matches \(f(x)=\cos x\) at the anchor point, \(a=0\). Draw the graphs of \(f(x)\) and \(p(x)\) and discuss the accuracy of the approximation, \(\cos \left(\frac{\pi}{4}\right) \doteq p\left(\frac{\pi}{4}\right)\).

Find \(p_{0}, p_{1}\), and \(p_{2}\) for which the polynomial, \(p(t)=p_{0}+p_{1} t+p_{2} t^{2}\), satisfies a. \(p(0)=5, \quad p^{\prime}(0)=-2,\) and \(p^{\prime \prime}(0)=\frac{1}{3}\). b. \(p(0)=1, \quad p^{\prime}(0)=0, \quad\) and \(\quad p^{\prime \prime}(0)=-\frac{1}{2}\). c. \(p(0)=0, \quad p^{\prime}(0)=1, \quad\) and \(\quad p^{\prime \prime}(0)=0\). d. \(p(0)=1, \quad p^{\prime}(0)=0, \quad\) and \(\quad p^{\prime \prime}(0)=-1\). e. \(p(0)=1, \quad p^{\prime}(0)=1, \quad\) and \(\quad p^{\prime \prime}(0)=1\). f. \(p(0)=17, p^{\prime}(0)=-15,\) and \(p^{\prime \prime}(0)=12\).

Show that the following functions are invertible. a. \(\quad F(t)=e^{t}\) b. \(\quad F(t)=\tan t, \quad-\frac{\pi}{2}

Add the two versions of Taylor's third order polynomial with error, Equation 12.17: $$ \begin{array}{l} f(a+h)=f(a)+f^{\prime}(a) h+\frac{f^{(2)}(a)}{2 !} h^{2}+\frac{f^{(3)}(a)}{3 !} h^{3}+\frac{f^{(4)}\left(c_{1}\right)}{3 !} h^{4} \\ f(a-h)=f(a)-f^{\prime}(a) h+\frac{f^{(2)}(a)}{2 !} h^{2}-\frac{f^{(3)}(a)}{3 !} h^{3}+\frac{f^{(4)}\left(c_{2}\right)}{4 !} h^{4}, \end{array} $$ and solve for \(f^{(2)}(a)\). You should find that (after a slight alteration) $$ f^{(2)}(a)=\frac{f(a+h)-2 f(a)+f(a-h)}{h^{2}}-\frac{f^{(4)}\left(c_{1}\right)+f^{(4)}\left(c_{2}\right)}{2} \frac{h^{2}}{12} $$ Use the intermediate value property to argue that there is a number \(c\) such that $$ f^{(2)}(a)=\frac{f(a+h)-2 f(a)+f(a-h)}{h^{2}}-f^{(4)}(c) \frac{h^{2}}{12} $$ This provides a good formula for approximating second derivatives.

Find the fourth degree polynomial, \(p(x)=p_{0}+p_{1} x+p_{2} x^{2}+p_{3} x^{3}+p_{4} x^{4}\), that matches \(f(x)=\cos x\) at the anchor point, \(a=0\). Draw the graphs of \(f(x)\) and \(p(x)\) and discuss the accuracy of the approximation, \(\cos \left(\frac{\pi}{4}\right) \doteq p\left(\frac{\pi}{4}\right)\).

Show that for the function $$ F(h)=(1+h)^{1 / h} \quad F^{\prime}(h)<0 \quad \text { for } \quad 0

Find a cubic polynomial, \(p(x)=p_{0}+p_{1} x+p_{2} x^{2}+p_{3} x^{3}\) that matches \(f(x)=e^{-x}\) at \(a=0\)

The graph of the function \(f\) defined by $$ f(t)=\left\\{\begin{array}{ll} t & \text { for } 0 \leq t<1 \\ 0 & \text { for } t=1 \end{array}\right. $$ is shown in Figure Ex. 12.1.6. It does not have a high point. Furthermore, it does not have a horizontal tangent; yet it satisfies all but one of the hypotheses of Rolle's Theorem. Which hypothesis of Rolle's Theorem does it not satisfy?

Assume that the sixth degree polynomials, \(S(x), C(x),\) and \(E(x)\) that match, respectively, \(\sin x, \cos x,\) and \(e^{x},\) are $$ \begin{array}{l} S(x)=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !} \\ C(x)=1-\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}-\frac{x^{6}}{6 !} \\ E(x)=1+x+\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}+\frac{x^{4}}{4 !}+\frac{x^{5}}{5 !}+\frac{x^{6}}{6 !} \end{array} $$ a. Compute \(S^{\prime}(x)\) and compare it with \(C(x)\). b. Compute \(C^{\prime}(x)\) and compare it with \(S(x)\). c. Compute \(E^{\prime}(x)\) and compare it with \(E(x)\). d. (Only for the adventurous.) Let \(i=\sqrt{-1}\). Note that \(i^{2}=-1, i^{3}=i^{2} \cdot i=-i,\) and \(i^{4}=i^{2} \cdot i^{2}=1,\) and continue this sequence. Compute \(E(i \cdot x)\) and write it in terms of \(S(x)\) and \(C(x)\).

Draw the graph of a continuous function \(F\) with domain [1,5] that has a non- negative derivative at every point between 1 and \(5,\) and for which 5 is a local minimum. The next three exercises demonstrate interesting function equivalents of the Axiom of Completeness 5.2 .1 from Section 5.2 .