@@ -635,4 +635,22 @@ Prove that the total number of maxima and minima possessed by a third-order poly is at most two. \end{hwwithsoln} +\begin{hwwithsoln}{pyramidal-tent} +Euclid proved that the volume of a pyramid equals $(1/3)bh$, where $b$ is the area of its +base, and $h$ its height. A pyramidal tent without tent-poles is erected by blowing air +into it under pressure. The area of the base is easy to measure accurately, because the +base is nailed down, but the height fluctuates somewhat and is hard to measure accurately. +If the amount of uncertainty in the measured height is plus or minus $e_h$, find the +amount of possible error $e_V$ in the volume. +\end{hwwithsoln} + +\begin{hwwithsoln}{rocket-height} +A hobbyist is going to measure the height to which her model rocket rises at the peak of +its trajectory. She plans to +take a digital photo from far away and then do trigonometry to determine the height, +given the baseline from the launchpad to the camera and the angular height of the rocket +as determined from analysis of the photo. Comment on the error incurred by the inability to +snap the photo at exactly the right moment. +\end{hwwithsoln} + \end{hwsection} ch01/ch01.tex @@ -374,16 +374,6 @@ for terms up to order $x^2$. \end{hw} \begin{hw} -Expand $(1+x)^{1/3}$ in a Taylor series around $x=0$. The value $x=28$ lies outside -this series' radius of convergence, but we can nevertheless use it to -extract the cube root of 28 by recognizing that $28^{1/3}=3(28/27)^{1/3}$. -Calculate the root to four significant figures of precision, and check it -in the obvious way. -\end{hw} - -\pagebreak - -\begin{hw} In classical physics, the kinetic energy $K$ of an object of mass $m$ moving at velocity $v$ is given by $K=\frac{1}{2}mv^2$. For example, if a car is to start from a stoplight and then accelerate up to $v$, this is @@ -407,6 +397,14 @@ the classical expression was inferred. \pagebreak \begin{hw} +Expand $(1+x)^{1/3}$ in a Taylor series around $x=0$. The value $x=28$ lies outside +this series' radius of convergence, but we can nevertheless use it to +extract the cube root of 28 by recognizing that $28^{1/3}=3(28/27)^{1/3}$. +Calculate the root to four significant figures of precision, and check it +in the obvious way. +\end{hw} + +\begin{hw} Find the Taylor series expansion of $\log_2 x$ around $x=1$, and use it to evaluate $\log_2 1.0595$ to four significant figures of precision. Check your result by using the fact that 1.0595 is approximately the twelfth root of 2. This number is the ratio of ch05/ch05.tex @@ -226,6 +226,7 @@ $k$ are irrelevant, and can be set to 1. We then have Plugging in $r=\pm 1$, we get a positive result, which confirms that the concavity is upward. \hwsolnhdr{prove-n-extrema} + Since polynomials don't have kinks or endpoints in their graphs, the maxima and minima must be points where the derivative is zero. Differentiation bumps down all the powers of a polynomial by one, so the derivative of a third-order polynomial is a second-order polynomial. A second-order polynomial @@ -234,6 +235,25 @@ quadratic formula. (If the number inside the square root in the quadratic formul there could be less than two real roots.) That means a third-order polynomial can have at most two maxima or minima. +\hwsolnhdr{pyramidal-tent} + +Considering $V$ as a function of $h$, with $b$ treated as a constant, we have for the slope +of its graph +\begin{align*} + \dot{V} &= \frac{e_V}{e_h} \qquad ,\\ +\intertext{so} + e_V &= \dot{V}\cdot e_h \\ + &= \frac{1}{3}b e_h +\end{align*} + +\hwsolnhdr{rocket-height} + +Thinking of the rocket's height as a function of time, we can see that goal is to measure +the function at its maximum. The derivative is zero at the maximum, so the error incurred +due to timing is approximately zero. She should not worry about the timing error too much. +Other factors are likely to be more important, e.g., the rocket may not rise exactly +vertically above the launchpad. + \beginsolutions{2} \hwsolnhdr{fourth-power} ch99/hwans.tex 56dddfdee02d6df5d2a11dd5d1a89bc47775b067 Ben Crowell githubcrowell09@lightandmatter.com http://github.com/bcrowell/calculus/commit/fe02eeeee2add45c71c28c7ed6464acc40a7abc0 fe02eeeee2add45c71c28c7ed6464acc40a7abc0 2009-04-15T15:25:53-07:00 2009-04-15T15:25:53-07:00 before renumbering chapters 4f827012c0818bdbd4c549cef4b826e922cf8c5e Ben Crowell githubcrowell09@lightandmatter.com