em dash; sentence case

quarto/ODEs/odes.qmd

@@ -149,7 +149,7 @@ $$
 U'(t) = -r U(t), \quad U(0) = U_0.
 $$
 
-This shows that the rate of change of $U$ depends on $U$. Large positive values indicate a negative rate of change - a push back towards the origin, and large negative values of $U$ indicate a positive rate of change - again, a push back towards the origin. We shouldn't be surprised to either see a steady decay towards the origin, or oscillations about the origin.
+This shows that the rate of change of $U$ depends on $U$. Large positive values indicate a negative rate of change---a push back towards the origin, and large negative values of $U$ indicate a positive rate of change---again, a push back towards the origin. We shouldn't be surprised to either see a steady decay towards the origin, or oscillations about the origin.
 
 
 What will we find? This equation is different from the previous two equations, as the function $U$ appears on both sides. However, we can rearrange to get:
@@ -177,7 +177,7 @@ $$
 In words, the initial difference in temperature of the object and the environment exponentially decays to $0$.
 
 
-That is, as $t > 0$ goes to $\infty$, the right hand will go to $0$ for $r > 0$, so $T(t) \rightarrow T_a$ - the temperature of the object will reach the ambient temperature. The rate of this is largest when the difference between $T(t)$ and $T_a$ is largest, so when objects are cooling the statement "hotter things cool faster" is appropriate.
+That is, as $t > 0$ goes to $\infty$, the right hand will go to $0$ for $r > 0$, so $T(t) \rightarrow T_a$---the temperature of the object will reach the ambient temperature. The rate of this is largest when the difference between $T(t)$ and $T_a$ is largest, so when objects are cooling the statement "hotter things cool faster" is appropriate.
 
 
 A graph of the solution for $T_0=200$ and $T_a=72$ and $r=1/2$ is made as follows. We've added a few line segments from the defining formula, and see that they are indeed tangent to the solution found for the differential equation.
@@ -403,7 +403,7 @@ To finish, we call `dsolve` to find a solution (if possible):
 out = dsolve(eqn)
 ```
 
-This answer - to a first-order equation - has one free constant, `C₁`, which can be solved for from an initial condition. We can see that when $a > 0$, as $x$ goes to positive infinity the solution goes to $1$, and when $x$ goes to negative infinity, the solution goes to $0$ and otherwise is trapped in between, as expected.
+This answer---to a first-order equation---has one free constant, `C₁`, which can be solved for from an initial condition. We can see that when $a > 0$, as $x$ goes to positive infinity the solution goes to $1$, and when $x$ goes to negative infinity, the solution goes to $0$ and otherwise is trapped in between, as expected.
 
 
 The limits are confirmed by investigating  the limits of the right-hand:
@@ -618,6 +618,7 @@ nothing
 ```
 
 ![The cables of an unloaded suspension bridge have a different shape than a loaded suspension bridge. As seen, the cables in this [figure](https://www.brownstoner.com/brooklyn-life/verrazano-narrows-bridge-anniversary-historic-photos/) would be modeled by a catenary.](./figures/verrazano-narrows-bridge-anniversary-historic-photos-2.jpeg)
+
 ---
 
 
@@ -641,7 +642,7 @@ $$
 x''(t) = 0, \quad y''(t) = -g.
 $$
 
-That is, the $x$ position - where no forces act - has $0$ acceleration, and the $y$ position - where the force of gravity acts - has constant acceleration, $-g$, where $g=9.8m/s^2$ is the gravitational constant. These equations can be solved to give:
+That is, the $x$ position---where no forces act---has $0$ acceleration, and the $y$ position---where the force of gravity acts---has constant acceleration, $-g$, where $g=9.8m/s^2$ is the gravitational constant. These equations can be solved to give:
 
 
 $$
@@ -957,7 +958,7 @@ radioq(choices, answ)
 ##### Question
 
 
-The example with projectile motion in a medium has a parameter $\gamma$ modeling the effect of air resistance. If `y` is the answer - as would be the case if the example were copy-and-pasted in - what can be said about `limit(y, gamma=>0)`?
+The example with projectile motion in a medium has a parameter $\gamma$ modeling the effect of air resistance. If `y` is the answer---as would be the case if the example were copy-and-pasted in---what can be said about `limit(y, gamma=>0)`?
 
 
 ```{julia}
@@ -966,7 +967,7 @@ The example with projectile motion in a medium has a parameter $\gamma$ modeling
 choices = [
 "The limit is a quadratic polynomial in `x`, mirroring the first part of that example.",
 "The limit does not exist, but the limit to `oo` gives a quadratic polynomial in `x`, mirroring the first part of that example.",
-"The limit does not exist -- there is a singularity -- as seen by setting `gamma=0`."
+"The limit does not exist---there is a singularity---as seen by setting `gamma=0`."
 ]
 answ = 1
 radioq(choices, answ)