Good lecture optimization problem involving $\ln x$ or $e^x$

Question

I am teaching a Calc 1 of sorts, like a slightly easier version of Calc 1 with no trig. I want a good optimization/practical problem to do in lecture that involves $\ln x$ or $e^x$, to combine review of optimization/practical problems and new learning of $\ln x$ or $e^x$.

I want the problem to be more complicated than, "Find the absolute extrema of this function on this interval." I want it to be a word problem where they need to get all the details out and turn it into a problem of the type I just mentioned. I can't think of anything that would be at the appropriate level. Can you help?

Answer 1

What's below is somewhat pure and it's not really a word problem, but it might lend itself to a good classroom discussion, especially one in which student calculator exploration is involved (in case that's desired).

Given a real number constant $k$, what are the $(x,y)$ coordinates of the point on the graph of $y = e^{kx}$ that is closest to the origin? Don't overlook the special case $k = 0.$ What can you determine about the location of this point for $k \rightarrow 0$? What can you determine about the location of this point for $k \rightarrow \infty$ and for $k \rightarrow -\infty$?

You might start with $k = 1$. A rough sketch by hand (this reviews what graphs of exponentials look like) shows that the point closest to the origin is likely to be in the 2nd quadrant. Setting the first derivative of the square of the distance to the origin equal to $0$ leads to the transcendental equation $e^{2x} = -x,$ which a rough sketch by hand shows has a solution in the 2nd quadrant. Approximations for this point can be found using standard graphing calculator methods. Can students show that this is an absolute minimum using the first derivative test? Can students show that this is a local minimum using the 2nd derivative test?

This can be continued by considering, for a real number $c$, the point on the graph of $y = c\ln x$ that is closest to the origin. For instance, the fact that the graphs of $y = e^{x}$ and $y = \ln x$ are reflections of each other about $y = x$ (this reviews some precalculus ideas involving inverse functions) can be brought up for consideration.

Answer 2

Bob set up an account with \$1,000 which earns 5% interest compounded continually. Money flows out of the account faster and faster. After 1 year it is flowing out at a rate of \$10 per year and after 2 years it is flowing out at a rate of \$20.

Assume that the rate at which money is flowing out of the account grows linearly and find the minimum amount in the account.

Answer: The balance in the account after $t$ years is $F(t) = 1000e^{0.05t}-100t$

$F'(t) = 50e^{0.05t}-100=0$ then $0.05t = \ln(2)$ and so $t \approx 13.863$ years with a minimum balance of $F(13.863) = $ \$613.71

Note: $F''(t)=2.5e^{0.05t}>0$ so the function is concave up and thus this is a min.

Wolfram Alpha plot of $F(t)$

Answer 3

Partition a number $X$ to $N$ terms so that the product of the terms are maximized. Extend the problem for $N \in \mathbb{R}.$