MIT 18.01 Practice Exam 2B

This lesson reproduces the Practice Exam “MIT 18.01 Practice Exam 2B” from MIT OpenCourseWare 18.01 Single Variable Calculus (Fall 2006) (https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/), licensed under CC BY-NC-SA 4.0. Work through 6 problems in order: type your reasoning in the answer box or write it on the handwriting pad, then submit for AI grading.

How the grading works Every problem is graded by an AI mathematics teacher that reads the official solution shipped with the course. It checks your work criterion by criterion, awards partial credit for correct steps, and returns written feedback. Equivalent mathematics and valid alternative methods are accepted — a different route to the same result can still earn full marks.

If your answer is incomplete, the feedback names exactly which steps are missing, so you can correct them and try again.

Find the local maxima and minima and points of inflection of

Then use this data to sketch its graph on the given axes, showing also where it is convex (concave up) or concave (down). (Note that different scales are used on the two axes.)

A new junk food — NoKarb PopKorn — is to be sold in large cylindrical metal cans with a removable plastic lid instead of a metal top. The metal side and bottom will be of uniform thickness, and the volume is fixed to be $64\pi$ cubic inches.

What base radius $r$ and height $h$ for the can will require the least amount of metal?

Show work, and include an argument to show your values for $r$ and $h$ really give a minimum.

Evaluate the following indefinite integrals:

a)

b)

c)

A searchlight $L$ is 100 meters from a prison wall. It is rotating at a constant rate of one revolution every 8 minutes. (How many radians/minute is that?)

Martha, an escaping prisoner, is running along the wall trying to keep just ahead of the beam of light.

At the moment when the searchlight angle $\theta$ is 60 degrees, how fast does she have to run?

a) What value for the constant $c$ will make the function $e^{-x}\sqrt{1+cx}$ approximately constant, for values of $x$ near $0$? (Show work.)

b) Find the solution $x(t)$ to the differential equation

which also satisfies the condition $x(0)=1$.

a) Use the Mean-value Theorem to show that $\ln(1+x)<x$, if $x>0$.

(You do not have to state the theorem.)

b) Let $c$ be any constant. Show that the function $f(x)=x^{3}+x+c$ cannot have two zeros.

(Use the Mean-value theorem, or some other argument.)