MIT 18.01 Practice Quiz 2A

This lesson reproduces the Practice Quiz “MIT 18.01 Practice Quiz 2A” 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 11 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.

For the function $3x^5-5x^3+1$, sketch the graph over a suitable interval showing all the local maximum and minimum points on the graph, the points of inflection, and the approximate location of its zeros (show on which intervals of the form $[n, n+1]$, ($n$ is an integer) they occur). Show work, or indicate reasoning.

Sketch the graph of $4x^2-\frac{1}{x}$ over an interval showing its interesting features – local maxima and minima, points of inflection, zeros, asymptotes.

A line of negative slope through (1, 2) cuts off a triangle in the first quadrant. For which such line will the triangle have least area? (Use its slope $m$ as the independent variable. Show that you get a minimum.)

The bottom of the legs of a three-legged table are the vertices of an isosceles triangle with sides 5, 5, and 6. The legs are to be braced at the bottom by three wires in the shape of a Y. What is the minimum length of wire needed? Show it is a minimum.

A 200 foot tree is falling in the forest; the sun is directly overhead. At the moment when the tree makes an angle of $30^\circ$ with the horizontal, its shadow is lengthening at the rate of 50 feet/sec. How fast is the angle changing at that moment?

A container in the shape of a right circular cone with vertex angle a right angle is partially filled with water.

a) Suppose water is added at the rate of 3 cu.cm./sec. How fast is the water level rising when the height $h=2$ cm.?

b) Suppose instead no water is added, but water is being lost by evaporation. Show the level falls at a constant rate. (You will have to make a reasonable physical assumption about the rate of water loss—state it clearly.)

How should the parameter $\lambda$ be chosen so that $f(x)=\frac{e^{-\lambda x}}{1+2\sin x}$ remain as close to 1 as possible, when $x \approx 0$? Using this value of $\lambda$, estimate $f(.1)$ to two decimal places.

State the Mean-value Theorem, and use it to prove that

a) if $f(x)$ is differentiable and $f'(x) > 0$ for all $x$, then $f(x)$ is an increasing function;

b) $e^x > 1+x$ for all $x > 0$.

Evaluate: a) $\int \frac{dx}{(3x+2)^2}$ b) $\int \sin 2x \sin x \, dx$ c) $\int \frac{\ln^2 x}{x} \, dx$

Find the function $y(x)$ satisfying $\frac{dy}{dx}=xy^2+x$, $y(0)=1$.

The rate at which a body heats up by conduction is proportional to the difference between its temperature $T$ and the temperature $T_e$ of its surroundings. A fish at room temperature ($20^\circ$) is cooked by putting it into boiling water at $100^\circ$. After 5 minutes its temperature has risen to $30^\circ$. How long will it take to be done ($60^\circ$)?

a) Set up a differential equation for the fish temperature $T(t)$. (Call the constant of proportionality $k$.)

b) Find the solution $T(t)$ satisfying the given data, then use it to answer the question.