MIT 18.01 Exam 2

This lesson reproduces the Exam “MIT 18.01 Exam 2” 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.

Estimate the following to two decimal places (show work).

a. (8 pts.) $\sin(\pi+\frac{1}{100})$

b. (7 pts.) $\sqrt{101}$

Sketch the graph of $y=\frac{4}{x}+x+1$ on $-\infty<x<\infty$ and label all critical points and inflection points with their coordinates on the graph along with the letter "C" or "I".

Write down on which intervals the function is:

Increasing:

Decreasing:

Concave down:

An architect plans to build a triangular enclosure with a fence on two sides and a wall on the third side. Each of the fence segments has fixed Length L. What is the length x of the third side if the region enclosed has the largest possible area? Show work and include an argument to show that your answer really gives the maximum area.

A rocket has launched straight up, and its altitude is $h=10t^2$ feet after $t$ seconds. You are on the ground 1000 feet from the launch site. The line of sight from you to the rocket makes an angle $\theta$ with the horizontal. By how many Radians per second is $\theta$ changing ten seconds after the launch?

a. (10 pts) Evaluate the following indefinite integrals

i. $\displaystyle\int \cos(3x)\,dx$

ii. $\displaystyle\int xe^{x^2}\,dx$

b. (10 pts) Find $y(x)$ such that $y'=\dfrac{1}{y^3}$ and $y(0)=1$.

Suppose that $f'(x)=e^{x^2}$, and $f(0)=10$. One can conclude from the mean value theorem that

for which numbers $A$ and $B$?