MIT 18.01 Exam 1

This lesson reproduces the Exam “MIT 18.01 Exam 1” 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 7 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.

(10 pts.) Find the tangent line to $y=\frac{1}{3}x^2$ at $x=1$.

Find the derivative of the following functions:

a. (7 pts.) $\frac{x}{\sqrt{1-x}}$

b. (8 pts.) $\frac{\cos(2x)}{x}$

c. (5 pts.) $e^{2f(x)}=g(x)$

d. (5 pts.) $\ln(\sin x)$

(15 pts.) Find $\frac{dy}{dx}$ for the function for the function $y$ defined implicitly by

(The phrase "for the function" is duplicated in the printed exam; reproduced verbatim.)

(15 pts.) Draw the graph of the derivative of the function (qualitatively accurate) directly under the graph of the function.


(15 pts.) Let

Find all $a$ and $b$ such that the function $f(x)$ is differentiable.

Evaluate these limits by relating them to a derivative.

a. (5 pts.) Evaluate $\lim_{x\to 0}\frac{(1+2x)^{10}-1}{x}$

b. (5 pts.) Evaluate $\lim_{x\to 0}\frac{\sqrt{\cos x}-1}{x}$

(10 pts.) Derive the formula $\frac{d}{dx}a^x=M(a)a^x$ directly from the definition of the derivative, and identify $M(a)$ as a limit.