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MIT 18.01 Practice Quiz 1A

Practice quiz on limits, the definition of the derivative, continuity and differentiability of piecewise functions.

1. MIT 18.01 Practice Quiz 1A — how this assessment works

This lesson reproduces the Practice Quiz “MIT 18.01 Practice Quiz 1A” 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 10 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.

2. MIT 18.01 PQ1A Problem 1 — Derivative, limit, and higher derivatives

Evaluate each of the following, simplifying where possible; for (b) indicate reasoning. The letters $a$ and $k$ represent constants.

a)\quad .\frac{d}{dt}(\frac{3t}{\ln t})|_{t=e^2} \qquad b)\quad \lim_{u\to 0}\frac{3u}{\tan 2u}

c)\quad \frac{d^3}{dx^3}\sin kx \qquad d)\quad \frac{d}{d\theta}\sqrt[3]{a+k\sin^2\theta}

3. MIT 18.01 PQ1A Problem 2 — Derivative of $x^3$ from the definition

Derive the formula for $\frac{d}{dx}x^3$ at the point $x=x_0$ directly from the definition of derivative.

4. MIT 18.01 PQ1A Problem 3 — Limit related to a derivative

Find $\displaystyle\lim_{h\to 0}\frac{1-\sqrt[3]{1+h}}{h}$ by relating it to a derivative. (Indicate reasoning.)

5. MIT 18.01 PQ1A Problem 4 — Inverse sine: graph and derivative

Sketch the curve $y=\sin^{-1}x$ , $-1\le x\le 1$ , and derive the formula for its derivative from that for the derivative of $\sin x$ .

6. MIT 18.01 PQ1A Problem 5 — Continuity and differentiability of a piecewise function

For the function

f(x)=\begin{cases} ax+b, & x>0\\ 1-x+x^2, & x\leq 0,\end{cases}\qquad a\text{ and }b\text{ constants},

\textbf{a)} find all values of $a$ and $b$ for which the function will be continuous;

\textbf{b)} find all values of $a$ and $b$ for which the function will be differentiable.

7. MIT 18.01 PQ1A Problem 6 — Points with horizontal tangent (implicit differentiation)

For the curve given by the equation

x^2y+y^3+x^2=8,

find all points on the curve where its tangent line is horizontal.

8. MIT 18.01 PQ1A Problem 7 — $x$-intercept of a tangent line

Where does the tangent line to the graph of $y=f(x)$ at the point $(x_0,y_0)$ intersect the $x$ -axis?

9. MIT 18.01 PQ1A Problem 8 — Deflating spherical balloon

The volume of a spherical balloon is decreasing at the instantaneous rate of $-10\ \mathrm{cm^3/sec}$ , at the moment when its radius is $20$ cm. At that moment, how rapidly is its radius decreasing?

10. MIT 18.01 PQ1A Problem 9 — Discontinuities

Where are the following functions discontinuous?

a)\ \sec x \qquad b)\ \frac{1+x^2}{1-x^2} \qquad c)\ \frac{d}{dx}|x|

11. MIT 18.01 PQ1A Problem 10 — Radioactive decay

A radioactive substance decays according to a law $A=A_0e^{-rt}$ , where $A(t)$ is the amount present at time $t$ , and $r$ is a positive constant.

\textbf{a)} Derive an expression in terms of $r$ for the time it takes for the amount to fall to one-quarter of the initial amount $A_0$ .

\textbf{b)} At the moment when the amount has fallen to $1/4$ the initial amount, how rapidly is the amount falling? (Units: grams, seconds.)

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