MIT 18.01 Review Problems A

This lesson reproduces the Review “MIT 18.01 Review Problems A” 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 24 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.

Evaluate the derivatives. Assume all letters represent constants, except for the independent and dependent variables occurring in the derivative.

a) $pV^{\gamma}=nRT$, $\dfrac{dp}{dV}=\,$?

b) $m=\dfrac{m_0}{\sqrt{1-v^{2}/c^{2}}}$, $\dfrac{dm}{dv}=\,$?

c) $R=\dfrac{\omega_0\sin\big((2k+1)\alpha\big)}{\alpha^{2}+\beta^{2}}$, $.\dfrac{dR}{d\alpha}|_{\alpha=0}=\,$?

Differentiate:

a) $\dfrac{\sin x}{x+1}$

b) $\sin^{2}(\sqrt{x})$

c) $x^{1/3}\tan x$

d) $\dfrac{x^{2}+2}{\sqrt{x+1}}$

e) $\cos(\sqrt{x^{2}+1})$

f) $\cos^{3}(\sqrt{x^{2}+1})$

g) $\tan(x^{3})$

h) $x\sec^{2}(3x+1)$

Consider $f(x)=2x^{2}+4x+3$. Where does the tangent line to the graph of $f(x)$ at $x=3$ cross the $y$-axis?

Find the equation of the tangent to the curve $2x^{2}+xy-y^{2}+2x-3y=20$ at the point $(3,2)$.

Define the derivative of $f(x)$. Directly from the definition, show that $f'(x)=\cos x$ if $f(x)=\sin x$. (You may use without proof: $\displaystyle\lim_{h\to0}\frac{\sin h}{h}=1$, $\displaystyle\lim_{h\to0}\frac{\cos h-1}{h}=0$.)

Find all real $x_0$ such that $f'(x_0)=0$:

a) $f(x)=\dfrac{x}{x^{2}+1}$

b) $f(x)=x^{2}+\cos x$

At what points is the tangent to the curve $y^{2}+xy+x^{2}-3=0$ horizontal?

State and prove the formula for $(uv)'$ in terms of the derivatives of $u$ and $v$. You may assume any theorems about limits that you need.

Derive a formula for $\big(x^{1/5}\big)'$.

a) What is the rate of change of the area $A$ of a square with respect to its side $x$?

b) What is the rate of change of the area $A$ of a circle with respect to its radius $r$?

c) Explain why one answer is the perimeter of the figure but the other answer is not.

Find all values of the constants $c$ and $d$ for which the function

will be (a) continuous, (b) differentiable.

Prove or give a counterexample:

a) If $f(x)$ is differentiable then $f(x)$ is continuous.

b) If $f(x)$ is continuous then $f(x)$ is differentiable.

Find all values of the constants $a$ and $b$ so that the function

will be (a) continuous; (b) differentiable.

Evaluate $\displaystyle\lim_{x\to0}\frac{\sin(4x)}{x}$. (Hint: Let $4x=t$.)

Sketch the graphs of the following functions, indicating maxima, minima, points of inflection, and concavity.

a) $f(x)=(x-1)^{2}(x+2)$

b) $f(x)=\sin^{2}x$, $0 \le x \le 2\pi$

c) $f(x)=x+1/x^{2}$

d) $f(x)=x+\sin 2x$

A baseball diamond is a 90 ft. square. A ball is batted along the third base line at a constant speed of 100 ft. per sec. How fast is its distance from first base changing when

a) it is halfway to third base,

b) it reaches third base?

If $x$ and $y$ are the legs of a right triangle whose hypotenuse is $\sqrt{5}$, find the largest value of $2x+y$.

Evaluate the following limits:

a) $\displaystyle\lim_{x\to\pi/2}\frac{\cos x}{\pi/2-x}$

b) $\displaystyle\lim_{x\to0}\frac{\sin x}{x}$

c) $\displaystyle\lim_{x\to\infty}\frac{x^{17}-4x^{3}+2x^{2}}{10x^{17}+6x^{10}-x^{3}-5x^{2}}$

Prove or give a counterexample:

a) If $f'(c)=0$ then $f$ has a minimum or a maximum at $c$.

b) If $f$ has a maximum at $c$ and if $f$ is differentiable at $c$, then $f'(c)=0$.

Let $f(x)=1-x^{2/3}$. Then $f(-1)=f(1)=0$ and yet $f'(x) \neq 0$ for $0<x<1$. Find the maximum value of $f(x)$ on the real line, nevertheless.

Why did the standard method fail?

A can is made in the shape of a right circular cylinder. What should its proportions be, if its volume is to be 1 and one wants to use the least amount of metal?

a) State the mean value theorem.

b) If $f'(x)=\dfrac{1}{1+x^{2}}$ and $f(1)=1$, use the mean value theorem to estimate $f(2)$. (Write your answer in the form $\alpha < f(2) < \beta$.)

One of these statements is false and one is true. Prove the true one, and give a counterexample to the false one. (Both statements refer to all $x$ in some interval $a<x<b$.)

a) If $f'(x) > 0$, then $f(x)$ is an increasing function.

b) If $f(x)$ is an increasing function, then $f'(x) > 0$.

Give examples (either by giving a formula or by a carefully drawn graph) of

a) A function with a relative minimum, but no absolute maximum on $0<x<1$.

b) A function with a relative maximum but no absolute maximum on the interval $0\le x\le1$.

c) A function $f(x)$ defined on $0\le x\le1$, with $f(0)<0$, $f(1)>0$, yet with no root on $0\le x\le1$.

d) A function $f(x)$ having a relative minimum at $0$, but the following is false: $f'(0)=0$.