MIT 18.01 Review Problems B

This lesson reproduces the Review “MIT 18.01 Review Problems B” 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 29 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:

Egbert, an MIT nerd bicyclist, is going down a steep hill. At time $t=0$, he starts from rest at the top of the hill; his acceleration while going down is $3t^2$ ft./sec$^2$, and the hill is 64 ft. long. If the fastest he can go without losing control is 64 ft./sec., will he survive this harrowing experience? (A nerd bicycle has no brakes.)

Evaluate $\int_0^2 x^2\,dx$ directly from the definition of the integral as the limit of a sum. You may use the fact that

If $f$ is a continuous function, find $f(2)$ if:

a) $\displaystyle\int_0^{x} f(t)\,dt=x^2(1+x)$

b) $\displaystyle\int_0^{x^2} f(t)\,dt=x^2(1+x)$

c) $\displaystyle\int_0^{f(x)} t^2\,dt=x^2(1+x)$

The area under the graph of $f(x)$ and over the interval $0 \le x \le a$ is

Find $f(\pi/2)$.

Use the trapezoidal rule to estimate the sum $\sqrt[3]{1}+\sqrt[3]{2}+\cdots+\sqrt[3]{10^6}$. Is your estimate high or low? Explain your reasoning.

Find the total area of the region above the graph of $y=-2x$ and below the graph of $y=x-x^2$.

Use the trapezoidal rule with 6 subintervals to estimate the area under the curve $y=\sqrt{1+x^2}$, $-3 \le x \le 3$. (You may use: $\sqrt{2} \approx 1.41$, $\sqrt{5} \approx 2.24$, $\sqrt{10} \approx 3.16$. Is your estimate too high or too low? Explain how you know.)

Fill in this outline of a proof that $\int_a^b F'(x)dx=F(b)-F(a)$. Supply reasons.

a) Put $G(x)=\int_a^x F'(t)dt$. Then $G'(x)=F'(x)$.

b) Therefore $G(x)=F(x)+c$, and one sees easily that $c=-F(a)$. We're done.

The table below gives the known values of a function $f(x)$:

Use Simpson's Rule to estimate the area under the curve $y=f(x)$ between $x=0$ and $x=6$.

Let $f(t)$ be a function, continuous and positive for all $t$. Let $A(x)$ be the area under the graph of $f$, between $t=0$ and $t=x$. Explain intuitively from the definition of derivative why $\frac{dA}{dx}=f(x)$.

Let $f(x)=\begin{cases} x+1, & 0 \le x \le 2 \\ x-2, & 2 < x \le 4 \end{cases}$ Evaluate $\int_0^4 f(x)dx$.

Suppose $F(x)$ is a function such that $F'(x)=\frac{\sin x}{x}$. In terms of $F(x)$, evaluate the indefinite integral $\int \frac{\sin 3x}{x} dx$.

Find a quadratic polynomial $ax^2+bx+c=f(x)$ such that $f(0)=0$, $f(1)=1$, and the area under the graph between $x=0$ and $x=1$ is 1.

The area in the first quadrant bounded by the lines $y=1$, $x=1$, $x=3$ and $f(x)=-x^2+15$ is rotated about the line $y=1$. Find the volume of the solid thus obtained.

Consider the circle $x^2+y^2=4$. A solid is formed with the given circle as base and such that every cross-section cut by a plane perpendicular to the x-axis is a square. Find the volume of this solid.

Find the length of the arc of $y=\frac{1}{3}(x^2+2)^{3/2}$ from $x=0$ to $x=3$

For a freely falling body, $s=\frac{1}{2}gt^2$, $v=gt=\sqrt{2gs}$. Show that:

a) the average value of $v$ over the interval $0 \leq t \leq t_1$ is one-half the final velocity;

b) the average value of $v$ over the interval $0 \leq s \leq s_1$ is two-thirds the final velocity.

A bag of sand originally weighing 144 pounds is lifted at a constant rate of 3 ft./min. the sand leaks out uniformly at such a rate that half the sand is lost when the bag has been lifted 18 feet. find the work done in lifting the bag this distance.

Find the area inside both loops of the lemniscate $r^2=2a^2 \cos 2\theta$.

Calculate the volume obtained when the region $(-2 \le x \le 2, 0 \le y \le x^2)$ is rotated about the y-axis.

The table below gives the known values of a function $f(x)$:

Use Simpson's Rule to estimate the volume obtained when the region below the graph of $y=f(x)$ and above the x-axis ($0 \le x \le 6$) is rotated about the x-axis.

Winnie the Pooh eats honey at a rate proportional to the amount he has left. If it takes him 1 hour to eat the first half of a pot of honey, how long will it take for him to eat another quarter of a pot? When will he finish?

a) Write down the definition of $\ln x$ as an integral.

b) Directly from the definition prove that:

i) $\ln(ax)=\ln a+\ln x$;

ii) $\ln x$ is an increasing function.

Differentiate:

a) $x^{1/x},\qquad e^{x^2}\cdot\ln(x^2)$

b) $\tan^{-1}(\frac{1+x}{1-x})$

Integrate:

a) $\int \sin^3 x \cos^2 x\,dx$

b) $\int e^x \sin x\,dx$

Integrate:

a) $\int \frac{e^x}{1+e^{2x}}\,dx$

b) $\int \frac{x+1}{x^3-1}\,dx$

c) $\int \frac{4x^2}{x-2}\,dx$

Integrate:

a) $\int \frac{x+1}{(1+x^2)^2}\,dx$

b) $\int x^2 \cos x\,dx$

a) Use the reduction formula

to evaluate $\int_0^{\pi/2} \cos^6 x\,dx$.

b) Derive the formula for $D\tan^{-1}x$ from the formula for $D\tan x$. What are the domain and range of $\tan^{-1}x$?