MIT 18.01 Problem Set 4

This lesson reproduces the Problem Set “MIT 18.01 Problem Set 4” 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.

(Lec 14, 10 pts: $2+2+2+2+2$)

a) Use the mean value property to show that if $f(0)=0$ and $f'(x) \geq 0$, then $f(x) \geq 0$ for all $x \geq 0$.

b) Deduce from part (a) that $\ln(1+x) \leq x$ for $x \geq 0$. Hint: Use $f(x)=x-\ln(1+x)$.

c) Use the same method as in (b) to show $\ln(1+x) \geq x-x^2/2$ and $\ln(1+x) \leq x-x^2/2+x^3/3$ for $x \geq 0$.

d) Find the pattern in (b) and (c) and make a general conjecture.

e) Show that $\ln(1+x) \leq x$ for $-1 < x \leq 0$. (Use the change of variable $u=-x$.)

(Lec 15, 4 pts: $2+2$)

a) Do 5.3/68

b) Show that both of the following integrals are correct, and explain.

(Lec 16, 6 pts: $3+3$)

a) Do 8.6/5 (answer in back of text)

b) Do 8.6/6 (optional?)

(Lec 16, 7 pts: $2+3+2$) Do 3F-5abc

The problem set prints only the reference; the exercise it points to, \textbf{3F-5 in Section E of the course reader}, is part of the course package, so its text is reproduced here (parts (a)–(c), the ones assigned):

\textbf{3F-5.} Air pressure satisfies the differential equation $dp/dh=-(0.13)p$, where $h$ is the altitude from sea level measured in kilometers.

a) At sea level the pressure is 1 kg/cm$^2$. Solve the equation and find the pressure at the top of Mt. Everest (10 km).

b) Find the difference in pressure between the top and bottom of the Green Building. (Pretend it's 100 meters tall starting at sea level.) Compute the numerical value using a calculator. Then use instead the linear approximation to $e^x$ near $x=0$ to estimate the percentage drop in pressure from the bottom to the top of the Green Building.

c) Use the linear approximation $\Delta p \approx p'(0)\Delta h$ and compute $p'(0)$ directly from the differential equation to find the drop in pressure from bottom to top of the Green Building. Notice that this gives an answer without even knowing the solution to the differential equation. Compare with the approximation in part (b). What does the linear approximation $p'(0)\Delta h$ give for the pressure at the top of Mt. Everest?

(Lec 18, 6 pts) Calculate $\int_0^1 e^x \, dx$ using lower Riemann sums. (You will need to sum a geometric series to get a usable formula for the Riemann sum. To take the limit of Riemann sums, you will need to evaluate $\lim_{n \to \infty} n(e^{1/n}-1)$, which can be done using the standard linear approximation to the exponential function.)

(Lec 16; extra credit: 10 pts: $2+2+3+3$) \textbf{More about the hypocycloid.} We use differential equations to find the curve with the property that the portion of its tangent line in the first quadrant has fixed length.

a) Suppose that a line through the point $(x_0, y_0)$ has slope $m_0$ and that the point is in the first quadrant. Let $L$ denote the length of the portion of the line in the first quadrant. Calculate $L^2$ in terms of $x_0$, $y_0$ and $m_0$. (Do not expand or simplify.)

b) Suppose that $y=f(x)$ is a graph on $0 \leq x \leq L$ satisfying $f(0)=L$ and $f(L)=0$ and such that the portion of each tangent line to the graph in the first quadrant has the same length $L$. Find the differential equation that $f$ satisfies. Express it in terms of $L$, $x$, $y$ and $y'=dy/dx$. (Hints: This requires only thought, not computation. Note that $y=f(x)$, $y'=f'(x)$. Don't take square roots, the expression using $L^2$ is much easier to use. Don't expand or simplify; that would make things harder in the next step.)

c) Differentiate the equation in part (b) with respect to $x$. Simplify and write in the form

(This starts out looking horrendous, but simplifies considerably.)

d) Show that one solution to the equation in part (c) is $x^{2/3}+y^{2/3}=L^{2/3}$. What about two other possibilities, namely, those solving $y''=0$ and $xy'-y=0$?