MIT 18.01 Problem Set 1

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

(Wed, 3 pts) Express $(x-1)/(x+1)$ as the sum of an even and an odd function. (Simplify as much as possible.)

(Thurs, 6 pts: $3+3$) Sensitivity of measurement: Suppose $f$ is a function of $x$. If $x=x_0+\Delta x$, then we define $\Delta f=f(x)-f(x_0)$ and $\Delta f/\Delta x$ measures how much changes in $x$ affect the value of $f$.

The planet Quirk is flat. GPS satellites hover over Quirk at an altitude of 20,000 km (unlike Earth where the satellites circle twice a day). See how accurately you can estimate the distance $L$ from the point directly below the satellite to a point on the planet surface knowing the distance $h$ from the satellite to the point on the surface in two cases. (The letter $h$ is for hypotenuse.)

a) Use a calculator to compute $\Delta L/\Delta h$ for $h=h_0 \pm \Delta h=25{,}000 \pm \Delta h$, and $\Delta h=1,\ 10^{-1},\ 10^{-2}$. Write an estimate for $L$ in the form

choosing the simplest round number $C$ that works for all three cases.

b) Do the same for $h=20{,}001 \pm \Delta h$, $\Delta h=1,\ 10^{-1},\ 10^{-2}$. Is the value of $L$ estimated more or less accurately than in part (a)? We will revisit this problem more systematically using calculus.

(Thurs, 4pts) On the planet Quirk, a cell phone tower is a 100-foot pole on top of a green mound 1000 feet tall whose outline is described by the parabolic equation $y=1000-x^2$. An ant climbs up the mound starting from ground level ($y=0$). At what height $y$ does the ant begin to see the tower?

(Thurs, 6 pts) 3.1/21 (parabolic mirrors)

(Thurs, 4pts: $2+2$)

a) A water cooler is leaking so that its volume at time $t$ in minutes is $(10-t)^2/5$ liters. Find the average rate at which water drains during the first 5 minutes.

b) At what rate is the water flowing out 5 minutes after the tank begins to drain.

Friday (8 pts: $1+1+1+1+1+1+2$) 2.5/19d (put $u=1/x$), 19f, 19g, 20c, 20g (show work); 22a (needs a calculator), 22b (see the proof on page 73).

Tuesday (6 pts: $2+4$)

a) If $u$, $v$ and $w$ are differentiable functions, find the formula for the derivative of their product, $D(uvw)$.

b) Generalize your work in part (a) by guessing the formula for $D(u_1 u_2 \cdots u_n)$—the derivative of the product of $n$ differentiable functions.

Then prove your formula by mathematical induction (i.e., prove its truth for the product of $n+1$ functions, assuming its truth for the product of $n$ functions).