MIT 18.01 Problem Set 6

This lesson reproduces the Problem Set “MIT 18.01 Problem Set 6” 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 5 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 22, 7 pts: 3+4) Do 7.4/12 and 13. (Course notation: 7.4/12 means Section 7.4, Problem 12 of the textbook, Simmons, \emph{Calculus with Analytic Geometry}, 2nd ed.)

The voltage $V$ of house current is given by

where $t$ is time, in seconds and $C$ is a constant amplitude. The square root of the average value of $V^2$ over one period of $V(t)$ (or cycle) is called the \emph{root-mean-square} voltage, abbreviated RMS. This is what the voltage meter on a house records. For house current, find the RMS in terms of the constant $C$. (The peak voltage delivered to the house is $\pm C$. The units of $V^2$ are square volts; when we take the square root again after averaging, the units become volts again.)

a) What is the probability that $x^2 < y$ if $(x, y)$ is chosen from the unit square $0 \leq x \leq 1$, $0 \leq y \leq 1$ with probability equal to the area.

b) What is the probability that $x^2 < y$ if $(x, y)$ is chosen from the square $0 \leq x \leq 2$, $0 \leq y \leq 2$ with probability \textbf{proportional} to the area. (Probability=Part/Whole).

c) Evaluate

This is known as an improper integral because it represents the area of an unbounded region. We are using the letter $W$ to signify "whole".

The probability that a radioactive particle will decay some time in the interval $0 \leq t \leq T$ is

Note that $P([0, \infty))=1=100\%$.

d) The half-life is the time $T$ for which $P([0, T])=1/2$. Find the value of $a$ and $W$ for which the half-life is $T=1$. Suppose that a radioactive particle has a half-life of 1 second. What is the probability that it survives to time $t=1$, but decays some time during the interval $1 \leq t \leq 2$? (Give an integral formula, and use a calculator to get an approximate numerical answer.)

The basis for Simpson's rule is the following formula. Let $x_1$ be the midpoint of the interval $[x_0, x_2]$, and denote its length by $2h$. Consider any three points $(x_0, y_0)$, $(x_1, y_1)$, $(x_2, y_2)$. There is a unique quadratic function (parabola)

whose graph passes through the three points. Simpson's rule says that the area under the parabola above $[x_0, x_2]$ is

This problem is devoted to proving this formula. It is significant because it illustrates how calculations can be simplified by using symmetry, and by looking ahead to see what you need.

Since the area will be the same if the parabola is translated to the left or right, we may assume that $x_0=-h$, $x_1=0$, and $x_2=h$. Then in terms of the rest of the data (i.e., $h$ and the $y_i$)

• make a sketch and determine $C$;
• show, by integrating, that to find the area we need only determine $A$ (or better, $2Ah^2$);
• determine $2Ah^2$ using the data;
• put the results together to establish the formula for area.

Use a calculator to make a table of values of the integrand and find approximations to the Fresnel integral $\int_0^a \cos(t^2)\,dt$ for $a=\sqrt{\pi/2}$, using Simpson's rule with four and eight intervals. (The exact answer to five decimal places is 1.22505. Record your approximations to six decimal places to compare.)