A series circuit contains a resistor and an inductor as shown in the figure. A rectangular circuit diagram has three items. The left side has a voltage source labeled E. The top has an inductor coil labeled L. The bottom has a resistor labeled R. Determine a differential equation for the current i(t) if the resistance is R, the inductance is L, and the impressed voltage is E(t). (Use i for i(t) and E for E(t)). di/dt =
R is the region bounded above by the function f(x) = x + 10 and below by the function g(x) = x/4 + 2 over the interval [a, b] where a = -6 and b = -2. Represent the area A of R by writing an integral with respect to x. You do not need to simplify.
Express the following statement in if-then form. Fix my ceiling or I won't pay my rent. If you fix my ceiling, then I won't pay my rent. If you don't fix my ceiling, then I will pay my rent. If you don't fix my ceiling, then I won't pay my rent. If I won't pay my rent, then you won't fix my ceiling. If I pay my rent, then you won't fix my ceiling.
For a poll, 43% of 21,944 people polled answered “yes” to the given question. Given that 43% is the best estimate of the population percentage, why would we need a confidence interval? That is, what additional information does the confidence interval provide? Choose the correct answer below. A. Information about the standard deviation of the population. B. Information about the mean of the population. C. Information about the accuracy of the estimate. D. Information about the mean of the sample.
Find the critical value zα/2 that corresponds to the given confidence level. 88% zα/2 = (Round to two decimal places as needed.) Find the critical value zα/2 that corresponds to α = 0.06. zα/2 = (Round to two decimal places as needed.)
For high-speed motion through the air - such as a skydiver shown in the figure, falling before the parachute is opened - air resistance is close to a power of the instantaneous velocity v(t). Determine a differential equation for the velocity v(t) of a falling body of mass m if air resistance is proportional to the square of the instantaneous velocity. Assume the downward direction is positive. (Use k > 0 for the constant of proportionality, g > 0 for acceleration due to gravity, and v for v(t).) dv/dt
R is the region bounded by the functions f(x) = 3x^2 + 27x + 54 and g(x) = - 3x^2/2 - 27x^2 - 27. Find the area A of R. Enter an exact answer.
Use truth tables to verify the following logical equivalences. Include a few words of explanation with your answers. (a) p→q ≡ ∼p ∨ q The truth table shows that p → q and ∼p ∨ q always do not always have the same truth values. Therefore they are are not logically equivalent. (b) ∼(p → q) ≡ p ∧ ∼q. (b) ∼(p → q) ≡ p ∧ ∼q. The truth table shows that ∼(p → q) and p ∧ ∼q always do not always have the same truth values. Therefore they are are not logically equivalent.
In the week before and the week after a holiday, there were 10,000 total deaths, and 4951 of them occurred in the week before the holiday. a. Construct a 90% confidence interval estimate of the proportion of deaths in the week before the holiday to the total deaths in the week before and the week after the holiday. b. Based on the result, does there appear to be any indication that people can temporarily postpone their death to survive the holiday? a. < p < (Round to three decimal places as needed.) b. Based on the result, does there appear to be any indication that people can temporarily postpone their death to survive the holiday? Yes, because the proportion could not easily equal 0.5. The interval is substantially less than 0.5 the week before the holiday. No, because the proportion could easily equal 0.5. The interval is not less than 0.5 the week before the holiday.
Write the negation, contrapositive, converse, and inverse for the following statement. (Assume that all variables represent fixed quantities or entities, as appropriate.) If the decimal expansion of r is terminating, then r is rational. (a) Negation The decimal expansion of r is terminating and r is not rational If the decimal expansion of r is not terminating, then r is rational If the decimal expansion of r is not terminating, then r is not rational. If r is rational, then the decimal expansion of r is terminating. If r is not rational, then the decimal expansion of r is not terminating (b) Contrapositive The decimal expansion of r is terminating and r is not rational If the decimal expansion of r is not terminating, then r is rational If the decimal expansion of r is not terminating, then r is not rational. If r is rational, then the decimal expansion of r is terminating. If r is not rational, then the decimal expansion of r is not terminating (c) Converse The decimal expansion of r is terminating and r is not rational. If the decimal expansion of r is not terminating, then r is rational. If the decimal expansion of r is not terminating, then r is not rational. If r is rational, then the decimal expansion of r is terminating If r is not rational, then the decimal expansion of r is not terminating (d) Inverse The decimal expansion of r is terminating and r is not rational If the decimal expansion of r is not terminating, then r is rational If the decimal expansion of r is not terminating, then r is not rational. If r is rational, then the decimal expansion of r is terminating If r is not rational, then the decimal expansion of r is not terminating
Reproduce the given computer-generated direction field. Then sketch an approximate solution curve that passes through each of the indicated points. dy dx = e -0.01xy^2
R is the region bounded by the functions f(x) = 5x/3 + 2 and g(x) = -2x + 5 over the interval [a, b] where a = 1 and b = 4. Represent the area A of R by writing an integral with respect to x. You do not need to simplify.
a. Write the following statements using and find the negation, the converse, the Inverse and the contrapositive If a natural number is even then it is divisible by two. Translate into symbols the following compound statement and give the form of the Compound statement. In each case, list the statements P, Q, R... b. If x is odd and y is odd then (x+1) is even and (y+1) is odd. c. It is raining but it is not hot.
Write each of the two statements in symbolic form and determine whether they are logically equivalent. Include a truth table and a few words of explanation to show that you understand what it means for statements to be logically equivalent. Let p = “2 is a factor of n”, q = “3 is a factor of n”, and r = “6 is a factor of n”. (a) If 2 is a factor of n and 3 is a factor of n, then 6 is a factor of n. p ∨ q ∼ p∧ ∼ q → r p∧ ∼ q p ∧ q → r p ∨ q → r (b) 2 is not a factor of n or 3 is not a factor of n or 6 is a factor of n. p∨ ∼ r ∼ p∧ ∼ q → r ∼ p∨ ∼ q ∨ r ∼ p∧ ∼ q ∧ r ∼ ρ ∧ r Construct a truth table for the statements. The truth table shows that p ∧ q → r and ∼ p∨ ∼ q ∨ r have the same truth values. Therefore they logically equivalent.
R is the region bounded above by the function f(x) = x + 4 and below by the function g(x) = -3x/4 - 1 over the interval [a, b] where a = -1 and b = 5. Represent R using the Desmos graph below. Submit your answer to this question by dragging the movable points so that the shaded region represents R.
If today is New Year's Eve, then tomorrow is January. (a) Negation If today is not New Year's Eve, then tomorrow is January. If today is not New Year's Eve, then tomorrow is not January. If tomorrow is January, then today is New Year's Eve. If tomorrow is not January, then today is not New Year's Eve. Today is New Year's Eve and tomorrow is not January. (b) Contrapositive If today is not New Year's Eve, then tomorrow is January. If today is not New Year's Eve, then tomorrow is not January. If tomorrow is January, then today is New Year's Eve. If tomorrow is not January, then today is not New Year's Eve. Today is New Year's Eve and tomorrow is not January. (c) Converse If today is not New Year's Eve, then tomorrow is January. If today is not New Year's Eve, then tomorrow is not January. If tomorrow is January, then today is New Year's Eve. If tomorrow is not January, then today is not New Year's Eve. Today is New Year's Eve and tomorrow is not January. (d) Inverse If today is not New Year's Eve, then tomorrow is January. If today is not New Year's Eve, then tomorrow is not January. If tomorrow is January, then today is New Year's Eve. If tomorrow is not January, then today is not New Year's Eve. Today is New Year's Eve and tomorrow is not January. Write the negation, contrapositive, converse, and inverse for the following statement. (Assume that all variables represent fixed quantities or entities, as appropriate.)
Assume that a random sample is used to estimate a population proportion p. Find the margin of error E that corresponds to the given statistics and confidence level. 95% confidence; the sample size is 2065, of which 20% are successes The margin of error E = . (Round to four decimal places as needed.)
R is the region bounded by the functions f(x) = 3 - 2cos(x) and g(x) = sin(x). Find the area of the region bounded by the functions on the interval [0, π/2]. Enter an exact answer.
Use the given data to find the minimum sample size required to estimate a population proportion or percentage. Margin of error: 0.03; confidence level 95%; pˆ and qˆ unknown n = (Round up to the nearest integer.)
Use the given data to find the minimum sample size required to estimate a population proportion or percentage. Margin of error: eight percentage points; confidence level 95%; from a prior study, pˆ is estimated by the decimal equivalent of 52% n = (Round up to the nearest integer.)
Write the integral required to find the area of the region pictured below that is bounded by the curves u(y) = √y + 3 and v(y) = y+3 / 4 when integrating with respect to y. Do not evaluate the integral.
Consider the following statement. If x is nonnegative, then x is positive or x is 0. In (a)-(d) below, select the negation, contrapositive, converse, and inverse for the statement. (Assume that all variables represent fixed quantities or entities, as appropriate.) (a) Negation If x is not positive and x is not 0, then x is not nonnegative. If x is positive or x is 0, then x is nonnegative. If x is not nonnegative, then both x is not positive and x is not 0. If x is not nonnegative, then x is positive or x is 0. x is nonnegative and x is not positive and x is not 0. (b) Contrapositive If x is not positive and x is not 0, then x is not nonnegative. If x is positive or x is 0, then x is nonnegative. If x is not nonnegative, then both x is not positive and x is not o. If x is not nonnegative, then x is positive or x is 0. x is nonnegative and x is not positive and x is not 0. (c) Converse If x is not positive and x is not 0, then x is not nonnegative. If x is positive or x is 0, then x is nonnegative. If x is not nonnegative, then both x is not positive and x is not 0. If x is not nonnegative, then x is positive or x is 0. x is nonnegative and x is not positive and x is not 0. (d) Inverse If x is not positive and x is not 0, then x is not nonnegative. If x is positive or x is0, then x is nonnegative. If x is not nonnegative, then both x is not positive and x is not 0. If x is not nonnegative, then x is positive or x is 0. x is nonnegative and x is not positive and x is not 0.
A clinical trial tests a method designed to increase the probability of conceiving a girl. In the study 600 babies were born, and 330 of them were girls. Use the sample data to construct a 99% confidence interval estimate of the percentage of girls born. Based on the result, does the method appear to be effective? < p < (Round to three decimal places as needed.) Does the method appear to be effective? Yes, the proportion of girls is significantly different from 0.5. No, the proportion of girls is not significantly different from 0.5.
A genetic experiment with peas resulted in one sample of offspring that consisted of 445 green peas and 153 yellow peas. a. Construct a 90% confidence interval to estimate of the percentage of yellow peas. b. It was expected that 25% of the offspring peas would be yellow. Given that the percentage of offspring yellow peas is not 25%, do the results contradict expectations? a. Construct a 90% confidence interval. Express the percentages in decimal form. < p < (Round to three decimal places as needed.) b. Given that the percentage of offspring yellow peas is not 25%, do the results contradict expectations? No, the confidence interval includes 0.25, so the true percentage could easily equal 25% Yes, the confidence interval does not include 0.25, so the true percentage could not equal 25%
In (14) of Section 1.3 we saw that a differential equation describing the velocity v of a falling mass subject to air resistance proportional to the instantaneous velocity is mdv/dt = mg - kv, where k > 0 is a constant of proportionality. The positive direction is downward. (a) Solve the equation subject to the initial condition v(0) = v0: v(t) = (b) Use the solution in part (a) to determine the limiting, or terminal, velocity of the mass. (c) If the distance s, measured from the point where the mass was released above ground, is related to velocity v by ds/dt = v(t), find an explicit expression for s(t) if s(0) = 0. s(t) =