Limits & Continuity Questions and Answers

Find the minimum of 3X+Y(3X-1+3Y-1-1) There is a hint given by teacher that is multiplying by 3 and add 1.

Find the values of a and b such that the following function 4a cos(x + 3)- b. 2x+5, if x < -3, f(x) = -4, if x = -3, ax² + bx - 7, is continuous at x = -3. Justify your answer by the definition of the continuity. if x > -3,

4. Extra Credit. (10 Points.) Consider the problem of solving for functions u(x, y), defined on x² + y² ≤ 1, such that (i) Au = 0, and (ii) u = 0 at all points in the boundary x² + y² = 1. Show that the only solution to this problem is the zero function: u(x, y) = 0. Hint: Consider the vector field F = uvu. Can you make the same conclusion if we replace replace condition (ii) with "Vun = 0 at all points on the boundary" (so the function becomes "flat" as you approach the boundary). If not, what is a conclusion about u you can make? Remark: The two "boundary conditions" in this problem are two very common ones you'll find in differen- tial equations.

Consider the angle depicted in standard position below. P(-7,9) Use the figure above to determine the following. Give exact values. (x,y) : ( y (a) cos(0) = (b) The coordinates of the terminal side of 0+ (c) tan(0+) = )

In Exercises 87-92, find the difference quotient and simplity S Slon your answer. 88. g(x) = 3x - 1, C c#0 - g(x + h) = g(x) him h #0 2009072

Date: Period Calculus AB Use the rectangles in the following graph to approximate the area of the region bounded by 4 y=y=0, x= 1, and x = 4 Round your final answer the nearest thousandth, square units

Determine the tax free Roc, capital Gain, Dividend, End basis, End basis shareholder basis= 130,000 AAA= 450,000 E&P= 350,000 Distribution=300,000

Solve the system of linear equations: y = 3x - 4 y = -2x + 1

Find the limit. (If an answer does not exist, enter DNE.) x² - 7x + 10 x-1 X-1

y = xe-2x, 0≤x≤ In(3) On this interval, the maximum value(s) occur at x = In(3) two of these are correct 0 1 2

Write the augmented matrix of the given system of equations. x - 4y = 9 6x + 7y = 4 The augmented matrix is

Prove that lim (5x-3)= 7, using the e-8 definition.

Two unfair coins, both favoring heads, are tossed simultaneously for many trials, with the results shown to the right. a) What is the probability that both coins show heads? b) What is the probability that at least one of the coins shows heads? c) What is the probability that the two coins show the same side? ... a) How is the probability that both coins show heads found? A. Find the sum of the probabilities of "two heads" and "two tails." B. Find the sum of the probabilities to "two heads" and "one head, one tail." C. Divide the frequency of "two heads" by the total. D. Subtract the probability of "two tails" from 1. A. P(at least one H) = P(two H) + P(one H, one T) - P(two T) B. P(at least one H) = 1 (P(two H) + P(one H, one T)) C. P(at least one H) = P(two H) + P(two T) - P(one H, one T) D. P(at least one H) = 1- P(two T) Outcome Frequency 61 103 34 Two heads One head, one tail Two tails The probability that both coins shown heads is (Simplify your answer.) b) Let H represents heads and T represent tails. What formula should be used to find the probability that at least one of the coins shows heads?

Find the value of the constant c so that the function below is a probability density function for a random variable over the specified interval. 1 f(x) = = x over [2,c]

52. Find m/ADC in Figure 20. Round to the nearest tenth. B 8 60° 9 10 C D M

What are the cylindrical coordinates of the point whose spherical coordinates are (p, 0, d) – (3, – 2,)? (r,0,2)=(

A spinner marked off in 5 equal-size regions, numbered 1 through 5, is spun twice. Let event E be that the outcome of both spins is 4 and event F be that the sum of the two outcomes is 7. a) Write the sample space, S. Use ordered pairs such as (2,3), and so on. b) Find P(E) and P (E'). c) Find P(F) and P (F'). d) Find P(EUF). OC. S= {(1,1), (1,2), (1,3), (1,4), (1,5), (2,2), (2,3), (2,4), (2,5), (3,3), (3,4), (3,5), (4,4), (4,5), (5,5)} D. S = {(1,2), (1,3), (1,4), (1,5), (2,1), (2,3), (2,4), (2,5), (3,1), (3,2), (3,4), (3,5), (4,1), (4,2), (4,3), (4,5), (5,1), (5,2), (5,3), (5,4)} b) P(E)= (Simplify your answer.) P(E') = (Simplify your answer.) c) P(F) = (Simplify your answer.) P(F') = (Simplify your answer.) d) P(EUF) = (Simplify your answer.)

Find the limit: Factor and simplify before substituting x by the value c: 2x²-5x+2 lim 2x-1

Evaluate the limit, if it exists. (If an answer does not exist, enter DNE.) lim x cos X-0

Use transformations to graph the function f(x) = -2+4*-2. Determine its domain, range, and horizontal asymptote. Which transformations must be applied to obtain the graph of f(x) = -2+4*-2? Select all that apply. A. Horizontal shift B. Reflection about the x-axis C. Horizontal compression or stretch D. Vertical shift E. Vertical compression or stretch F. Reflection about the y-axis Use the graphing tool to graph the function. * Click to enlarge graph What is the domain of f(x) = -2+4*-22

A graphing calculator is recommended. Use the Squeeze Theorem to show that lim x cos(9x) = 0. x10 Illustrate by graphing the functions f(x) = -x², g(x) = x² cos(9x), and h(x) = x² on the same screen. Let f(x) = -x², g(x) = x² cos(9x), and h(x) = x². Then ?s cos(9x) ≤ ? ✓ = 2≤ x² cos(9x) ≤ 2 ✓ Since lim x)= lim h(x) = X-0 X10 by the Squeeze Theorem we have lim g(x) = 110

A spinner marked off in four equal-sized regions, numbered 1, 2, 3, and 4, is spun twice. Let the random variable X be the nonnegative difference between the numbers of the regions where the pointer stops, and define a probability mass function g(x)=P(X=X). a) State the range of X. b) Write g as a set of ordered pairs. c) Draw a probability distribution as a histogram. d) Find g(3), and explain what this number represents. e) Find µ, and explain what this number represents. a) What is the range of X? OA. (1, 2, 3, 4, 5) OB. (1,2,3) O C. (0, 1, 2, 3) OD. (0, 1, 2, 3, 4, 5) b) Write g as a set of ordered pairs. Choose the correct answer below. 4 6 2 (0) (¹) 1, (2) (376) 16 16 16 OA. OB. O C. 4 2 (1.6) (2. 6) (3, ²76) 16 16 4 4 2 (¹6) (2) 96) (3, 76). (4, ²76) 16 16 16 16 Next

1 ->(x+6). Consider the functions f(x) = 7x-6 and g(x)= (a) Find f(g(x)). (b) Find g(f(x)). (c) Determine whether the functions f and g are inverses of each other. (a) What is f(g(x))? f(g(x)) = (Simplify your answer.) Give any values of x that need to be excluded from f(g(x)). Select the correct choice below and fill in any answer boxes within your choice. A. x# (Use a comma to separate answers as needed.) B. No values should be excluded from the domain. (b) What is g(f(x))? g(f(x)) = (Simplify your answer.) Give any values of x that need to be excluded from g(f(x)). Select the correct choice below and fill in any answer boxes within your choice. OA. x# (Use a comma to separate answers as needed.)

Evaluate the limit using the appropriate Limit Law(s). (If an answer does not exist, enter DNE.) 4-5 lim t-2 21²-3t+8

Express the following set in set-builder notation. (8, 10, 12, 14, 16, 18, 20) Select the correct answer below and fill in the answer boxes within your choice. OA. {x|x is an even integer and SXS OB. {x|x is an integer and SXS OC. (x | x is an odd integer and SXS

#64 An airplane flies 220 miles with a heading of 40°, and then flies 180 miles with a heading of 170°. How far is the plane from its starting point? Round answers to the nearest tenth. Based on the picture: α = 40 B = LB = 50 CVV OF av of The plane is 172.9 A of degrees degrees = 220 miles = 180 miles 40° o degrees C b 4 B 170° a miles from its starting point. C

1 Consider the functions f(x) = 7x − 6 and g(x) = (x + 6). (a) Find f(g(x)). (b) Find g(f(x)). (c) Determine whether the functions f and g are inverses of each other. (a) What is f(g(x))? f(g(x)) = (Simplify your answer.) Give any values of x that need to be excluded from f(g(x)). Select the correct choice below and fill in any answer boxes within your choice. OA. x# (Use a comma to separate answers as needed.) OB. No values should be excluded from the domain. (b) What is g(f(x))? g(f(x)) = (Simplify your answer.) Give any values of x that need to be excluded from g(f(x)). Select the correct choice below and fill in any answer boxes within your choice. OA. x# (Use a comma to separate answers as needed.) OR No values should be excluded from the domain

An article contains 1297 words. The frequencies of words with various lengths are given in the table. A word in the article is randomly chosen. Using the information in the table, find the following. a) The probability that the word has 4 letters. b) The probability that the word has 5 or fewer letters. c) The probability that the word has 8 or more letters. Click the icon to view the table. a) How is the probability that the word has 4 letters found? OA. Subtract the probability of 4 letters from 1. OB. Divide the frequency of 4 letters by the total. OC. Divide the total by the frequency of 4 letters. OD. Find the sum of the probabilities of 1 letter, 2 letters, and 3 letters The probability that the word has 4 letters is (Simplify your answer.) b) How is the probability that the word has 5 or fewer letters found? OA. Divide the total by the frequency of 5 letters. OB. Find the sum of the probabilities of 1 letter, 2 letters, 3 letters, 4 letters, and 5 letters. OC. Divide the frequency of 5 letters by the total. OD. Subtract the probability of 5 letters from 1. The probability that the word has 5 or fewer letters is GEILE Data table Length (number of letters) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Frequency 34 220 287 166 175 121 86 72 68 36 16 8 5 0 3 W

11. Find the limits algebraically:lim sin t t-0 St

In September (30 days), the temperature in a particular city reached 100°F on 24 days. Also, measureable rain fell at the city's airport on 5 days, and on 2 days, both events occurred. a) On how many days did at least one of the events occur? b) On how many days did the temperature reach 100°F without measurable rain? c) On how many days did the temperature not reach 100°F and rain did not fall? a) At least one of the events occurred on (Simplify your answer.) day(s). b) The temperature reached 100°F without measurable rain on (Simplify your answer.) c) The temperature did not reach 100°F and rain did not fall on (Simplify your answer.) day(s). day(s). ...

A test has many multiple-choice items, each with four answer choices. A student receives 1 point for every correct answer, and each incorrect answer brings a penalty 1 (loss) of of a point. If Chris guesses on every question, what score can he expect on the test? 3 Chris can expect to get a score of (Simplify your answer.) .…..

Express the following set using set-builder notation. {March, May} Choose the correct answer below. A. {x|x is a month in the first quarter of the year} B. {x|x is a month of the year} C. {x|x is a month that begins with M} OD. The set has no elements.

The function f(x) = 6x is one-to-one. (a) Find the inverse of f and check the answer. (b) Find the domain and the range of f and f (c) Graph f, f1, and y=x on the same coordinate axes. X .. (a) f(x)=y= (Simplify your answer. Use integers or fractions for any numbers in the expression.) (b) Find the domain of f. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. OA. The domain is {x|xs. B. The domain is {x|x*}. OC. The domain is {x|x²}. OD. The domain is the set of all real numbers. Find the range of f. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

5. Evaluate the limits and justify each step:lim h-0 h www Messe

Julia invests a sum of money in a retirement account with a fixed annual interest rate of 6% compounded continuously. After 17 years, the balance reaches $14,262.54. What was the amount of the initial investment?

For the following exercises, determine whether the graphs of the polar equation are symmetric with respect to the x -axis, the y -axis, or the origin. r² = 6 cos(20) 2 x-axis y-axis origin

A random inspection of 100 vehicles showed that 67 had tires with low pressure and 35 had tires with a bald tread. Furthermore, 20 had tires with both low pressure and bald tread. a) How many vehicles had tires with low pressure or bald tread? b) How many vehicles had tires with good tread? c) How many vehicles had tires with good tread and good pressure? a) The number of vehicles having tires with low pressure or bald tread is (Simplify your answer.) b) The number of vehicles having tires with good tread is (Simplify your answer.) c) The number of vehicles having tires with good tread and good pressure is (Simplify your answer.)

Find dy/dx by implicit differentiation. 7²-7y² = 16 dy/dx = X

Consider the function A. The function f(x) has vertical asympototes at x= B. f(x) is concave up on the region C. The inflection point for this function is the direction of concavity also changes at on the interval [-20, 15]. Answer the following questions for f (enter points in ascending order, i.e. smallest x values first; enter intervals in ascending order also): to and f(x) = and x3 x²-1 and x= and to where the function is undefined.

For a week, a clothing company tracks the amounts spent by its customers, with the results shown to the right. a) What is the probability that a randomly chosen customer spent $120 or more? b) What is the probability that a randomly chosen customer did not spend less than $80? c) What is the probability that a randomly chosen customer spent between $40 and $159.99? a) What formula should be used to find the probability that a randomly chosen customer spent $120 or more? A. P($120 or more) = P($120-$159.99) B. P($120 or more) = 1 - P($120 - $159.99) O C. P($120 or more) = P($120-$159.99) + P($160-$199.99) + P($200 or more) O D. P($120 or more) = P($120-$159.99) + P($160 - $199.99) - P($200 or more) The probability that a randomly chosen customer spent $120 or more is (Simplify your answer.) b) What formula should be used to find the probability that a randomly chosen customer did not spend less than $80? A. P(not less than $80)=P($0-$39.99)+ P($40-$79.99) + P($80 - $119.99) B. P(not less than $80)=P($80 - $119.99) DIADO 6440 001 Amount spent Frequency 37 57 94 97 42 11 $0-$39.99 $40-$79.99 $80-$119.99 $120-$159.99 $160 $199.99 $200 or more

A spinner marked off in eight equal-sized regions, numbered 1, 2, 3, 4, 5, 6, 7, and 8, is spun once. Let the random variable X be the number of the region where the pointer stops, and define a probability mass function f(x)=P(X=x). a) State the range of X. b) Write f as a set of ordered pairs. c) Draw the probability distribution as a histogram. d) Find f(8), and explain what this number represents. e) Find μ, and explain what this number represents.

Part 1 of 4 A box with a square base and open top must have a volume of 415292 cm³. We wish to find the dimensions of the box that minimize the amount of material used. First, find a formula for the surface area of the box in terms of only a, the length of one side of the square base. A(x)= = I Question Help: 55 Video & Written Example 2 Message instructor

A survey of 100 people showed that 26 own a car, 62 own a bicycle, and 12 own both a car and a bicycle. How many people own a car or a bicycle? The number of people that own a car or a bicycle is (Simplify your answer.) ...

Use roster form to write the elements of the following set, or state that the set has no elements. The seasons Choose the correct answer below. OA. (Spring, Summer} B. (Fall, Winter} C. (Spring, Summer, Fall, Winter} D. (Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday} OE. Ø

7. Evaluate the limits and justify each step: lim √x-12 x-144 x-144

Use continuity to evaluate the limit. lim xv 13-x X-3

The coefficient of xkyk in the expansion of (x + y)" equals OA. True B. False k n

Suppose the heights of the members of a population follow a normal distribution. If the mean height of the population is 68 inches and the standard deviation is 4 inches, 99.7% of the population will have a height within which of the following ranges? OA. 56 inches to 80 inches OB. 60 inches to 76 inches C. 64 inches to 72 inches D. 52 inches to 84 inches

If fis continuous on (-00, 00), what can you say about its graph? (Select all that apply.) The graph of f has a hole. The graph of f has a jump. The graph of f has a vertical asymptote. Onone of these

The grap the relation S is shown below. -3 Give the domain and range of S. Write your answers using set notation. domain = range = 00 2 0