Limits & Continuity Questions and Answers

In(1/200) Step 1 Apply the logarithmic property, In(a/b) = In(a) – In(b), to simplify. In(1/200)=ln ) - ln(200 Write the factors of 200. In(1/200) = lnn(1) - In(8 )

Find the mass and center of mass of the lamina that occupies the region D and has the given density function p. D = {(x, y) | 1 ≤ x ≤ 9, 1 ≤ y ≤ 4}; p(x, y) = ky² m = (x, y) =

Use long division to find the quotient and remainder. (3x² + 25x+28) = (x + 7) What is the quotient? What is the remainder?

Write the factors of 40, In(40) = In(8 x )

After the announcement of a sale, a bookstore sold % of all its books in stock. One the following day, this bookstore sold 4,000 more books. Now only 1/10 of the number of books in stock before the sale remain in the store. How many books were in stock before the announcement of the sale? a. 8,000 b. 10,000 c. 12,000 d. 15,000 e. 20,000

Use the given polynomial function to answer the questions. What is the degree of the polynomial? ƒ (x) = −x² (3x − 2)² What is the leading coefficient? Which describes the end behavior of the function? As x→-∞, f(x) →→∞; as x →∞, f(x) →→∞ As →-∞o, f(x) →∞o; as x →∞o, f(x) → ∞ As x→-∞, f(x) → -∞; as x → ∞, f (x) → ∞ As a →-∞, f(x) → ∞; as x → ∞o, f(x) → -∞

What kind of transformation converts the graph of f(x) = 2x + 9 into the graph of g(x) = 8x + 9? horizontal stretch vertical shrink vertical stretch horizontal shrink

Let v > 1. Show that 8v - 8 v - 1 > 8 ln 8 Justify your answer. Hint: Apply the Mean Value Theorem to f(x) = 8x on [1, v].

The pH of a solution ranges from 0 to 14. An acid solution has a pH less than 7. Pure water is neutral and has a pH of 7. The pH of a solution is given by pH = log x, where x represents the concentration of the hydrogen ions in the solution in moles per liter. Find the pH of a solution if its hydrogen ion concentration is 5.9 × 10-4 moles per liter. Round to the nearest hundredth.

Consider the equation below. (If an answer does not exist, enter DNE.) f(x) = 5 cos²(x) - 10 sin(x), 0≤x≤2π (a) Find the interval on which f is increasing. (Enter your answer using interval notation.) Find the interval on which f is decreasing. (Enter your answer using interval notation.) (b) Find the local minimum and maximum values of f. local minimum value local maximum value (c) Find the inflection points. (Order your answers from smallest to largest x, then from smallest to largest y.) (x, y) = (x, y) = Find the interval on which fis concave up. (Enter your answer using interval notation.) Find the interval on which f is concave down. (Enter your answer using interval notation.)

Use Newton's method with initial approximation x1 = -1 to find x₂ the second approximation to the solution of the following equation. x³ + x + 7 = 0

A radioactive material has a decay rate of 0.019% per year. Suppose that a nuclear accident causes this material to be released into the atmosphere perpetually at the rate of 3 lb each year. What is the limiting value of the radioactive buildup? The limiting value of the radioactive buildup is lb.

For the probability density function, over the given interval, find E(x), E(x2), the mean, the variance, and the standard deviation. f(x) = 1 18x2, [-3,3]

A number x is selected at random from the interval [4,20]. The probability density function for x is given by the following function. Find the probability that a number selected is in the subinterval [8,17]. f(x) = 1 / 16, for 4 ≤ x ≤ 20. How is the probability that a number selected is in the subinterval [8,17] calculated?

Fill in the following blanks with the greatest digit that makes each statement true. a. 3 | 53_ b. 9 | 809_7 c. 11 | 54_9 d. 5 | 347_ e. 6 | 4_18 f. 8 | 425

Use the graph of f in the figure to the right to complete parts (A) through (M) (assume that f''(0) <0, f''(b)>0, and f''(g) > 0). (L) Find the horizontal asymptote(s). Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one horizontal asymptote, (Type an equation.) B. The function has two horizontal asymptotes. The top asymptote is and the bottom asymptote is (Type equations.) C. The function has no horizontal asymptotes. (M) Find the vertical asymptote(s). Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one vertical asymptote, (Type an equation.) B. The function has two vertical asymptotes. The leftmost asymptote is and the rightmost asymptote is (Type equations.) C. The function has no vertical asymptotes.

A survey found that 10 out of 15 students like pizza. If four students are chosen at random what is the probability that all the four students like pizza?

Of all numbers whose difference is 2, find the two that have the minimum product. What are the two numbers?

A group of researchers wanted to measure the impact of playing classical music at bedtime to infants. They choose 20 infants to play music from 6:30 - 7:30 pm at bedtime and 20 infants to not play music at all. For each day for 3 months they observed them for 4 hours and recorded exactly how long it took for them to fall asleep. At the end of the 3 months they calculated the percentage of awake time to the time they slept for the 4 hours and compared the percentages to infants that had music played versus infants that did not. Experimental Survey Observational Study None of the answer choices

Monthly sales of a particular computer are expected to decline at the following rate of S'(t) computers per month, where t is time in months and S(t) is the number of computers sold each month. S'(t)= - 40t2/3 - 80 The company plans to stop manufacturing this computer when monthly sales reach 700 computers. If monthly sales now (t = 0) are 2,140 computers, find S(t). Use a graphing calculator to approximate the solution of the equation S(t) = 700. S(t) = Use a graphing calculator to approximate the solution of the equation S(t) = 700. t≈ (Round to two decimal places as needed.)

For the quadratic function f(x)=-(x+4)²-3, determine the largest open interval of the domain (a) over which the function is increasing and (b) over which the function is decreasing. A) The largest open interval of the domain over which the function is increasing is. B) The largest open interval over which the function is decreasing is

Find the following product, and write the result in rectangular form using exact values. (8 cis 30°)(3 cis 195°)

Consider the exponential function f(x) = 10.8(0.65)x. Using a graphing calculator, determine the function's half-life (the change in z for the output value to cut in half).

Solve the following system of equations using the substitution method. 2x - y = 4 (1) 2y = 4x-8 (2) What is the solution of the system? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution of the system is (Simplify your answer. Type an ordered pair.) B. There are infinitely many solutions. C. There is no solution.

Solve by the elimination method. Also, determine whether the system is consistent or inconsistent, and whether the equations are dependent or independent. 7/2x + 7/3y = 259/6 1/4x+1/3y=47/12 (Hint: First multiply by the least common denominator to clear fractions.) Select the correct choice below and fill in any answer boxes in your choice A. There is one solution. The solution is (Simplify your answer. Type an ordered pair.) B. There are infinitely many solutions of the form (x). (Simplify your answer.) C. There is no solution. Is the system consistent or inconsistent? inconsistent consistent Are the equations dependent or independent? dependent independent

For the graph of y = f(x) shown to the right, find the absolute minimum and the absolute maximum over the interval [1,11]. Identify the absolute minimum. Select the correct choice below fill in any answer boxes within your choice. A. The absolute minimum is at x = and x = (Round to the nearest integer as needed. Use ascending order.) B. The absolute minimum is at x = (Round to the nearest integer as needed.) C. There is no absolute minimum. Identify the absolute maximum. Select the correct choice below fill in any answer boxes within your choice. A. The absolute maximum is at x = and x = (Round to the nearest integer as needed. Use ascending order.) B. The absolute maximum is at x = (Round to the nearest integer as needed.) C. There is no absolute maximum.

Solve using the elimination method. 0.2x-0.3y = -0.4 0.3x-0.2y = -0.6 (Hint: Since each coefficient has one decimal place, first multiply each equation by 10 to clear the decimals.) Select the correct choice below and fill in any answer boxes in your choice. A. The solution is The system is consistent and independent. (Type an ordered pair.) B. There are infinitely many solutions in the form (x): The system is consistent and dependent. C. There is no solution. The system is inconsistent and independent.

Write the function as a sum of terms of the form axn, where a is a constant. f(x)=10/x4

Suppose f(x) = -1/2 Determine and graph four distinct antiderivatives of f.

A study of consumer smoking habits includes 193 people in the 18-22 age bracket (60 of whom smoke), 127 people in the 23-30 age bracket (35 of whom smoke), and 96 people in the 31-40 age bracket (23 of whom smoke). If one person is randomly selected from this sample, find the probability of getting someone who is age 23-30 or smokes. A. 0.084 B. 0.276 C. 0.589 D. 0.505

The Topology Taxi Company charges 2.50 for the first quarter of a mile and 0.45 for each additional quarter of a mile. Find a linear function which models the taxi fare F as a function of the number of miles driven, m.

The following involves a trigonmetric equation in quadratic form. Solve the equation on the interval [0,2π). 6 tan2 x-18=0

Solve the inequality and enter the solution set in interval notation. Type infinity for and negative infinity for -∞. Do not enter blank spaces in your answer. 12x - 48 ≥ 0 Type your answer and submit

Find the exact value of cos θ. sin θ= -12/13, π<θ<3π/2

Let f (x) = 3 (1/2) ². Evaluate the following: f(-1) = f(0) = f(2)= Is f(x) increasing or decreasing? (Hint: Type in "inc" for increasing and "dec" for decreasing.)

Let f(x) = 2(3)x. Evaluate the following: f(-1) = f(0) = f(2)= Is f(x) increasing or decreasing? (Hint: Type in "inc" for increasing and "dec" for decreasing.)

An investment of $45,000 earns 6% annual interest and is compounded semiannually. If no funds are added or removed from this account, what is the future value of the investment after 3 years? Round your answer to the nearest penny.

Graphical Behavior of the Log to the Base 10 (Common Log) Function: a. Plot both the exponential function f(x) = 10 * and the logarithmic function g(x) = log10 x in Mathematica. (Notation: In Mathematica the notation for log10 X is Log[10,x].) b. Is f(x), the exponential function, increasing or decreasing? faster and faster or slower and slower? c. Is g(x), the logarithmic function, increasing or decreasing? faster and faster or slower and slower?

Suppose you have a non-right triangle such that angle A is opposite of side a, angle B is opposite of side b, and angle C is opposite of side c. If side b = 3.3, side c = 5.6, and angle A = 87, solve the triangle for the missing values: Side a = Angle B = Angle C = Note: Round your answers to 2 places after the decimal when applicable

Find the domain and intercepts. f(x) =42/x-3 A. The x-intercept(s) of the graph is (are) x = (Simplify your answer. Type an integer or a decimal. Use a comma to separate answers as needed.) B. There is no x-intercept. Find the y-intercept(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The y-intercept(s) of the graph is (are) y = (Simplify your answer. Type an integer or a decimal. Use a comma to separate answers as needed.) B. There is no y-intercept.

Find all real solutions of the equation (x - 5)² = 36. x1 = and x2 = with x1<x2

The temperature of a pond has been fluctuating abnormally over the past three days. A computer program has calculated the deviation from the normal temperature to follow the function Δ(t)=t²-1 in degrees Celsius per day. We want to calculate the total deviation from the norm over the past three days.

The polynomial function fis defined by f(x) = −2x²-3x³+4x²+5x+2. Use the ALEKS graphing calculator to find all the points (x, f(x)) where there is a local minimum Round to the nearest hundredth. If there is more than one point, enter them using the "and" button.

Suppose that the supply and demand equations for printed T-shirts for a particular week are given as follows, where p is the price in dollars and q is the quantity in hundreds. p = 0.9q + 1 Price-supply equation p= -1.9q + 18 Price-demand equation Answer parts (A) through (D) below. (A) Find the supply and demand (to the nearest unit) if T-shirts are $5 each. To use the price-supply and price-demand equations to find the requested quantities, substitute $5 for The number of units supplied or demanded will be given by The supply when T-shirts are $5 is units. (Round to the nearest whole number as needed.)

Below is the graph of y = 3x Translate it to become the graph of y=3x-4 +2.

A principal of $3000 is invested at 6.5% interest, compounded annually. How much will the investment be worth after 13 years? Use the calculator provided and round your answer to the nearest dollar.

The student council at a college is made up of five freshmen, six sophomores, seven juniors, and eight seniors. A yearbook photographer would like to line up two council members from each class for a picture. How many different pictures are possible if each group of classmates stands together? Which expression below results in the number of different pictures that are possible? A. C(5.2).C6,2).C(7,2).C(8,2) B. C(4,4).C(5.2).C6,2).C(7,2).C(8.2) C. P(5,2).P(6,2).P(7.2).P(8.2) D. P(4,4).P(5.2).P(6.2).P(7,2).P(8.2) The number of possible pictures is

For z = 3-4 i and w = 3+2 i calculate a 10 significant figure approximation to the argument of (w13+z11) / (z18+w17)

Verify the following identity. -tan α/2 = tan α / sec α + 1

Write the polynomial inequality in the form p(x)<0(x)<0, p(x)≤0(x)≤0, p(x)>0(x)>0, or p(x)≥0(x)≥0; then find the real zeros of p(x) (x).