Questions and Answers of Calculus

1. The starting salary for a job is $43,800 with a guaranteed increase of $1950 per year. Determine (a) The salary during the fifth year (b) The total compensation through five full years of
Determine whether the sequence is geometric. If so, find the common ratio. 1. 2, 6, 18, 54, 162, . . . 2. 48, −24, 12, −6, . . . 3. 1/5, -3/5, 9/5, -27/5, . . . .
Write the first five terms of the geometric sequence. 1. a1 = 2, r = 15 2. a1 = 6, r = -1/3
Write an expression for the nth term of the geometric sequence. Then find the 10th term of the sequence. 1. a1 = 100, r = 1.05 2. a1 = 5, r = 0.2 3. a1 = 18, a2 = −9
Write an expression for the apparent nth term (an) of the sequence. (Assume that n begins with 1.) 1. −2, 2, −2, 2, −2, . . . 2. −1, 2, 7, 14, 23, . . .
Find the sum of the finite geometric sequence.1.2.3.4.
Find the sum of the infinite geometric series.1.2. 3. 4.
A paper manufacturer buys a machine for $120,000. It depreciates at a rate of 30% per year. (In other words, at the end of each year the depreciated value is 70% of what it was at the beginning of
Use mathematical induction to prove the formula for all integers n ≥1. 1. 3 + 5 + 7 + . . . + (2n + 1) = n (n + 2)
Find a formula for the sum of the first n terms of the sequence. Prove the validity of your formula. 9, 13, 17, 21, . . .
Find the sum using the formulas for the sums of powers of integers.1.2.
Write the first five terms of the sequence beginning with the term a1. Then calculate the first and second differences of the sequence. State whether the sequence has a perfect linear model, a
Find the binomial coefficient. 1. 6C4 2. 12C3
Evaluate using Pascal's Triangle.1.2.
Use the Binomial Theorem to write the expansion of the expression. 1. (x + 4)4 2. (5 + 2z)4
Determine the number of ways a computer can generate the sum using randomly selected integers from 1 through 14. 1. Two distinct integers whose sum is 7 2. Two distinct integers whose sum is 12
1. All of the landline telephone numbers in a small town use the same three-digit prefix. How many different telephone numbers are possible by changing only the last four digits? 2. A college student
Simplify the factorial expression. 1. 3! / 5! 2. 7! / 3! · 4! 3. (n - 1)! / (n + 1)!
1. There are 10 bicyclists entered in a race. In how many different ways can the top three places be decided? 2. In how many different ways can a jury of 12 people be randomly selected from a group
A local sandwich shop offers five different breads, four different meats, three different cheeses, and six different vegetables. A customer can choose one bread, one or no meat, one or no cheese, and
1. A drawer contains six white socks, two blue socks, and two gray socks.(a) What is the probability of randomly selecting one blue sock?(b) What is the probability of randomly selecting one white
1. At a university, 31% of the students are freshmen, 26% are sophomores, 25% are juniors, and 18% are seniors. One student receives a cash scholarship randomly by lottery. Find the probability that
1. You toss a six-sided die four times. What is the probability of getting four 5's? 2. You toss a six-sided die six times. What is the probability of getting each number exactly once? 3. You draw
1. To rewrite the expression 3/x5 using negative exponents, move x5 to the ________ and change the sign of the exponent. 2. When dividing fractions, multiply by the ________.
Factor the expression. 1. 2x(x + 2)−1/2 + (x + 2)1/2 2. x2(x2 + 1)−5 − (x2 + 1)−4
Complete the factored form of the expression. 1. (5x + 3) / 4 = 1/4 ( ) 2. 7x2 / 10 = 7/10 ( ) 3. 2/3 x2 + 1/3 x + 5 = 1/3 ( )
Insert the required factor in the parentheses. 1. x2(x3 − 1)4 = ( )(x3 − 1)4(3x2) 2. x(1 − 2x2)3 = ( )(1 − 2x2)3(−4x) 3. (4x + 6) / (x2 + 3x + 7)3 = ( ) 1 / (x2 + 3x + 7)3
Show that the two expressions are equivalent. 1. 4x2 + 6y2 / 10 x2 / (1/4) + 3y2/5 2. 4x2/14 - 2y2 = 2x2/7 - y2 / (1/2) 3. 25x2 / 36 + 4y2 / 9 = x2 / (36/25) + y2 / (9/4)
Describe and correct the error.
Rewrite the expression using negative exponents. 1. 7 / (x + 3)5 2. (2 - x) / (x + 1)3/2 3. 2x5 / (3x + 5)4 4. (x + 1) / x (6 - x)1/2
Rewrite the fraction as a sum of two or more terms. 1. (x2 + 6x + 12) / 3x 2. (x3 - 5x2 + 4) / x2 3. (4x3 - 7x2 + 1) / x1/3
Simplify the expression.1.2. 3.
(a) Verify that y1 = y2 analytically.y1 = x2(1/3) (x2 + 1)ˆ’2/3(2x) + (x2 + 1)1/3(2x)y2 = 2x(4x2 + 3) / 3(x2 + 1)2/3(b) Complete the table and demonstrate the equality in part (a)
An athlete has set up a course in which she is dropped off by a boat 2 miles from the nearest point on shore. Once she reaches the shore, she must run to a point 4 miles down the coast and 2 miles
1. Write a paragraph explaining to a classmate why 1 / [(x - 2)1/2 + x4 ≠ (x - 2)-1/2 + x-4 2. You are taking a course in calculus, and for one of the homework problems you obtain the following

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