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mathematics
precalculus

Questions and Answers of Precalculus

In Problems 9–42, find each limit algebraically. x² + x - 6 lim x-3x² + 2x - 3
In Problems 11 – 54, simplify each expression. Assume that all variables are positive when they appear. 3-40x14y10
In Problems 25–34, replace the question mark by < ,> or =, whichever is correct. ? 0.33
In Problems 13–32, use the accompanying graph of y = f(x).Is f continuous at 5? x=-6 I -8 -6 (-4,2) -4 -2 У 4 2 -2 -4 2 (2, 3) 6 (6,2) X
In Problems 27 – 36, factor the perfect squares.x2 + 10x + 25
In Problems 30–32, find the slope of the tangent line to the graph of f at the given point. Graph f and the tangent line.f (x) = x3 + x2 at (2, 12)
In Problems 23–42, graph each function. Use the graph to investigate the indicated limit. lim f(x), f(x) = ex x-0
In Problems 25–32, write each inequality using interval notation, and illustrate each inequality using the real number line. x < 5
In Problems 11 – 54, simplify each expression. Assume that all variables are positive when they appear. √9x5
In Problems 21–32, find the derivative of each function at the given number.f (x) = x3 − 2x2 + x at −1
In Problems 25–34, replace the question mark by < ,> or =, whichever is correct. 2? 1.41
In Problems 27 – 36, factor the perfect squares.x2 − 2x + 1
In Problems 23–42, graph each function. Use the graph to investigate the indicated limit. lim f(x), f(x) = X-T = cos x
In Problems 23–30, an integral is given.(a) What area does the integral represent?(b) Graph the function, and shade the region represented by the integral.(c) Use a graphing utility to approximate
In Problems 19 – 36, perform the indicated operation and simplify the result. Leave your answer in factored form. 4- x 4 + x 4x 2 x² - 16 X
In Problems 11 – 68, solve each equation.w (4 − w2) = 8 − w3
In Problems 11 – 68, solve each equation.x2 = 9x
In Problems 11–48, write each expression in the standard form a + bi. + √3 2 2
In Problems 30–32, find the slope of the tangent line to the graph of f at the given point. Graph f and the tangent line.f (x) = x2 + 2x − 3 at (−1, −4)
In Problems 9–42, find each limit algebraically. lim x--3 x- - x - 12 - 9 x2 x²
In Problems 11 – 54, simplify each expression. Assume that all variables are positive when they appear. 4/162x⁹y12
In Problems 25–34, replace the question mark by < ,> or =, whichever is correct. ? 0.5
Confirm the entries in Table 7.Data from table 7 Using left endpoints: Using right endpoints: n Area n Area 2 0.5 2 1.5 4 0.75 4 1.25 10 0.9 10 1.1 100 0.99 100 1.01
In Problems 19 – 36, perform the indicated operation and simplify the result. Leave your answer in factored form. 3 + x 3 x 2-9 x² 9x³
In Problems 11 – 68, solve each equation.x3 = 4x2
In Problems 11–48, write each expression in the standard form a + bi. 今 1 2 2 2
In Problems 1 – 22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 1) = 1⁄n(3n 2 + 5 + 8 + + (3n— 1) = (3n+1)
In Problems 5 – 16, evaluate each expression. 7 3
Expressusing summation notation. 1-1/2+ 3 1/3+ 4 .. + 1 13
In Problems 1 – 22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 1 + 4 + 7 + ··· + (3n - 2) = · + (3n − 2) = —⁄n(3n −
In Problems 5 – 16, evaluate each expression. 7 5
The notationis an example of________ notation. ai + a tag + + απ || Σακ k=1
Solve the each equation Σκ=1+2+3 + + n = k=1
In Problems 7–12, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the
True or False For a geometric sequence with first term a1 and common ratio r , where r ≠ 0, r ≠ 1, the sum of the first n terms is Sn = a₁ 1- pn 1-r
In Problems 5 – 16, evaluate each expression. 7
The sequence a1 = 5, an = 3an−1 is an example of a(n) sequence.(a) Alternating(b) Recursive(c) Fibonacci(d) Summation
In Problems 7–12, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the
In Problems 1 – 22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 1+ 3 + 3² + .. +3n-1 = (3″ − 1) -
In Problems 1 – 22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 1+ 4 + 4² + ... + 4n-1 (4) (4" - 1)
In Problems 9 – 18, show that each sequence is geometric. Then find the common ratio and list the first four terms. {n} = {4"}
In Problems 9 – 18, show that each sequence is geometric. Then find the common ratio and list the first four terms. {$₂} = {(-5)"}
In Problems 5 – 16, evaluate each expression. 50 49
In Problems 7–12, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference and the sum of the first n terms. If the
In Problems 5 – 16, evaluate each expression. 100 98
In Problems 1 – 22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.1 + 2 + 22 + . . . + 2n−1 = 2n − 1
Solve the equation 2 sin² x sin x 3 = 0, 0 < x < 2π
In Problems 9 – 18, show that each sequence is geometric. Then find the common ratio and list the first four terms. {„(7)ε-} = {"}
In Problems 5 – 16, evaluate each expression. 1000 1000
In Problems 9 – 18, show that each sequence is geometric. Then find the common ratio and list the first four terms. {Cn} (2n-1 4 = = {²
In Problems 9 – 18, show that each sequence is geometric. Then find the common ratio and list the first four terms. {bn} = = {({)"}
In Problems 5 – 16, evaluate each expression. 1000 0
In Problems 13–16, find each sum. 30 Σ (k2 + 2) k=1
In Problems 5 – 16, evaluate each expression. 55 23
In Problems 5 – 16, evaluate each expression. 60 20
In Problems 9–14, evaluate each factorial expression. 5! - 8! 3!
In Problems 5 – 16, evaluate each expression. 47 25
In Problems 15–26, list the first five terms of each sequence. {bn}: = 2n {²n +1}
In Problems 1 – 22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. 4+3+2+ +(5-n) = + (5 − n) = ½1⁄n(9 — , n) ...
Find the exact value of cos−1 (−0.5).
In Problems 1 – 22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n. -2-3-4- - (n + 1) = -27(n in + 3)
In Problems 5 – 16, evaluate each expression. 37 19
In Problems 15–26, list the first five terms of each sequence. 1-1 n {dn} = {(-1) "-¹ (2₂² 2²-1)} 2n
In Problems 22–25, determine whether each infinite geometric series converges or diverges. If it converges, find its sum. +=+=+1+E
In Problems 26–28, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers. 2 + 6 + 18 + +2.3n-1 = 3n 1
In Problems 15–26, list the first five terms of each sequence. {$n} {(3)"}
In Problems 17 – 28, expand each expression using the Binomial Theorem. 4 (√x - √3)*
In Problems 26–28, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers. 1² +4² +7²+...+(3n - 2)² = = žn(6n2 — 31 3n - 1)
In Problems 17 – 28, expand each expression using the Binomial Theorem. (ax-by)4
In Problems 27 – 32, find the indicated term of each geometric sequence. 7th term of 1,1.1.
In Problems 26–28, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers.Evaluate: 5
In Problems 17 – 28, expand each expression using the Binomial Theorem. (ax + by)5
In Problems 27–34, the given pattern continues. Write down the nth term of a sequence {an } suggested by the pattern. 1 1 2' 4 8
In Problems 17–24, find the nth term of the arithmetic sequence {an} whose first term a1 and common difference d are given. What is the 51st term?a1 = 6; d = −2
Use mathematical induction to prove that a + (a + d) + (a + 2d) + + [a + (n − 1)d] - n(n-1) 2 = na + d² dn
In Problems 1 – 22, use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers n.n2 − n + 2 is divisible by 2.
In Problems 25–30, find the indicated term in each arithmetic sequence. 70th term of 2√√5, 4√5, 6√5,...
In Problems 23–27, prove each statement.If x > 1, then xn > 1.
In Problems 33 – 40, find the nth term an of each geometric sequence. When given, r is the common ratio. 4, 1, 1/16 4'
Use the Binomial Theorem to find the numerical value of (1.001)5 correct to five decimal places. (1.001)5 = (1 + 10−3)5 THEOREM Binomial Theorem Let x and a be real numbers. For any positive
In Problems 69–80, find the sum of each sequence. 60 Σ (2k) k=10
Problems 75–84. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as
Ivan bought a car by taking out a loan for $18,500 at 0.5% interest per month. Ivan’s normal monthly payment is $434.47 per month, but he decides that he can afford to pay $100 extra toward the
In Problems 30 and 31, expand each expression using the Binomial Theorem.(3x − 4)4
In Problems 27–34, the given pattern continues. Write down the nth term of a sequence {an} suggested by the pattern.1, −1, 1, −1, 1, −1, . . .
In Problems 27 – 32, find the indicated term of each geometric sequence.7th term of 0 .1, 1.0, 10.0, . . .
A pond currently contains 2000 trout. A fish hatchery decides to add 20 trout each month. It is also known that the trout population is growing at a rate of 3% per month. The size of the population
In Problems 31–38, find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term.9th term is −5;
If a sheet of paper is folded in half by folding the top edge down to the bottom edge, one crease will result. If the folded paper is folded in the same manner, the result is three creases. With each
Problems 113–122. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as
Problems 53–62. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus.If
If {an} is an arithmetic sequence with 100 terms where a1 = 2 and a2 = 9, and {bn} is an arithmetic sequence with 100 terms where b1 = 5 and b2 = 11, how many terms are the same in each sequence?
In Problems 69–82, determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio.
Solve the given equation. 0! ;1!
Problems 113–122. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as
The binomial coefficientequals
Find a rectangular equation of the plane curve with parametric equations x(t) = t + 5 and y(t) = √t for t ≥ 0.
Find the function g whose graph is the graph of y = √x but is stretched vertically by a factor of 7 and shifted left 5 units.
Problems 113–122. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as
Solve: 3x2 − 2x = −1

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