Chapter 11

Evaluate the area under each curve for \(-1 \leq x \leq 2\) $$ f(x)=-x^{4}+2 x^{3}+3 $$

Find the missing terms of each geometric sequence. (Hint: The geometric mean of the first and fifth terms is the third term. Some terms might be negative.) \(-4\), \(\quad\),\(\quad\),\(\quad\),\(-30 \frac{3}{8}, \ldots\)

Find the next two terms in each sequence. Write a formula for the \(n\) th term. Identify each formula as explicit or recursive. $$ -75,-68,-61,-54, \dots $$

Open-Ended Write an infinite geometric series that converges to \(3 .\) Use the formula to evaluate the series.

Open-Ended Write equations for three curves that are positive for \(1 \leq x \leq 3\) . Use your graphing calculator to find the area under each curve for this domain.

Technology A school committee has decided to spend a large portion of its annual technology budget on graphing calculators. This year, the technolagy cooodinator bought 75 calculators, and plans to buy 25 new calculators each year from now on. a. Suppose the school committee has decided that each student in the school should have access to a graphing calculator within seven years. The school population is 500 . Will the technology coordinator meet this goal? Explain your reasoning. b. Writing What are some pros and cons of buying calculators in this manner? If you could change the plan, would you? If so, how would you change it?

a. Open-Ended Choose two positive numbers. Find their geometric mean. b. Find the common ratio for a gometric sequence that includes the terms from part (a) in order from least to greatest or from greatest to least. c. Find the 9 th term of the geometric sequence from part \((b) .\) d. Find the geometric mean of the term from part (c) and the first term of your sequence. What term of the sequence have you just found?

Find the next two terms in each sequence. Write a formula for the \(n\) th term. Identify each formula as explicit or recursive. $$ 21,13,5,-3, \dots $$

a. A classmate uses the formula for the sum of an infinite geometric series to evaluate \(1+1.1+1.21+1.331+\ldots\) and gets \(-10 .\) Is your classmate's answer reasonable? Explain. b. Error Analysis What did your classmate fail to check before using the formula?

Approximate the area under the curve \(f(x)=x^{2}\) for the interval \(0 \leq x \leq 4\) by evaluating each sum. Use inscribed rectangles. a. \(\sum_{n=1}^{8}(0.5) f\left(a_{n}\right) \quad\) b. \(\sum_{n=1}^{4}(1) f\left(a_{n}\right)\) c. Which estimate is closer to the actual area under the curve? Explain.

a. Open-Ended Write two explicit formulas for arithmetic sequences. b. Write the first five terms of each related series. c. Use summation notation to rewrite each series. d. Evaluate each series.

For the geometric sequence \(3,12,48,192, \ldots,\) find the indicated term. 5th term

Critical Thinking Find the specified value for each infinite geometric series. $$ a_{1}=12, S=96 ; \text { find } r $$

a. Graph \(y=\frac{1}{4} x^{3}+1\) and \(y=1\) over the domain \(-4.7 \leq x \leq 4.7\) b. critical thinking Evaluate the area under eurve for the interval \(-1.5 \leq x \leq 1.5 .\) What do you notice? Explain.

For the geometric sequence \(3,12,48,192, \ldots,\) find the indicated term. 7 th term

Writing Explain the difference between a recursive formula and an explicit formula.

Critical Thinking Find the specified value for each infinite geometric series. $$ S=12, r=\frac{1}{6} ; \text { find } a_{1} $$

critical Thinking Use your graphing calculator to find the area of the triangle with vertices \((-3,0),(-1,3),\) and \((1,0) .\) (Hint. First find the function whose graph makes a peak at \((-1,3) . )\)

Use the values of \(a_{1}\) and \(S_{n}\) to find the value of \(a_{n}\) $$ a_{1}=-6 \text { and } S_{50}=-5150 ; a_{50} $$

For the geometric sequence \(3,12,48,192, \ldots,\) find the indicated term. 10 th term

a. Use your calculator to generate an arithmetic sequence with a common difference of - \(7 .\) How could you se a calculator to find the 6th term? The 8th term? The 20th term? b. Explain how your answer to part (a) relates to the explicit formula \(a_{n}=a_{1}+(n-1) d\)

a. Open-Ended Write four terms of a sequence of numbers that you can describe both recursively and explicitly. b. Write a recursive formula and an explicit formula for your sequence. c. Find the 20 th term of the sequence by evaluating one of your formulas. Use the other formula to check your work.

Writing Suppose you are to receive an allowance each week for the next 26 weeks. Would you rather receive (a) \(\$ 1000\) per week or (b) 2\(c\) the first week, 4\(c\) the second week, 8\(c\) the third week, and so on for the 26 weeks?

a. Write the equation \(\frac{x^{2}}{25}+\frac{y^{2}}{9}=1\) in calculator-ready form. b. Graph the top half of the ellipse. Calculate the area under the curve for the interval \(-5 \leq x \leq 5 .\) c. Use symmetry to find the area of the entire ellipse. d. Open-Ended Find the area of another symmetric shape by graphing part of it. Sketch your graph and show your calculations.

For the geometric sequence \(3,12,48,192, \ldots,\) find the indicated term. 14 th term

Use the given rule to write the \(4 \mathrm{th}, 5 \mathrm{th}, 6 \mathrm{th},\) and 7 th terms of each sequence. $$ a_{1}=-1, a_{n}=a_{n-1}+n^{2} $$

The sum of an infinite geometric series is twice its first term. a. Error Analysis A student says the common ratio of the series is \(\frac{3}{2} \cdot\) What is the student's error? b. Find the common ratio of the series.

For the geometric sequence \(3,12,48,192, \ldots,\) find the indicated term. 17 th term

Use the given rule to write the \(4 \mathrm{th}, 5 \mathrm{th}, 6 \mathrm{th},\) and 7 th terms of each sequence. $$ a_{1}=-2, a_{n}=3\left(a_{n-1}+2\right) $$

Find the 17th term of each sequence. \(a_{16}=18, d=5\)

Technology Create a spreadsheet to evaluate the first \(n\) terms of each series. Determine whether each infinite series converges to a sum. If so, estimate the sum. $$ \sum_{n=1}^{\infty} \frac{1}{2^{n}} $$

Use the given rule to write the \(4 \mathrm{th}, 5 \mathrm{th}, 6 \mathrm{th},\) and 7 th terms of each sequence. $$ a_{n}=(n+1)^{2} $$

Use the graph of \(f(x)=\sqrt{x}+2\) for Exercises \(46-49\) Which of the following is the most accurate value of the area under \(f(x)=\sqrt{x}+2\) for \(0 \leq x \leq 4 ?\) \(\begin{array}{llll}{\text { A. } 12.1} & {\text { B. } 14.1} & {\text { C. } 16.1} & {\text { D. } 24.0}\end{array}\)

Find the 10 th term of each geometric sequence. $$ a_{9}=8, r=\frac{1}{2} $$

Find the 17th term of each sequence. \(a_{16}=18, d=\frac{1}{2}\)

Use the given rule to write the \(4 \mathrm{th}, 5 \mathrm{th}, 6 \mathrm{th},\) and 7 th terms of each sequence. $$ a_{n}=2(n-1)^{3} $$

Technology Create a spreadsheet to evaluate the first \(n\) terms of each series. Determine whether each infinite series converges to a sum. If so, estimate the sum. $$ \sum_{n=1}^{\infty} \frac{1}{(n-1) !} $$

Which expression represents the sum \(10+20+30+40 ?\) I. \(\sum_{n=1}^{4} 10 n\) II. \(\sum_{n=10}^{40} 10 n\) III. \(10\left(\sum_{n=1}^{4} n\right)\) A. I and II only B. I and III only C. II and III only D. \(1, \|\), and \(\| 1\)

Find the 10 th term of each geometric sequence. $$ a_{11}=8, r=\frac{1}{2} $$

Use the given rule to write the \(4 \mathrm{th}, 5 \mathrm{th}, 6 \mathrm{th},\) and 7 th terms of each sequence. $$ a_{n}=\frac{n^{2}}{n+1} $$

Physics Because of friction and air resistance, each swing of a pendulum is a little shorter than the previous one. The lengths of the swings form a geometric sequence. Suppose the first swing of a pendulum has a length of 100 \(\mathrm{cm}\) and the return swing is 99 \(\mathrm{cm} .\) a. On which swing will the arc first have a length less than 50 \(\mathrm{cm} ?\) b. Find the total distance traveled by the pendulum until it comes to rest.

The area under a curve is estimated using inscribed rectangles and circumscribed rectangles. Explain why the mean of these two values might be a more accurate estimate than either one.

What is the value of \(\sum_{n=1}^{5}(2 n-3) ?\) \(\begin{array}{lllll}{\text { F. } 6} & {\text { G. } 15} & {\text { H. } 17} & {\text { J. } 10 n-15}\end{array}\)

Find the 10 th term of each geometric sequence. $$ a_{9}=-5, r=-\frac{1}{2} $$

Use the given rule to write the \(4 \mathrm{th}, 5 \mathrm{th}, 6 \mathrm{th},\) and 7 th terms of each sequence. $$ a_{n}=\frac{n+1}{n+2} $$

a. Show that the infinite geometric series \(0.142857+\) \(0.000000142857+\ldots\) has a sum of \(\frac{1}{7}\) b. Find the fraction form of the repeating decimal 0.428571428571\(\ldots\)

Determine whether the sum of each infinite geometric series exists. $$ 4+2+1+\frac{1}{2}+\frac{1}{4}+\ldots $$

Which expression defines the series \(14+20+26+32+38+44+50 ?\) A. \(\sum_{n=2}^{8}(7 n-1) \quad\) B. \(\sum_{n=3}^{8}(6 n-4) \quad\) C. \(\sum_{n=3}^{9}(6 n-4) \quad\) D. \(\sum_{n=8}^{14}(n+6)\)

Find the 10 th term of each geometric sequence. $$ a_{11}=-5, r=-\frac{1}{2} $$

The function \(S(n)=\frac{10\left(1-0.8^{n}\right)}{0.2}\) represents the sum of the first \(n\) terms of an infinite geometric series. a. What is the domain of the function? b. Find \(S(n)\) for \(n=1,2,3, \ldots, 10 .\) Sketch the graph of the function. c. Find the sum \(S\) of the infinite geometric series.