Chapter 13

Find the vertical asymptotes, if any, of the graph of $$ f(x)=\frac{3 x^{2}}{(x-3)(x+1)} $$

Express each sum using summation notation. \(\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\cdots+\frac{13}{13+1}\)

Drury Lane Theater The Drury Lane Theater has 25 seats in the first row and 30 rows in all. Each successive row contains one additional seat. How many seats are in the theater?

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ \sum_{k=1}^{\infty} 8\left(\frac{1}{3}\right)^{k-1} $$

If \(f(x)=\frac{x^{2}+1}{2 x+5},\) find \(f(-2) .\) What is the corresponding point on the graph of \(f ?\)

Express each sum using summation notation. \(1+3+5+7+\cdots+[2(12)-1]\)

Seats in an Amphitheater An outdoor amphitheater has 35 seats in the first row, 37 in the second row, 39 in the third row, and so on. There are 27 rows altogether. How many can the amphitheater seat?

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ \sum_{k=1}^{\infty} \frac{1}{2} \cdot 3^{k-1} $$

Express each sum using summation notation. \(1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\cdots+(-1)^{6}\left(\frac{1}{3^{6}}\right)\)

Football Stadium The corner section of a football stadium has 15 seats in the first row and 40 rows in all. Each successive row contains two additional seats. How many seats are in this section?

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ \sum_{k=1}^{\infty} 3\left(\frac{3}{2}\right)^{k-1} $$

Express each sum using summation notation. \(\frac{2}{3}-\frac{4}{9}+\frac{8}{27}-\cdots+(-1)^{12}\left(\frac{2}{3}\right)^{11}\)

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ \sum_{k=1}^{\infty} 6\left(-\frac{2}{3}\right)^{k-1} $$

Express each sum using summation notation. \(3+\frac{3^{2}}{2}+\frac{3^{3}}{3}+\cdots+\frac{3^{n}}{n}\)

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ \sum_{k=1}^{\infty} 4\left(-\frac{1}{2}\right)^{k-1} $$

Express each sum using summation notation. \(\frac{1}{e}+\frac{2}{e^{2}}+\frac{3}{e^{3}}+\cdots+\frac{n}{e^{n}}\)

Stadium Construction How many rows are in the corner section of a stadium containing 2040 seats if the first row has 10 seats and each successive row has 4 additional seats?

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ \sum_{k=1}^{\infty} 3\left(\frac{2}{3}\right)^{k} $$

Express each sum using summation notation. \(a+(a+d)+(a+2 d)+\cdots+(a+n d)\)

Express each sum using summation notation. \(a+a r+a r^{2}+\cdots+a r^{n-1}\)

Old Faithful Old Faithful is a geyser in Yellowstone National Park named for its regular eruption pattern. Past data indicates that the average time between eruptions is \(1 \mathrm{~h} 35 \mathrm{~m}\) (a) Suppose rangers log the first eruption on a given day at 12: 57 am. Using \(a_{1}=57,\) write a prediction formula for the sequence of eruption times that day in terms of the number of minutes after midnight. (b) At what time of day (e.g., \(7: 15 \mathrm{am})\) is the 10 th eruption expected to occur? (c) At what time of day is the last eruption expected to occur?

Cooling Air As a parcel of air rises (for example, as it is pushed over a mountain), it cools at the dry adiabatic lapse rate of \(5.5^{\circ} \mathrm{F}\) per 1000 feet until it reaches its dew point. If the ground temperature is \(67^{\circ} \mathrm{F},\) write a formula for the sequence of temperatures, \(\left\\{T_{n}\right\\},\) of a parcel of air that has risen \(n\) thousand feet. What is the temperature of a parcel of air if it has risen 5000 feet?

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. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. $$ \\{2 n-5\\} $$

Citrus Ladders Ladders used by fruit pickers are typically tapered with a wide bottom for stability and a narrow top for ease of picking. If the bottom rung of such a ladder is 49 inches wide and the top rung is 24 inches wide, how many rungs does the ladder have if each rung is 2.5 inches shorter than the one below it? How much material would be needed to make the rungs for the ladder described?

Challenge Problem If \(\left\\{a_{n}\right\\}\) is an arithmetic sequence with 100 terms where \(a_{1}=2\) and \(a_{2}=9,\) and \(\left\\{b_{n}\right\\}\) is an arithmetic sequence with 100 terms where \(b_{1}=5\) and \(b_{2}=11,\) how many terms are the same in each sequence?

Challenge Problem Suppose \(\left\\{a_{n}\right\\}\) is an arithmetic sequence. If \(S_{n}\) is the sum of the first \(n\) terms of \(\left\\{a_{n}\right\\},\) and \(\frac{S_{2 n}}{S_{n}}\) is a positive constant for all \(n,\) find an expression for the \(n\) th term, \(a_{n}\), in terms of only \(n\) and the common difference, \(d\).

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. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. $$ \left\\{3-\frac{2}{3} n\right\\} $$

Find the sum of each sequence. \(\sum_{k=1}^{20}(5 k+3)\)

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. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. $$ \left\\{8-\frac{3}{4} n\right\\} $$

Find the sum of each sequence. \(\sum_{k=1}^{26}(3 k-7)\)

Describe the similarities and differences between arithmetic sequences and linear functions.

Find the sum of each sequence. \(\sum_{k=1}^{16}\left(k^{2}+4\right)\)

Problems \(75-84\) are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. If a credit card charges \(15.3 \%\) interest compounded monthly, find the effective rate of interest.

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. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. $$ 2,4,6,8, \ldots $$

Find the sum of each sequence. \(\sum_{k=0}^{14}\left(k^{2}-4\right)\)

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. The vector \(\mathbf{v}\) has initial point \(P=(-1,2)\) and terminal point \(Q=(3,-4) .\) Write \(\mathbf{v}\) in the form \(a \mathbf{i}+b \mathbf{j} ;\) that is, find its position vector.

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. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. $$ \left\\{\left(\frac{2}{3}\right)^{n}\right\\} $$

Find the sum of each sequence. \(\sum_{k=10}^{60}(2 k)\)

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Analyze and graph the equation: \(25 x^{2}+4 y^{2}=100\)

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. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. $$ \left\\{\left(\frac{5}{4}\right)^{n}\right\\} $$

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the inverse of the matrix \(\left[\begin{array}{rr}2 & 0 \\ 3 & -1\end{array}\right],\) if it exists; otherwise, state that the matrix is singular.

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. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. $$ -1,2,-4,8, \ldots $$

Find the sum of each sequence. \(\sum_{k=5}^{20} k^{3}\)

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the partial fraction decomposition of \(\frac{3 x}{x^{3}-1}\)

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the exact value of \(\sin ^{2} \frac{5 \pi}{8}-\cos ^{2} \frac{5 \pi}{8}\).

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. If the sequence is arithmetic or geometric, find the sum of the first 50 terms. $$ \left\\{3^{n / 2}\right\\} $$

Credit Card Debt John has a balance of \(\$ 3000\) on his Discover card, which charges \(1 \%\) interest per month on any unpaid balance from the previous month. John can afford to pay \(\$ 100\) toward the balance each month. His balance each month after making a \(\$ 100\) payment is given by the recursively defined sequence \(B_{0}=\$ 3000 \quad B_{n}=1.01 B_{n-1}-100\)

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. If \(g\) is a function with domain \([-4,10],\) what is the domain of the function \(2 g(x-1) ?\)

Trout Population 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 after \(n\) months is given by the recursively defined sequence \(p_{0}=2000 \quad p_{n}=1.03 p_{n-1}+20\)

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.Find the real zeros of $$ h(x)=\frac{\left(x^{4}+1\right) \cdot 2 x-\left(x^{2}-1\right) \cdot 4 x^{3}}{\left(x^{4}+1\right)} $$