🎉 Announcing Numerade's $26M Series A, led by IDG Capital!Read how Numerade will revolutionize STEM Learning # Calculus for Business, Economics, Life Sciences, and Social Sciences 13th ## Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen ## Chapter 1 ## Functions and Graphs ## Educators ### Problem 1 Use point-by-point plotting to sketch the graph of each equation. $$y=x+1$$ Wendi Z. Numerade Educator ### Problem 2 Use point-by-point plotting to sketch the graph of each equation. $$x=y+1$$ Wendi Z. Numerade Educator ### Problem 3 Use point-by-point plotting to sketch the graph of each equation. $$x=y^{2}$$ Wendi Z. Numerade Educator ### Problem 4 Use point-by-point plotting to sketch the graph of each equation. $$y=x^{2}$$ Wendi Z. Numerade Educator ### Problem 5 Use point-by-point plotting to sketch the graph of each equation. $$y=x^{3}$$ Wendi Z. Numerade Educator ### Problem 6 Use point-by-point plotting to sketch the graph of each equation. $$x=y^{3}$$ Wendi Z. Numerade Educator ### Problem 7 Use point-by-point plotting to sketch the graph of each equation. $$x y=-6$$ Wendi Z. Numerade Educator ### Problem 8 Use point-by-point plotting to sketch the graph of each equation. $$x y=12$$ Wendi Z. Numerade Educator ### Problem 9 Indicate whether each table specifies a function. Wendi Z. Numerade Educator ### Problem 10 Indicate whether each table specifies a function. Wendi Z. Numerade Educator ### Problem 11 Indicate whether each table specifies a function. Wendi Z. Numerade Educator ### Problem 12 Indicate whether each table specifies a function. Wendi Z. Numerade Educator ### Problem 13 Indicate whether each table specifies a function. Wendi Z. Numerade Educator ### Problem 14 Indicate whether each table specifies a function. Wendi Z. Numerade Educator ### Problem 15 Indicate whether each graph specifies a function. Wendi Z. Numerade Educator ### Problem 16 Indicate whether each graph specifies a function. Wendi Z. Numerade Educator ### Problem 18 Indicate whether each graph specifies a function. Wendi Z. Numerade Educator ### Problem 19 Indicate whether each graph specifies a function. Wendi Z. Numerade Educator ### Problem 20 Indicate whether each graph specifies a function. Wendi Z. Numerade Educator ### Problem 22 Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither. $$y=10-3 x$$ Wendi Z. Numerade Educator ### Problem 23 Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither. $$x y-4=0$$ Wendi Z. Numerade Educator ### Problem 24 Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither. $$x^{2}-y=8$$ Wendi Z. Numerade Educator ### Problem 25 Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither. $$y=5 x+\frac{1}{2}(7-10 x)$$ Wendi Z. Numerade Educator ### Problem 26 Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither. $$y=\frac{2+x}{3}+\frac{2-x}{3}$$ Wendi Z. Numerade Educator ### Problem 27 Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither. $$3 x+4 y=5$$ Wendi Z. Numerade Educator ### Problem 28 Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither. $$9 x-2 y+6=0$$ Wendi Z. Numerade Educator ### Problem 29 Use point-by-point plotting to sketch the graph of each function. $$f(x)=1-x$$ Wendi Z. Numerade Educator ### Problem 30 Use point-by-point plotting to sketch the graph of each function. $$f(x)=\frac{x}{2}-3$$ Wendi Z. Numerade Educator ### Problem 31 Use point-by-point plotting to sketch the graph of each function. $$f(x)=x^{2}-1$$ Wendi Z. Numerade Educator ### Problem 32 Use point-by-point plotting to sketch the graph of each function. $$f(x)=3-x^{2}$$ Check back soon! ### Problem 33 Use point-by-point plotting to sketch the graph of each function. $$f(x)=4-x^{3}$$ Check back soon! ### Problem 34 Use point-by-point plotting to sketch the graph of each function. $$f(x)=x^{3}-2$$ Check back soon! ### Problem 35 Use point-by-point plotting to sketch the graph of each function. $$f(x)=\frac{8}{x}$$ Check back soon! ### Problem 36 Use point-by-point plotting to sketch the graph of each function. $$f(x)=\frac{-6}{x}$$ Check back soon! ### Problem 37 The three points in the table are on the graph of the indicated function$f .$Do these three points provide sufficient information for you to sketch the graph of$y=f(x)$? Add more points to the table until you are satisfied that your sketch is a good representation of the graph of$y=f(x)$for$-5 \leq x \leq 5$. $$\begin{array}{lccc} x & -1 & 0 & 1 \\ f(x) & -1 & 0 & 1 \end{array} \quad f(x)=\frac{2 x}{x^{2}+1}$$ Check back soon! ### Problem 38 The three points in the table are on the graph of the indicated function$f .$Do these three points provide sufficient information for you to sketch the graph of$y=f(x)$? Add more points to the table until you are satisfied that your sketch is a good representation of the graph of$y=f(x)$for$-5 \leq x \leq 5$. $$\begin{array}{lccc} x & 0 & 1 & 2 \\ f(x) & 0 & 1 & 2 \end{array} \quad f(x)=\frac{3 x^{2}}{x^{2}+2}$$ Check back soon! ### Problem 39 Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some problems may have more than one answer. $$y=f(-5)$$ Check back soon! ### Problem 40 Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some problems may have more than one answer. $$y=f(4)$$ Check back soon! ### Problem 41 Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some problems may have more than one answer. $$y=f(5)$$ Check back soon! ### Problem 42 Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some problems may have more than one answer. $$y=f(-2)$$ Check back soon! ### Problem 43 Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some problems may have more than one answer. $$0=f(x)$$ Check back soon! ### Problem 44 Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some problems may have more than one answer. $$3=f(x), x<0$$ Check back soon! ### Problem 45 Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some problems may have more than one answer. $$-4=f(x)$$ Check back soon! ### Problem 46 Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some problems may have more than one answer. $$4=f(x)$$ Check back soon! ### Problem 47 Find the domain of each function. $$F(x)=2 x^{3}-x^{2}+3$$ Check back soon! ### Problem 48 Find the domain of each function. $$H(x)=7-2 x^{2}-x^{4}$$ Check back soon! ### Problem 49 Find the domain of each function. $$f(x)=\frac{x-2}{x+4}$$ Check back soon! ### Problem 50 Find the domain of each function. $$g(x)=\frac{x+1}{x-2}$$ Check back soon! ### Problem 51 Find the domain of each function. $$g(x)=\sqrt{7-x}$$ Check back soon! ### Problem 52 Find the domain of each function. $$F(x)=\frac{1}{\sqrt{5+x}}$$ Check back soon! ### Problem 53 Does the equation specify a function with independent variable$x$? If so, find the domain of the function. If not, find a value of$x$to which there corresponds more than one value of$y$. $$2 x+5 y=10$$ Check back soon! ### Problem 54 Does the equation specify a function with independent variable$x$? If so, find the domain of the function. If not, find a value of$x$to which there corresponds more than one value of$y$. $$6 x-7 y=21$$ Check back soon! ### Problem 55 Does the equation specify a function with independent variable$x$? If so, find the domain of the function. If not, find a value of$x$to which there corresponds more than one value of$y$. $$y(x+y)=4$$ Check back soon! ### Problem 56 Does the equation specify a function with independent variable$x$? If so, find the domain of the function. If not, find a value of$x$to which there corresponds more than one value of$y$. $$x(x+y)=4$$ Check back soon! ### Problem 57 Does the equation specify a function with independent variable$x$? If so, find the domain of the function. If not, find a value of$x$to which there corresponds more than one value of$y$. $$x^{-3}+y^{3}=27$$ Check back soon! ### Problem 58 Does the equation specify a function with independent variable$x$? If so, find the domain of the function. If not, find a value of$x$to which there corresponds more than one value of$y$. $$x^{2}+y^{2}=9$$ Check back soon! ### Problem 59 Does the equation specify a function with independent variable$x$? If so, find the domain of the function. If not, find a value of$x$to which there corresponds more than one value of$y$. $$x^{3}-y^{2}=0$$ Check back soon! ### Problem 60 Does the equation specify a function with independent variable$x$? If so, find the domain of the function. If not, find a value of$x$to which there corresponds more than one value of$y$. $$\sqrt{x}-y^{3}=0$$ Check back soon! ### Problem 61 Find and simplify the expression if$f(x)=x^{2}-4$. $$f(4)$$ Check back soon! ### Problem 62 Find and simplify the expression if$f(x)=x^{2}-4$. $$f(-5)$$ Check back soon! ### Problem 63 Find and simplify the expression if$f(x)=x^{2}-4$. $$f(x+1)$$ Check back soon! ### Problem 64 Find and simplify the expression if$f(x)=x^{2}-4$. $$f(x-2)$$ Check back soon! ### Problem 65 Find and simplify the expression if$f(x)=x^{2}-4$. $$f(-6 x)$$ Check back soon! ### Problem 66 Find and simplify the expression if$f(x)=x^{2}-4$. $$f(10 x)$$ Check back soon! ### Problem 67 Find and simplify the expression if$f(x)=x^{2}-4$. $$f\left(x^{3}\right)$$ Check back soon! ### Problem 68 Find and simplify the expression if$f(x)=x^{2}-4$. $$f(\sqrt{x})$$ Check back soon! ### Problem 69 Find and simplify the expression if$f(x)=x^{2}-4$. $$f(2)+f(h)$$ Check back soon! ### Problem 70 Find and simplify the expression if$f(x)=x^{2}-4$. $$f(-3)+f(h)$$ Check back soon! ### Problem 71 Find and simplify the expression if$f(x)=x^{2}-4$. $$f(2+h)$$ Check back soon! ### Problem 72 Find and simplify the expression if$f(x)=x^{2}-4$. $$f(-3+h)$$ Check back soon! ### Problem 73 Find and simplify the expression if$f(x)=x^{2}-4$. $$f(2+h)-f(2)$$ Check back soon! ### Problem 74 Find and simplify the expression if$f(x)=x^{2}-4$. $$f(-3+h)-f(-3)$$ Check back soon! ### Problem 75 Find and simplify each of the following, assuming$h \neq 0$in$(C) .$(A)$f(x+h)$(B)$f(x+h)-f(x)$(C)$\frac{f(x+h)-f(x)}{h}$$$f(x)=4 x-3$$ Check back soon! ### Problem 76 Find and simplify each of the following, assuming$h \neq 0$in$(C) .$(A)$f(x+h)$(B)$f(x+h)-f(x)$(C)$\frac{f(x+h)-f(x)}{h}$$$f(x)=-3 x+9$$ Check back soon! ### Problem 77 Find and simplify each of the following, assuming$h \neq 0$in$(C) .$(A)$f(x+h)$(B)$f(x+h)-f(x)$(C)$\frac{f(x+h)-f(x)}{h}$$$f(x)=4 x^{2}-7 x+6$$ Check back soon! ### Problem 78 Find and simplify each of the following, assuming$h \neq 0$in$(C) .$(A)$f(x+h)$(B)$f(x+h)-f(x)$(C)$\frac{f(x+h)-f(x)}{h}$$$f(x)=3 x^{2}+5 x-8$$ Check back soon! ### Problem 79 Find and simplify each of the following, assuming$h \neq 0$in$(C) .$(A)$f(x+h)$(B)$f(x+h)-f(x)$(C)$\frac{f(x+h)-f(x)}{h}$$$f(x)=x(20-x)$$ Check back soon! ### Problem 80 Find and simplify each of the following, assuming$h \neq 0$in$(C) .$(A)$f(x+h)$(B)$f(x+h)-f(x)$(C)$\frac{f(x+h)-f(x)}{h}$$$f(x)=x(x+40)$$ Check back soon! ### Problem 81 Refer to the area A and perimeter$P$of a rectangle with length$l$and width$w$(see the figure). The area of a rectangle is$25 \mathrm{sq}$in. Express the perimeter$P(w)$as a function of the width$w,$and state the domain of this function. Check back soon! ### Problem 82 Refer to the area A and perimeter$P$of a rectangle with length$l$and width$w$(see the figure). The area of a rectangle is 81 sq in. Express the perimeter$P(l)$as a function of the length$l,$and state the domain of this function. Check back soon! ### Problem 83 Refer to the area A and perimeter$P$of a rectangle with length$l$and width$w$(see the figure). The perimeter of a rectangle is$100 \mathrm{~m}$. Express the area$A(l)$as a function of the length$l,$and state the domain of this function. Check back soon! ### Problem 84 Refer to the area A and perimeter$P$of a rectangle with length$l$and width$w$(see the figure). The perimeter of a rectangle is$160 \mathrm{~m}$. Express the area$A(w)$as a function of the width$w,$and state the domain of this function. Check back soon! ### Problem 85 A company manufactures memory chips for microcomputers. Its marketing research department, using statistical techniques, collected the data shown in Table 8 where$p$is the wholesale price per chip at which$x$million chips can be sold. Using special analytical techniques (regression analysis), an analyst produced the following price-demand function to model the data: $$p(x)=75-3 x \quad 1 \leq x \leq 20$$ $$\begin{array}{cc} \hline x \text { (millions) } & p(\) \\ 1 & 72 \\ 4 & 63 \\ 9 & 48 \\ 14 & 33 \\ 20 & 15 \end{array}$$ (A) Plot the data points in Table 8 , and sketch a graph of the price-demand function in the same coordinate system. (B) What would be the estimated price per chip for a demand of 7 million chips? For a demand of 11 million chips? Check back soon! ### Problem 86 A company manufactures notebook computers. Its marketing research department, using statistical techniques, collected the data shown in Table$9,$where$p$is the wholesale price per computer at which$x$thousand computers can be sold. Using special analytical techniques (regression analysis), an analyst produced the following price-demand function to model the data: $$p(x)=2,000-60 x \quad 1 \leq x \leq 25$$ $$\begin{array}{cc} \hline \boldsymbol{x} \text { (thousands) } & p(\) \\ 1 & 1,940 \\ 8 & 1,520 \\ 16 & 1,040 \\ 21 & 740 \\ 25 & 500 \\ \hline \end{array}$$ (A) Plot the data points in Table 9 , and sketch a graph of the price-demand function in the same coordinate system. (B) What would be the estimated price per computer for a demand of 11,000 computers? For a demand of 18,000 computers? Check back soon! ### Problem 87 (A) Using the price-demand function $$p(x)=75-3 x \quad 1 \leq x \leq 20$$ from Problem$85,$write the company's revenue function and indicate its domain. (B) Complete Table 10 , computing revenues to the nearest million dollars. (C) Plot the points from part (B) and sketch a graph of the revenue function using these points. Choose millions for the units on the horizontal and vertical axes. Check back soon! ### Problem 88 (A) Using the price-demand function $$p(x)=2,000-60 x \quad 1 \leq x \leq 25$$ from Problem 86 , write the company's revenue function and indicate its domain. (B) Complete Table 11 , computing revenues to the nearest thousand dollars. (C) Plot the points from part (B) and sketch a graph of the revenue function using these points. Choose thousands for the units on the horizontal and vertical axes. Check back soon! ### Problem 89 The financial department for the company in Problems 85 and 87 established the following cost function for producing and selling$x$million memory chips:$C(x)=125+16 x$million dollars (A) Write a profit function for producing and selling$x$million memory chips and indicate its domain. (B) Complete Table 12 , computing profits to the nearest million dollars. (C) Plot the points in part (B) and sketch a graph of the profit function using these points. Check back soon! ### Problem 90 The financial department for the company in Problems 86 and 88 established the following cost function for producing and selling$x$thousand notebook computers: $$C(x)=4,000+500 x \text { thousand dollars }$$ (A) Write a profit function for producing and selling$x$thousand notebook computers and indicate its domain. (B) Complete Table 13 , computing profits to the nearest thousand dollars. (C) Plot the points in part (B) and sketch a graph of the profit function using these points. Check back soon! ### Problem 91 A candy box will be made out of a piece of cardboard that measures 8 by 12 in. Equal-sized squares$x$inches on a side will be cut out of each corner, and then the ends and sides will be folded up to form a rectangular box. (A) Express the volume of the box$V(x)$in terms of$x$. (B) What is the domain of the function$V$(determined by the physical restrictions)? (C) Complete Table 14 . (D) Plot the points in part (C) and sketch a graph of the volume function using these points. Check back soon! ### Problem 92 Refer to Problem$91 .$(A) Table 15 shows the volume of the box for some values of$x$between 1 and 2 . Use these values to estimate to one decimal place the value of$x$between 1 and 2 that would produce a box with a volume of$65 \mathrm{cu}$in. $$\begin{array}{cc} \boldsymbol{x} & V(x) \\ 1.1 & 62.524 \\ 1.2 & 64.512 \\ 1.3 & 65.988 \\ 1.4 & 66.976 \\ 1.5 & 67.5 \\ 1.6 & 67.584 \\ 1.7 & 67.252 \\ \hline \end{array}$$ (B) Describe how you could refine this table to estimate$x$to two decimal places. (C) Carry out the refinement you described in part (B) and approximate$x$to two decimal places. Check back soon! ### Problem 93 Refer to Problems 91 and$92 .$(A) Examine the graph of$V(x)$from Problem$91 \mathrm{D}$and discuss the possible locations of other values of$x$that would produce a box with a volume of$65 \mathrm{cu}$in. (B) Construct a table like Table 15 to estimate any such value to one decimal place. (C) Refine the table you constructed in part (B) to provide an approximation to two decimal places. Check back soon! ### Problem 94 A parcel delivery service will only deliver packages with length plus girth (distance around) not exceeding 108 in. A rectangular shipping box with square ends$x$inches on a side is to be used. (A) If the full 108 in. is to be used, express the volume of the box$V(x)$in terms of$x$. (B) What is the domain of the function$V$(determined by the physical restrictions)? (C) Complete Table 16 . (D) Plot the points in part (C) and sketch a graph of the volume function using these points. Check back soon! ### Problem 95 In a study of the speed of muscle contraction in frogs under various loads, British biophysicist A. W. Hill determined that the weight$w$(in grams) placed on the muscle and the speed of contraction$v$(in centimeters per second) are approximately related by an equation of the form $$(w+a)(v+b)=c$$ where$a, b,$and$c$are constants. Suppose that for a certain muscle,$a=15, b=1,$and$c=90 .$Express$v$as a function of$w$. Find the speed of contraction if a weight of$16 \mathrm{~g}$is placed on the muscle. Check back soon! ### Problem 96 The percentage$s$of seats in the House of Representatives won by Democrats and the percentage$v$of votes cast for Democrats (when expressed as decimal fractions) are related by the equation $$5 v-2 s=1.4 \quad 0<s<1, \quad 0.28<v<0.68$$ (A) Express$v$as a function of$s$and find the percentage of votes required for the Democrats to win$51 \%$of the seats. (B) Express$s$as a function of$v$and find the percentage of seats won if Democrats receive$51 \%\$ of the votes. Wendi Z.