1.E: Analytic Geometry (Exercises)
- Page ID
- 3086
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)These are homework exercises to accompany David Guichard's "General Calculus" Textmap. Complementary General calculus exercises can be found for other Textmaps and can be accessed here.
1.1: Lines
Ex 1.1.1 Find the equation of the line through \((1,1)\) and \((-5, -3)\) in the form \(y=mx+b\). (answer)
Ex 1.1.2 Find the equation of the line through \((-1,2)\) with slope \(-2\) in the form \(y=mx+b\). (answer)
Ex 1.1.3 Find the equation of the line through \((-1,1)\) and \((5, -3)\) in the form \(y=mx+b\). (answer)
Ex 1.1.4 Change the equation \(y-2x=2\) to the form \(y=mx+b\), graph the line, and find the \(y\)-intercept and \(x\)-intercept. (answer)
Ex 1.1.5 Change the equation \(x+y=6\) to the form \(y=mx+b\), graph the line, and find the \(y\)-intercept and \(x\)-intercept. (answer)
Ex 1.1.6 Change the equation \(x=2y-1\) to the form \(y=mx+b\), graph the line, and find the \(y\)-intercept and \(x\)-intercept. (answer)
Ex 1.1.7 Change the equation \(3=2y\) to the form \(y=mx+b\), graph the line, and find the \(y\)-intercept and \(x\)-intercept. (answer)
Ex 1.1.8 Change the equation \(2x+3y+6=0\) to the form \(y=mx+b\), graph the line, and find the \(y\)-intercept and \(x\)-intercept. (answer)
Ex 1.1.9 Determine whether the lines \(3x+6y=7\) and \(2x+4y=5\) are parallel. (answer)
Ex 1.1.10 Suppose a triangle in the \(x,y\)--plane has vertices \((-1,0)\), \((1,0)\) and \((0,2)\). Find the equations of the three lines that lie along the sides of the triangle in \(y=mx+b\) form. (answer)
Ex 1.1.11 Suppose that you are driving to Seattle at constant speed. After you have been traveling for an hour you pass a sign saying it is 130 miles to Seattle, and after driving another 20 minutes you pass a sign saying it is 105 miles to Seattle. Using the horizontal axis for the time \(t\) and the vertical axis for the distance \(y\) from your starting point, graph and find the equation \(y=mt+b\) for your distance from your starting point. How long does the trip to Seattle take? (answer)
Ex 1.1.12 Let \(x\) stand for temperature in degrees Celsius (centigrade), and let \(y\) stand for temperature in degrees Fahrenheit. A temperature of \(0^\circ\) C corresponds to \(32^\circ \) F, and a temperature of \(100^\circ\)C corresponds to \(212^\circ\)F. Find the equation of the line that relates temperature Fahrenheit \(y\) to temperature Celsius \(x\) in the form \(y=mx+b\). Graph the line, and find the point at which this line intersects \(y=x\). What is the practical meaning of this point? (answer)
Ex 1.1.13 A car rental firm has the following charges for a certain type of car: $25 per day with 100 free miles included, $0.15 per mile for more than 100 miles. Suppose you want to rent a car for one day, and you know you'll use it for more than 100 miles. What is the equation relating the cost \(y\) to the number of miles \(x\) that you drive the car? (answer)
Ex 1.1.14 A photocopy store advertises the following prices: 5\cents per copy for the first 20 copies, 4\cents per copy for the 21st through 100th copy, and 3\cents per copy after the 100th copy. Let \(x\) be the number of copies, and let \(y\) be the total cost of photocopying. (a) Graph the cost as \(x\) goes from 0 to 200 copies. (b) Find the equation in the form \(y=mx+b\) that tells you the cost of making \(x\) copies when \(x\) is more than 100. (answer)
Ex 1.1.15 In the Kingdom of Xyg the tax system works as follows. Someone who earns less than 100 gold coins per month pays no tax. Someone who earns between 100 and 1000 gold coins pays tax equal to 10% of the amount over 100 gold coins that he or she earns. Someone who earns over 1000 gold coins must hand over to the King all of the money earned over 1000 in addition to the tax on the first 1000. (a) Draw a graph of the tax paid \(y\) versus the money earned \(x\), and give formulas for \(y\) in terms of \(x\) in each of the regions \(0\le x\le 100\), \(100\le x\le 1000\), and \(x\ge 1000\). (b) Suppose that the King of Xyg decides to use the second of these line segments (for \(100\le x\le 1000\)) for \(x\le 100\) as well. Explain in practical terms what the King is doing, and what the meaning is of the \(y\)-intercept. (answer)
Ex 1.1.16 The tax for a single taxpayer is described in the figure 1.1.3. Use this information to graph tax versus taxable income (i.e., \(x\) is the amount on Form 1040, line 37, and \(y\) is the amount on Form 1040, line 38). Find the slope and \(y\)-intercept of each line that makes up the polygonal graph, up to \(x=97620\). (answer)
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Ex 1.1.17 Market research tells you that if you set the price of an item at $1.50, you will be able to sell 5000 items; and for every 10 cents you lower the price below $1.50 you will be able to sell another 1000 items. Let \(x\) be the number of items you can sell, and let \(P\) be the price of an item. (a) Express \(P\) linearly in terms of \(x\), in other words, express \(P\) in the form \(P=mx+b\). (b) Express \(x\) linearly in terms of \(P\). (answer)
Ex 1.1.18 An instructor gives a 100-point final exam, and decides that a score 90 or above will be a grade of 4.0, a score of 40 or below will be a grade of 0.0, and between 40 and 90 the grading will be linear. Let \(x\) be the exam score, and let \(y\) be the corresponding grade. Find a formula of the form \(y=mx+b\) which applies to scores \(x\) between 40 and 90. (answer)
1.2: Distance Between Two Points; Circles
Ex 1.2.1Find the equation of the circle of radius 3 centered at:
| a) \((0,0)\) | d) \((0,3)\) |
| b) \((5,6)\) | e) \((0,-3)\) |
| c) \((-5,-6)\) | f) \((3,0)\) |
(answer)
Ex 1.2.2 For each pair of points \(A(x_1,y_1)\) and \(B(x_2,y_2)\) find (i) \(\Delta x\) and \(\Delta y\) in going from \(A\) to \(B\), (ii) the slope of the line joining \(A\) and \(B\), (iii) the equation of the line joining \(A\) and \(B\) in the form \(y=mx+b\), (iv) the distance from \(A\) to \(B\), and (v) an equation of the circle with center at \(A\) that goes through \(B\).
| a) \(A(2,0)\), \(B(4,3)\) | d) \(A(-2,3)\), \(B(4,3)\) |
| b) \(A(1,-1)\), \(B(0,2)\) | e) \(A(-3,-2)\), \(B(0,0)\) |
| c) \(A(0,0)\), \(B(-2,-2)\) | f) \(A(0.01,-0.01)\), \(B(-0.01,0.05)\) |
( (b) \(\Delta x=-1\), \(\Delta y = 3\), \(m=-3\), \(y=-3x+2\), \(\sqrt{10}\) </p>
(c) \(\Delta x=-2\), \(\Delta y = -2\), \(m=1\), \(y=x\), \(\sqrt{8}\)">answer
)Ex 1.2.3 Graph the circle \(x^2+y^2+10y=0\).
Ex 1.2.4 Graph the circle \(x^2-10x+y^2=24\).
Ex 1.2.5 Graph the circle \(x^2-6x+y^2-8y=0\).
Ex 1.2.6 Find the standard equation of the circle passing through \((-2,1)\) and tangent to the line \(3x-2y =6\) at the point \((4,3)\). Sketch. (Hint: The line through the center of the circle and the point of tangency is perpendicular to the tangent line.) (answer)
1.3: Functions
Find the domain of each of the following functions:
Ex 1.3.1 \( y=f(x)=\sqrt{2x-3}\) (answer)
Ex 1.3.2 \(y=f(x)=1/(x+1)\) (answer)
Ex 1.3.3 \(y=f(x)=1/(x^2-1)\) (answer)
Ex 1.3.4 \(y=f(x)=\sqrt{-1/x}\) (answer)
Ex 1.3.5 \(y=f(x)={\root 3 \of x}\) (answer)
Ex 1.3.6 \(y=f(x)={\root 4 \of x}\) (answer)
Ex 1.3.7 \(y=f(x)=\sqrt{r^2-(x-h)^2 }\), where \(r\) and \(h\) are positive constants. (answer)
Ex 1.3.8 \(y=f(x)=\sqrt{1-(1/x)}\) (answer)
Ex 1.3.9 \(y=f(x)=1/\sqrt{1-(3x)^2}\) (answer)
Ex 1.3.10 \(y=f(x)=\sqrt{x}+1/(x-1)\) (answer)
Ex 1.3.11 \(y=f(x)=1/(\sqrt{x}-1)\) (answer)
Ex 1.3.12 Find the domain of \(h(x) = \cases{ (x^2-9)/(x-3)& x\neq 3\cr 6& if \(x=3\).\cr}\) (answer)
Ex 1.3.13 Suppose \(f(x) = 3x-9\) and \( g(x) = \sqrt{x}\). What is the domain of the composition \((g\circ f)(x)\)? (Recall that composition is defined as \((g\circ f)(x) = g(f(x))\).) What is the domain of \((f\circ g)(x)\)? (answer)
Ex 1.3.14 A farmer wants to build a fence along a river. He has 500 feet of fencing and wants to enclose a rectangular pen on three sides (with the river providing the fourth side). If \(x\) is the length of the side perpendicular to the river, determine the area of the pen as a function of \(x\). What is the domain of this function? (answer)
Ex 1.3.15 A can in the shape of a cylinder is to be made with a total of 100 square centimeters of material in the side, top, and bottom; the manufacturer wants the can to hold the maximum possible volume. Write the volume as a function of the radius \(r\) of the can; find the domain of the function. (answer)
Ex 1.3.16 A can in the shape of a cylinder is to be made to hold a volume of one liter (1000 cubic centimeters). The manufacturer wants to use the least possible material for the can. Write the surface area of the can (total of the top, bottom, and side) as a function of the radius \(r\) of the can; find the domain of the function. (answer)
1.4: Shifts and Dilations
Starting with the graph of \( y=\sqrt{x}\), the graph of \( y=1/x\), and the graph of \( y=\sqrt{1-x^2}\) (the upper unit semicircle), sketch the graph of each of the following functions:
Ex 1.4.1 \(f(x)=\sqrt{x-2}\)
Ex 1.4.2 \(f(x)=-1-1/(x+2)\)
Ex 1.4.3 \(f(x)=4+\sqrt{x+2}\)
Ex 1.4.4 \(y=f(x)=x/(1-x)\)
Ex 1.4.5 \( y=f(x)=-\sqrt{-x}\)
Ex 1.4.6 \( f(x)=2+\sqrt{1-(x-1)^2}\)
Ex 1.4.7 \(f(x)=-4+\sqrt{-(x-2)}\)
Ex 1.4.8 \(f(x)=2\sqrt{1-(x/3)^2}\)
Ex 1.4.9 \(f(x)=1/(x+1)\)
Ex 1.4.10 \(f(x)=4+2\sqrt{1-(x-5)^2/9}\)
Ex 1.4.11 \(f(x)=1+1/(x-1)\)
Ex 1.4.12 \(f(x)=\sqrt{100-25(x-1)^2}+2\)
The graph of \(f(x)\) is shown below. Sketch the graphs of the following functions.
Ex 1.4.13 \( y=f(x-1)\)
Ex 1.4.14 \(y=1+f(x+2)\)
Ex 1.4.15 \(y=1+2f(x)\)
Ex 1.4.16 \(y=2f(3x)\)
Ex 1.4.17 \(y=2f(3(x-2))+1\)
Ex 1.4.18 \(y=(1/2)f(3x-3)\)
Ex 1.4.19 \(y=f(1+x/3)+2\)